Physics Book - User contributions [en]

3-Dimensional Position and Motion

2016-11-28T00:45:11Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016

Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and there are several conceptual models in physics such as kinematics or the momentum principle which describe motion. Because kinematics and the momentum principle are both vector quantities, it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. The kinematic equations are most useful when the object under observation is subject to a constant force <math> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\vec{F}_{net} \over\ m </math> is constant.

The use of the momentum principle is most applicable because we can apply it to any situation. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object.

For both models, an object in motion has properties along each axis which are independent of other axes allowing us to decompose three-dimensional motion into three one-dimensional problