Physics Book - User contributions [en]

Center of Mass

2018-04-18T22:25:40Z

Cvizon3: /* History */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple Problem with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Solution===
Starting from the equation
Ptotal = Mtotal(Vcm)

We have Ptotal, which is <-242, 214, 89>, and Mtotal, which is 2 + 5 + 9 = 16.
Substituting these values into the center of mass momentum equation, we see that:

<-242, 214, 89> = (2 + 5 + 9)(Vcm)

and Vcm = <-242, 214, 89> / 16

Therefore, Vcm = <-15.125, 13.375, 5.56>.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. Additionally, Euler's first law of motion states exactly what was what mentioned in the "Velocity, Momentum, and Kinetic Energy with Center of Mass" section. Leonhard Euler formulated his laws of motion after Newton did, and stated that the linear momentum of a multiparticle system (Ptotal) is equal to the total mass of the multiparticle system (Mtotal) times the velocity of the center of mass of the multiparticle system (Vcm). Therefore, Ptotal = Mtotal(Vcm). [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T22:22:19Z

Cvizon3: /* History */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple Problem with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Solution===
Starting from the equation
Ptotal = Mtotal(Vcm)

We have Ptotal, which is <-242, 214, 89>, and Mtotal, which is 2 + 5 + 9 = 16.
Substituting these values into the center of mass momentum equation, we see that:

<-242, 214, 89> = (2 + 5 + 9)(Vcm)

and Vcm = <-242, 214, 89> / 16

Therefore, Vcm = <-15.125, 13.375, 5.56>.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. Additionally, Euler's first law of motion states exactly what was what mentioned in the [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T22:14:41Z

Cvizon3: /* Simple with Velocity of Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple Problem with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Solution===
Starting from the equation
Ptotal = Mtotal(Vcm)

We have Ptotal, which is <-242, 214, 89>, and Mtotal, which is 2 + 5 + 9 = 16.
Substituting these values into the center of mass momentum equation, we see that:

<-242, 214, 89> = (2 + 5 + 9)(Vcm)

and Vcm = <-242, 214, 89> / 16

Therefore, Vcm = <-15.125, 13.375, 5.56>.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T22:14:18Z

Cvizon3: /* Simple with Velocity of Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Solution===
Starting from the equation
Ptotal = Mtotal(Vcm)

We have Ptotal, which is <-242, 214, 89>, and Mtotal, which is 2 + 5 + 9 = 16.
Substituting these values into the center of mass momentum equation, we see that:

<-242, 214, 89> = (2 + 5 + 9)(Vcm)

and Vcm = <-242, 214, 89> / 16

Therefore, Vcm = <-15.125, 13.375, 5.56>.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T22:08:40Z

Cvizon3: /* Simple with Velocity of Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum Ptotal = <-242, 214, 89>, find the velocity of the center of mass.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:Velocm.jpg

2018-04-18T22:06:59Z

Cvizon3: Cvizon3 uploaded a new version of "File:Velocm.jpg"

File:Kineticenergy.jpg

2018-04-18T22:06:07Z

Cvizon3:

File:Momentumcm.jpg

2018-04-18T22:05:49Z

Cvizon3:

File:Velocm.jpg

2018-04-18T22:05:32Z

Cvizon3: Cvizon3 uploaded a new version of "File:Velocm.jpg"

File:Velocm.jpg

2018-04-18T22:05:04Z

Cvizon3:

File:Velocmi.jpg

2018-04-18T22:03:32Z

Cvizon3:

Center of Mass

2018-04-18T22:01:30Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.jpg]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumcm.jpg]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:kineticenergy.jpg]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:Vcm.JPG

2018-04-18T21:53:18Z

Cvizon3: Cvizon3 uploaded a new version of "File:Vcm.JPG"

File:Ktrans.JPG

2018-04-18T21:47:22Z

Cvizon3:

File:Psys.JPG

2018-04-18T21:47:05Z

Cvizon3:

File:Vcm.JPG

2018-04-18T21:45:43Z

Cvizon3: Cvizon3 uploaded a new version of "File:Vcm.JPG"

Center of Mass

2018-04-18T21:45:04Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:vcm.JPG]]

If the velocity of the center of mass is much less than the speed of light:
[[File:psys.JPG]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:ktrans.JPG]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T21:44:25Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:]]

If the velocity of the center of mass is much less than the speed of light:
[[File:]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:Vcm.JPG

2018-04-18T21:01:31Z

Cvizon3:

Center of Mass

2018-04-18T21:00:33Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:vcm.JPG]]

If the velocity of the center of mass is much less than the speed of light:
[[File:psys.JPG]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:ktrans.JPG]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T20:55:28Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:IMG_2121.JPG]]

If the velocity of the center of mass is much less than the speed of light:
[[File:IMG_2120.JPG]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:IMG_2119.JPG]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T20:48:47Z

Cvizon3: /* Simple */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.png]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumSys.png]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:KE.png]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Simple with Velocity of Center of Mass===
Considering a system with the three particles:
m1 = 2 kg, v1 = < 10, -5, 14 > m/s
m2 = 9 kg, v2 = < -13, 6, -6 > m/s
m3 = 5 kg, v3 = < -29, 34, 23 > m/s

and given the momentum p =

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:MomentumSys.png

2018-04-18T19:58:02Z

Cvizon3: Cvizon3 uploaded a new version of "File:MomentumSys.png"

File:MomentumSys.png

2018-04-18T19:57:52Z

Cvizon3: Cvizon3 uploaded a new version of "File:MomentumSys.png"

File:MomentumSys.png

2018-04-18T19:51:45Z

Cvizon3:

Center of Mass

2018-04-18T19:50:27Z

Cvizon3: /* Velocity, Momentum, and Kinetic Energy with Center of Mass */

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.png]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentumSys.png]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:KE.png]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:KE.png

2018-04-18T19:01:02Z

Cvizon3:

Center of Mass

2018-04-18T18:59:47Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.png]]

If the velocity of the center of mass is much less than the speed of light:
[[File:momentum.png]]

To extend this knowledge to translational kinetic energy for a multiparticle system:
[[File:KE.png]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

File:Velocm.png

2018-04-18T18:52:14Z

Cvizon3:

Center of Mass

2018-04-18T18:51:47Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
[[File:velocm.png]]
----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T18:50:13Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T18:49:06Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----
===Velocity, Momentum, and Kinetic Energy with Center of Mass===

If you know the center of mass, you can use this center of mass to find other quantities associated with the motion of the multiparticle system.
===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T18:46:30Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T18:45:18Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----
===Velocity and Momentum with Center of Mass===
===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Center of Mass

2018-04-18T14:54:06Z

Cvizon3:

Claimed by Connor Vizon (Spring 2018)

==The Main Idea==

The center of mass of an object is the point in space where if a force was applied, the object moves according to Newton's laws without rotation. At the center of mass, the distribution of mass is balanced, and the average of the weighted position coordinates of the distributed mass defines its coordinates. Mechanical calculations are often simplified with respect to the center of mass of objects. [5]

[[File:Ahuynh5.png]]

When a force is applied not to the center of mass:

[[File:Ahuynh6.png]]
----

===A Mathematical Model===

The position of the center of mass of an object can be found by summating the product of each point mass and its relative linear position and dividing by the sum by the total mass.

[[File:Ahuynh3.png]]

This equation can then be extended to three dimensions.

[[File:Ahuynh1.gif]]

When calculating the center of mass for a continuous distribution of mass, the expression becomes an infinite sum that can be expressed in the form of an integral. [6]

[[File:Ahuynh4.gif]]

----

==Examples==

===Simple===
A 10 kg point mass and a 5 kg point mass are connected by a 10 m massless rod.

[[File:1D_COM.png]]

Find the center of mass. This is example of a 1-dimensional center of mass problem.

====Solution====
First, we will need to establish an origin. For simplicity, we will establish the left end, the position of the 10 kg point mass, as the origin. Relative to the origin, the position of the 10 kg mass is 0 m and the position of the 5 kg mass is 10 m. Placing these values into our formula for 1-dimensional center of mass, we get:

[[File:1D_COM_SOLN.png]]

Therefore, the center of mass is 10/3 m from the left end of the rod. It is important to state your distances relative to your stated origin.

===Moderate===
3 point masses are placed on the x-y plane as shown:

[[File:2D_COM.png]]

Find the center of mass. This is an example of a 2-dimensional center of mass problem.

====Solution====
This problem is similar to the last problem except that we will need to find BOTH the x and y components of the center of mass. Like before, we need to establish an origin. For simplicity, we will establish the lower left corner, the position of the 3 kg mass as the origin. Then, we need to solve the two directions separately. We will start with the x direction. Relative to the origin, the 3 kg mass is at 0 m, the 8 kg mass is at 1 m, and the 4 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN1.png]]

Therefore, the center of mass lies 16/15 m from the origin in the x direction.

Next, we will solve for the y direction. Relative to the origin, the 3 kg mass is at 0 m, the 4 kg mass is at 1 m, and the 8 kg mass is at 2 m. Placing these values into our center of mass of point masses formula, we get:

[[File:2D_COM_SOLN2.png]]

Therefore, the center of mass lies 4/3 m from the origin in the y direction.

The center of mass lies at the coordinates (16/15 m, 4/3 m).

===Difficult===
Find the x component of the center of mass of a uniform rod of length L and mass M with continuous, uniform mass. This is an example of a continuous mass problem.

[[File:Continuous_mass_COM_2.png]]

====Solution====

For this problem, we need to use the continuous mass formula. As shown in the diagram, the dm component can be represented by M/L dx because the mass in the rod is uniformly distributed. Therefore, our equation becomes:

[[File:Continuous_mass_COM_soln1.png]]

Therefore, the center of mass lies at L/2, which agrees with the rule of symmetry for uniform masses.

==Connectedness==
The main purpose of using the center of mass is to simplify irregular objects into point masses. Force diagrams are based on this idea, and this aids in the calculation of motion of complex objects. For biomedical engineering majors, we can use the center of mass of body parts in order to simplify the motion of body parts.

==History==

The concept of center of mass was first introduced by Archimedes while he worked to simplify assumptions about gravity that amounted to a uniform field, thus arriving at the mathematical properties of the center of mass. In an experiment, Archimedes observed that the torque exerted on a lever by weights resting at various points along the lever is the same as what it would be if all of the weights were moved to a single point—their center of mass. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes. [7]

== See also ==

===Further reading===

Mass [http://www.physicsbook.gatech.edu/Mass]

Force [http://www.physicsbook.gatech.edu/Net_Force]

Moment of Inertia [http://www.physicsbook.gatech.edu/The_Moments_of_Inertia]

Torque [http://www.physicsbook.gatech.edu/Torque]

==References==

PhysicsLab [http://dev.physicslab.org/document.aspx?doctype=3&filename=rotarymotion_centermass.xml]

HyperPhysics [http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html]

Wikipedia [https://en.wikipedia.org/wiki/Center_of_mass]

[[Category:Momentum]]

Main Page

2018-04-18T14:48:22Z

Cvizon3: /* Energy Principle */

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A page for review of [[Vectors]] and vector operations
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">
==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Help with VPython====
<div class="mw-collapsible-content">
*[[Python Syntax]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Momentum Principle]]
*[[Inertia]]
*[[Net Force]]
*[[Derivation of the Momentum Principle]]
*[[Impulse Momentum]]
*[[Acceleration]]
*[[Momentum with respect to external Forces]]
*[[Relativistic Momentum]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Newton’s Second Law of Motion]]
*[[Iterative Prediction]]
*[[Kinematics]]
*[[Newton’s Laws and Linear Momentum]]
*[[Projectile Motion]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Analytical Prediction]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Predicting Change in multiple dimensions]]
*[[Spring Force]]
*[[Hooke's Law]]
*[[Simple Harmonic Motion]]
*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">

==Main Idea==

*[[Gravitational Force]]
*[[Fluid Mechanics]]
*[[An Application of Gravitational Potential]]
*[[Electric Force]]
*[[Reciprocity]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[The Energy Principle]]
*[[Conservation of Energy]]
*[[Kinetic Energy]]
*[[Work]]
*[[Power (Mechanical)]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Momentum with respect to external Forces]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Heat Capacity]]
*[[Calorific Value(Heat of combustion)]]
*[[Specific Heat Capacity]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Models of Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotation]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Moment of Inertia for a cylinder]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Quantized energy levels part II]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Insulators====
<div class="mw-collapsible-content">
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Conductors====
<div class="mw-collapsible-content">
*[[Conductivity]]
*[[Charge Transfer]]
*[[Resistivity]]
*[[Polarization of a conductor]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Node rule====
<div class="mw-collapsible-content">
*[[Node Rule]]

====The Main Idea====
*'''Mathematical Model'''

**In this image, you can see what our equations are based on: [[File:noderule.jpg]]
**The node rules can be written as I_total = I_1 + I_2 and I_total = I_3 + I_4. It is also true that I_1 + I_2 = I_3 + I_4.
**However, each of these currents are different because each point has a different resistance. The current is different for each because it is equal to V/R, and in a parallel circuit, the voltage drop across each point is equal.
**An easy way to know when to use node rule is by seeing if there are three connections or more. That is when node rule is most helpful.

*'''Computational Model'''

**In an electric circuit in series, electrons flow from the negative end of a power source, creating a constant current. This current remains consistent at each point in the circuit in series. Sometimes, a circuit is not simply one constant path and may include parts that are in parallel, where the current must travel down two paths such as this:
**[[File:noderule.jpg]]
**In this case, when the current enters a portion of the circuit where the items are in parallel, the total amount of current in must equal the total amount of current out. Therefore, the currents in each branch of the parallel portion must sum up to the amount of current at any other point in series in the circuit.
**People also call this the "Junction Rule"
**Another important point is that this comes from the Kirchoff's Circuit Laws

====Examples====

*'''Simple'''
**Here is an example of a simple circuit problem: [[File:SimpleNodeRule.jpg]]

*'''Medium'''
**Here is an example of a medium circuit problem: [[File:MediumNodeRule.jpg]]

*'''Difficult'''
**Here is an example of a difficult circuit problem: [[File:DifficultNodeRule.jpg]]

====Connectedness====

*'''To other topics:'''
**Many times when you use Node Rule you will also use the Loop Rule. The Loop Rule states that the sum of voltage will equal zero. So using this concept and the Node Rule, you are usually able to figure out missing variables in circuit problems.

*'''To majors:'''
**Node rule is important in all and any major. More specifically, electrical engineering because of the constant need to look, analyze, and understand circuits. However, in general, any major that involves some sort of circuitry will need this. It is the basis to making an effective circuit.

*'''To industrial application:'''
**If you go into robots, engineering, or really anything that involves wires and batteries. You will need to know this.

====History====

*Basic History
**Gustav Kirchoff was the man who discovered this rule while studying electrical currents. He was also the first person to confirm an electrical impulse moves at the speed of light.

====External Resources and Information====

*Sources like Khan Academy and simple YouTube searches can be very helpful in learning more about this topic.

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>

Talk:Energy of a Single Particle

2018-04-18T14:41:30Z

Cvizon3: Created page with "'''Claimed by Connor Vizon Spring 2018''' ''Energy of a Single Particle'' There are two ways of envisioning systems in physics: the point particle method and the extended sys..."

'''Claimed by Connor Vizon Spring 2018'''

''Energy of a Single Particle''
There are two ways of envisioning systems in physics: the point particle method and the extended system method. In this page, we will discuss the point particle method, more specifically the energy of this single point particle. This point particle may represent anything from a standard atomic particle (proton, neutron, etc.) to a planet or a star, if we're only choosing to analyze its translational kinetic energy.

==The Main Idea==

Point particle systems can have two forms of energy, rest energy and kinetic energy. Even if a particle is still, it still has (rest) energy. Kinetic energy is the energy associated with a moving particle.

===A Mathematical Model===

[[File:particleEnergy.jpg]]

===A Computational Model===

How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
===Middling===
===Difficult===

==Connectedness==
#How is this topic connected to something that you are interested in?
#How is it connected to your major?
#Is there an interesting industrial application?

==History==

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

Internet resources on this topic

==References==

This section contains the the references you used while writing this page

[[Category:Energy Principle[[File:Example.jpg]]]]

File:ParticleEnergy.jpg

2018-04-18T14:40:15Z

Cvizon3:

Main Page

2018-04-18T03:32:48Z

Cvizon3:

__NOTOC__
= '''Georgia Tech Student Wiki for Introductory Physics.''' =

This resource was created so that students can contribute and curate content to help those with limited or no access to a textbook. When reading this website, please correct any errors you may come across. If you read something that isn't clear, please consider revising it for future students!

Looking to make a contribution?
#Pick one of the topics from intro physics listed below
#Add content to that topic or improve the quality of what is already there.
#Need to make a new topic? Edit this page and add it to the list under the appropriate category. Then copy and paste the default [[Template]] into your new page and start editing.

Please remember that this is not a textbook and you are not limited to expressing your ideas with only text and equations. Whenever possible embed: pictures, videos, diagrams, simulations, computational models (e.g. Glowscript), and whatever content you think makes learning physics easier for other students.

== Source Material ==
All of the content added to this resource must be in the public domain or similar free resource. If you are unsure about a source, contact the original author for permission. That said, there is a surprisingly large amount of introductory physics content scattered across the web. Here is an incomplete list of intro physics resources (please update as needed).
* A physics resource written by experts for an expert audience [https://en.wikipedia.org/wiki/Portal:Physics Physics Portal]
* A wiki written for students by a physics expert [http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes MSU Physics Wiki]
* A wiki book on modern physics [https://en.wikibooks.org/wiki/Modern_Physics Modern Physics Wiki]
* The MIT open courseware for intro physics [http://ocw.mit.edu/resources/res-8-002-a-wikitextbook-for-introductory-mechanics-fall-2009/index.htm MITOCW Wiki]
* An online concept map of intro physics [http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html HyperPhysics]
* Interactive physics simulations [https://phet.colorado.edu/en/simulations/category/physics PhET]
* OpenStax intro physics textbooks: [https://openstax.org/details/books/university-physics-volume-1 Vol1], [https://openstax.org/details/books/university-physics-volume-2 Vol2], [https://openstax.org/details/books/university-physics-volume-3 Vol3]
* The Open Source Physics project is a collection of online physics resources [http://www.opensourcephysics.org/ OSP]
* A resource guide compiled by the [http://www.aapt.org/ AAPT] for educators [http://www.compadre.org/ ComPADRE]
* The Feynman lectures on physics are free to read [http://www.feynmanlectures.caltech.edu/ Feynman]

== Resources ==
* Commonly used wiki commands [https://en.wikipedia.org/wiki/Help:Cheatsheet Wiki Cheatsheet]
* A guide to representing equations in math mode [https://en.wikipedia.org/wiki/Help:Displaying_a_formula Wiki Math Mode]
* A page to keep track of all the physics [[Constants]]
* A page for review of [[Vectors]] and vector operations
* A listing of [[Notable Scientist]] with links to their individual pages

<div style="float:left; width:30%; padding:1%;">
==Physics 1==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Help with VPython====
<div class="mw-collapsible-content">
*[[Python Syntax]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====

<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====VPython====

<div class="mw-collapsible-content">
*[[VPython]]
*[[VPython basics]]
*[[VPython Common Errors and Troubleshooting]]
*[[VPython Functions]]
*[[VPython Lists]]
*[[VPython Loops]]
*[[VPython Multithreading]]
*[[VPython Animation]]
*[[VPython Objects]]
*[[VPython 3D Objects]]
*[[VPython Reference]]
*[[VPython MapReduceFilter]]
*[[VPython GUIs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Vectors and Units====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[SI Units]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Interactions====
<div class="mw-collapsible-content">
*[[Types of Interactions and How to Detect Them]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Velocity and Momentum====
<div class="mw-collapsible-content">
*[[Newton's First Law of Motion]]
*[[Velocity]]
*[[Mass]]
*[[Speed and Velocity]]
*[[Relative Velocity]]
*[[Derivation of Average Velocity]]
*[[2-Dimensional Motion]]
*[[3-Dimensional Position and Motion]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Momentum and the Momentum Principle====
<div class="mw-collapsible-content">
*[[Momentum Principle]]
*[[Inertia]]
*[[Net Force]]
*[[Derivation of the Momentum Principle]]
*[[Impulse Momentum]]
*[[Acceleration]]
*[[Momentum with respect to external Forces]]
*[[Relativistic Momentum]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Newton’s Second Law of Motion]]
*[[Iterative Prediction]]
*[[Kinematics]]
*[[Newton’s Laws and Linear Momentum]]
*[[Projectile Motion]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">

====Analytic Prediction with a Constant Force====
<div class="mw-collapsible-content">
*[[Analytical Prediction]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Iterative Prediction with a Varying Force====
<div class="mw-collapsible-content">
*[[Predicting Change in multiple dimensions]]
*[[Spring Force]]
*[[Hooke's Law]]
*[[Simple Harmonic Motion]]
*[[Iterative Prediction of Spring-Mass System]]
*[[Terminal Speed]]
*[[Determinism]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Fundamental Interactions====
<div class="mw-collapsible-content">

==Main Idea==

*[[Gravitational Force]]
*[[Fluid Mechanics]]
*[[An Application of Gravitational Potential]]
*[[Electric Force]]
*[[Reciprocity]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Conservation of Momentum====
<div class="mw-collapsible-content">
*[[Conservation of Momentum]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Properties of Matter====
<div class="mw-collapsible-content">
*[[Kinds of Matter]]
*[[Ball and Spring Model of Matter]]
*[[Density]]
*[[Length and Stiffness of an Interatomic Bond]]
*[[Young's Modulus]]
*[[Speed of Sound in Solids]]
*[[Malleability]]
*[[Ductility]]
*[[Weight]]
*[[Hardness]]
*[[Boiling Point]]
*[[Melting Point]]
*[[Change of State]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Identifying Forces====
<div class="mw-collapsible-content">
*[[Free Body Diagram]]
*[[Inclined Plane]]
*[[Compression or Normal Force]]
*[[Tension]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Curving Motion====
<div class="mw-collapsible-content">
*[[Curving Motion]]
*[[Centripetal Force and Curving Motion]]
*[[Perpetual Freefall (Orbit)]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Energy Principle====
<div class="mw-collapsible-content">
*[[The Energy Principle]]
*[[Conservation of Energy]]
*[[Kinetic Energy]]
*[[Work]]
*[[Power (Mechanical)]]
*[[Energy of a Single Particle]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Work by Non-Constant Forces====
<div class="mw-collapsible-content">
*[[Work Done By A Nonconstant Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential Energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
*[[Potential Energy of Macroscopic Springs]]
*[[Spring Potential Energy]]
*[[Ball and Spring Model]]
*[[Gravitational Potential Energy]]
*[[Energy Graphs]]
*[[Escape Velocity]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Multiparticle Systems====
<div class="mw-collapsible-content">
*[[Center of Mass]]
*[[Multi-particle analysis of Momentum]]
*[[Momentum with respect to external Forces]]
*[[Potential Energy of a Multiparticle System]]
*[[Work and Energy for an Extended System]]
*[[Internal Energy]]
**[[Potential Energy of a Pair of Neutral Atoms]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Choice of System====
<div class="mw-collapsible-content">
*[[System & Surroundings]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Thermal Energy, Dissipation and Transfer of Energy====
<div class="mw-collapsible-content">
*[[Thermal Energy]]
*[[Specific Heat]]
*[[Heat Capacity]]
*[[Calorific Value(Heat of combustion)]]
*[[Specific Heat Capacity]]
*[[First Law of Thermodynamics]]
*[[Second Law of Thermodynamics and Entropy]]
*[[Temperature]]
*[[Predicting Change]]
*[[Energy Transfer due to a Temperature Difference]]
*[[Transformation of Energy]]
*[[The Maxwell-Boltzmann Distribution]]
*[[Air Resistance]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">

====Rotational and Vibrational Energy====
<div class="mw-collapsible-content">
*[[Translational, Rotational and Vibrational Energy]]
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Different Models of a System====
<div class="mw-collapsible-content">
*[[Point Particle Systems]]
*[[Real Systems]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Models of Friction====
<div class="mw-collapsible-content">
*[[Friction]]
*[[Static Friction]]
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====Collisions====
<div class="mw-collapsible-content">
*[[Newton's Third Law of Motion]]
*[[Collisions]]
*[[Elastic Collisions]]
*[[Inelastic Collisions]]
*[[Maximally Inelastic Collision]]
*[[Head-on Collision of Equal Masses]]
*[[Head-on Collision of Unequal Masses]]
*[[Scattering: Collisions in 2D and 3D]]
*[[Rutherford Experiment and Atomic Collisions]]
*[[Coefficient of Restitution]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Rotations====
<div class="mw-collapsible-content">
*[[Rotation]]
*[[Angular Velocity]]
*[[Eulerian Angles]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Angular Momentum====
<div class="mw-collapsible-content">
*[[Total Angular Momentum]]
*[[Translational Angular Momentum]]
*[[Rotational Angular Momentum]]
*[[The Angular Momentum Principle]]
*[[Angular Momentum Compared to Linear Momentum]]
*[[Angular Impulse]]
*[[Predicting the Position of a Rotating System]]
*[[Angular Momentum of Multiparticle Systems]]
*[[The Moments of Inertia]]
*[[Moment of Inertia for a cylinder]]
*[[Right Hand Rule]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Analyzing Motion with and without Torque====
<div class="mw-collapsible-content">
*[[Torque]]
*[[Torque 2]]
*[[Systems with Zero Torque]]
*[[Systems with Nonzero Torque]]
*[[Torque vs Work]]
*[[Gyroscopes]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
====Introduction to Quantum Concepts====
<div class="mw-collapsible-content">
*[[Bohr Model]]
*[[Energy graphs and the Bohr model]]
*[[Quantized energy levels]]
*[[Quantized energy levels part II]]
*[[Entropy]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 2==
===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====3D Vectors====
<div class="mw-collapsible-content">
*[[Vectors]]
*[[Right-Hand Rule]]
*[[Right Hand Rule]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Field and Electric Potential]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric force====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric field of a point particle====
<div class="mw-collapsible-content">
*[[Point Charge]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Superposition====
<div class="mw-collapsible-content">
*[[Superposition Principle]]
*[[Superposition principle]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Dipoles====
<div class="mw-collapsible-content">
*[[Electric Dipole]]
*[[Magnetic Dipole]]
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Interactions of charged objects====
<div class="mw-collapsible-content">
*[[Electric Field]]
*[[Electric Potential]]
*[[Electric Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Tape experiments====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Polarization====
<div class="mw-collapsible-content">
*[[Polarization]]
*[[Electric Polarization]]
*[[Polarization of an Atom]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Insulators====
<div class="mw-collapsible-content">
*[[Insulators]]
*[[Potential Difference in an Insulator]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Conductors====
<div class="mw-collapsible-content">
*[[Conductivity]]
*[[Charge Transfer]]
*[[Resistivity]]
*[[Polarization of a conductor]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Charging and discharging====
<div class="mw-collapsible-content">
*[[Charge Transfer]]
*[[Electrostatic Discharge]]
*[[Charged Conductor and Charged Insulator]]
*[[Charged conductor and charged insulator]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged rod====
<div class="mw-collapsible-content">
*[[Charged Rod]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged ring/disk/capacitor====
<div class="mw-collapsible-content">
*[[Charged Ring]]
*[[Charged Disk]]
*[[Charged Capacitor]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Field of a charged sphere====
<div class="mw-collapsible-content">
*[[Charged Spherical Shell]]
*[[Field of a Charged Ball]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Potential energy====
<div class="mw-collapsible-content">
*[[Potential Energy]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric potential====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
*[[Potential Difference in a Uniform Field]]
*[[Potential Difference of Point Charge in a Non-Uniform Field]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Sign of a potential difference====
<div class="mw-collapsible-content">
*[[Sign of a Potential Difference]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Potential at a single location====
<div class="mw-collapsible-content">
*[[Electric Potential]]
*[[Potential Difference at One Location]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Path independence and round trip potential====
<div class="mw-collapsible-content">
*[[Path Independence of Electric Potential]]
*[[Potential Difference Path Independence, claimed by Aditya Mohile]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in an insulator====
<div class="mw-collapsible-content">
*[[Potential Difference in an Insulator]]
*[[Electric Field in an Insulator]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges in a magnetic field====
<div class="mw-collapsible-content">
*[[Magnetic Field]]
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Biot-Savart Law====
<div class="mw-collapsible-content">
*[[Biot-Savart Law]]
*[[Biot-Savart Law for Currents]]
</div>
</div>
<div class="toccolours mw-collapsible mw-collapsed">
====Moving charges, electron current, and conventional current====
<div class="mw-collapsible-content">
*[[Moving Point Charge]]
*[[Current]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a wire====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Long Straight Wire]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a current-carrying loop====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Loop]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic field of a Charged Disk====
<div class="mw-collapsible-content">
*[[Magnetic Field of a Disk]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic dipoles====
<div class="mw-collapsible-content">
*[[Magnetic Dipole Moment]]
*[[Bar Magnet]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Atomic structure of magnets====
<div class="mw-collapsible-content">
*[[Atomic Structure of Magnets]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Steady state current====
<div class="mw-collapsible-content">
*[[Steady State]]
*[[Non Steady State]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Node rule====
<div class="mw-collapsible-content">
*[[Node Rule]]

====The Main Idea====
*'''Mathematical Model'''

**In this image, you can see what our equations are based on: [[File:noderule.jpg]]
**The node rules can be written as I_total = I_1 + I_2 and I_total = I_3 + I_4. It is also true that I_1 + I_2 = I_3 + I_4.
**However, each of these currents are different because each point has a different resistance. The current is different for each because it is equal to V/R, and in a parallel circuit, the voltage drop across each point is equal.
**An easy way to know when to use node rule is by seeing if there are three connections or more. That is when node rule is most helpful.

*'''Computational Model'''

**In an electric circuit in series, electrons flow from the negative end of a power source, creating a constant current. This current remains consistent at each point in the circuit in series. Sometimes, a circuit is not simply one constant path and may include parts that are in parallel, where the current must travel down two paths such as this:
**[[File:noderule.jpg]]
**In this case, when the current enters a portion of the circuit where the items are in parallel, the total amount of current in must equal the total amount of current out. Therefore, the currents in each branch of the parallel portion must sum up to the amount of current at any other point in series in the circuit.
**People also call this the "Junction Rule"
**Another important point is that this comes from the Kirchoff's Circuit Laws

====Examples====

*'''Simple'''
**Here is an example of a simple circuit problem: [[File:SimpleNodeRule.jpg]]

*'''Medium'''
**Here is an example of a medium circuit problem: [[File:MediumNodeRule.jpg]]

*'''Difficult'''
**Here is an example of a difficult circuit problem: [[File:DifficultNodeRule.jpg]]

====Connectedness====

*'''To other topics:'''
**Many times when you use Node Rule you will also use the Loop Rule. The Loop Rule states that the sum of voltage will equal zero. So using this concept and the Node Rule, you are usually able to figure out missing variables in circuit problems.

*'''To majors:'''
**Node rule is important in all and any major. More specifically, electrical engineering because of the constant need to look, analyze, and understand circuits. However, in general, any major that involves some sort of circuitry will need this. It is the basis to making an effective circuit.

*'''To industrial application:'''
**If you go into robots, engineering, or really anything that involves wires and batteries. You will need to know this.

====History====

*Basic History
**Gustav Kirchoff was the man who discovered this rule while studying electrical currents. He was also the first person to confirm an electrical impulse moves at the speed of light.

====External Resources and Information====

*Sources like Khan Academy and simple YouTube searches can be very helpful in learning more about this topic.

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric fields and energy in circuits====
<div class="mw-collapsible-content">
*[[Series circuit]]
*[[Node Rule]]
*[[Loop Rule]]
*[[Electric Potential Difference]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Macroscopic analysis of circuits====
<div class="mw-collapsible-content">
*[[Series Circuits]]
*[[Parallel Circuits]]
*[[Parallel Circuits vs. Series Circuits*]]
*[[Loop Rule]]
*[[Node Rule]]
*[[Fundamentals of Resistance]]
*[[Problem Solving]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Electric field and potential in circuits with capacitors====
<div class="mw-collapsible-content">
*[[Charging and Discharging a Capacitor]]
*[[RC Circuit]]
*[[R Circuit]]
*[[AC and DC]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic forces on charges and currents====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[Motors and Generators]]
*[[Applying Magnetic Force to Currents]]
*[[Magnetic Force in a Moving Reference Frame]]
*[[Right-Hand Rule]]
*[[Analysis of Railgun vs Coil gun technologies]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Electric and magnetic forces====
<div class="mw-collapsible-content">
*[[Electric Force]]
*[[Magnetic Force]]
*[[Lorentz Force]]
*[[VPython Modelling of Electric and Magnetic Forces]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Velocity selector====
<div class="mw-collapsible-content">
*[[Lorentz Force]]
*[[Combining Electric and Magnetic Forces]]
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">

====Hall Effect====
<div class="mw-collapsible-content">
*[[Hall Effect]]
*[[Right-Hand Rule]]
*[[Motional Emf]]
*[[Magnetic Force]]
*[[Magnetic Torque]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Motional EMF====
<div class="mw-collapsible-content">
*[[Motional Emf]]
*[[Motional Emf using Faraday's Law]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic force====
<div class="mw-collapsible-content">
*[[Magnetic Force]]
*[[Lorentz Force]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Magnetic torque====
<div class="mw-collapsible-content">
*[[Magnetic Torque]]
*[[Right-Hand Rule]]
</div>
</div>

===Week 12===

<div class="toccolours mw-collapsible mw-collapsed">
====Gauss's Law====
<div class="mw-collapsible-content">
*[[Gauss's Flux Theorem]]
*[[Gauss's Law]]
*[[Magnetic Flux]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Ampere's Law====
<div class="mw-collapsible-content">
*[[Ampere's Law]]
*[[Ampere-Maxwell Law]]
*[[Magnetic Field of Coaxial Cable Using Ampere's Law]]
*[[Magnetic Field of a Long Thick Wire Using Ampere's Law]]
*[[Magnetic Field of a Toroid Using Ampere's Law]]
*[[Magnetic Field of a Solenoid Using Ampere's Law]]
*[[The Differential Form of Ampere's Law]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Semiconductors====
<div class="mw-collapsible-content">
*[[Semiconductor Devices]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Faraday's Law====
<div class="mw-collapsible-content">
*[[Faraday's Law]]
*[[Motional Emf using Faraday's Law]]
*[[Lenz's Law]]

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Maxwell's equations====
<div class="mw-collapsible-content">
*[[Gauss's Law]]
*[[Magnetic Flux]]
*[[Ampere's Law]]
*[[Faraday's Law]]
*[[Maxwell's Electromagnetic Theory]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Circuits revisited====
<div class="mw-collapsible-content">

</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">

====Inductors====
<div class="mw-collapsible-content">
*[[Inductors]]
*[[Current in an LC Circuit]]
*[[Current in an RL Circuit]]
</div>
</div>

===Week 15===
<div class="toccolours mw-collapsible mw-collapsed">
==== Electromagnetic Radiation ====
<div class="mw-collapsible-content">
*[[Electromagnetic Radiation]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Sparks in the air====
<div class="mw-collapsible-content">
*[[Sparks in Air]]
*[[Spark Plugs]]
</div>
</div>

<div class="toccolours mw-collapsible mw-collapsed">
====Superconductors====
<div class="mw-collapsible-content">
*[[Superconducters]]
*[[Superconductors]]
*[[Meissner effect]]
</div>
</div>
</div>

<div style="float:left; width:30%; padding:1%;">

==Physics 3==

===Week 1===
<div class="toccolours mw-collapsible mw-collapsed">
====Classical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 2===
<div class="toccolours mw-collapsible mw-collapsed">
====Special Relativity====
<div class="mw-collapsible-content">
*[[Frame of Reference]]
*[[Einstein's Theory of Special Relativity]]
*[[Time Dilation]]
*[[Einstein's Theory of General Relativity]]
*[[Albert A. Micheleson & Edward W. Morley]]
*[[Magnetic Force in a Moving Reference Frame]]
</div>
</div>

===Week 3===
<div class="toccolours mw-collapsible mw-collapsed">
====Photons====
<div class="mw-collapsible-content">
*[[Spontaneous Photon Emission]]
*[[Light Scattering: Why is the Sky Blue]]
*[[Lasers]]
*[[Electronic Energy Levels and Photons]]
*[[Quantum Properties of Light]]
</div>
</div>

===Week 4===
<div class="toccolours mw-collapsible mw-collapsed">
====Matter Waves====
<div class="mw-collapsible-content">
*[[Wave-Particle Duality]]
</div>
</div>

===Week 5===
<div class="toccolours mw-collapsible mw-collapsed">
====Wave Mechanics====
<div class="mw-collapsible-content">
*[[Standing Waves]]
*[[Wavelength]]
*[[Wavelength and Frequency]]
*[[Mechanical Waves]]
*[[Transverse and Longitudinal Waves]]
</div>
</div>

===Week 6===
<div class="toccolours mw-collapsible mw-collapsed">
====Rutherford-Bohr Model====
<div class="mw-collapsible-content">
*[[Rutherford Experiment and Atomic Collisions]]
*[[Bohr Model]]
*[[Quantized energy levels]]
*[[Energy graphs and the Bohr model]]
</div>
</div>

===Week 7===
<div class="toccolours mw-collapsible mw-collapsed">
====The Hydrogen Atom====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
</div>
</div>

===Week 8===
<div class="toccolours mw-collapsible mw-collapsed">
====Many-Electron Atoms====
<div class="mw-collapsible-content">
*[[Quantum Theory]]
*[[Atomic Theory]]
*[[Pauli exclusion principle]]
</div>
</div>

===Week 9===
<div class="toccolours mw-collapsible mw-collapsed">
====Molecules====
<div class="mw-collapsible-content">
</div>
</div>

===Week 10===
<div class="toccolours mw-collapsible mw-collapsed">
====Statistical Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 11===
<div class="toccolours mw-collapsible mw-collapsed">
====Condensed Matter Physics====
<div class="mw-collapsible-content">
</div>
</div>

===Week 12===
<div class="toccolours mw-collapsible mw-collapsed">
====The Nucleus====
<div class="mw-collapsible-content">
*[[Nucleus]]
</div>
</div>

===Week 13===
<div class="toccolours mw-collapsible mw-collapsed">
====Nuclear Physics====
<div class="mw-collapsible-content">
*[[Nuclear Fission]]
*[[Nuclear Energy from Fission and Fusion]]
</div>
</div>

===Week 14===
<div class="toccolours mw-collapsible mw-collapsed">
====Particle Physics====
<div class="mw-collapsible-content">
*[[Elementary Particles and Particle Physics Theory]]
*[[String Theory]]
</div>
</div>
</div>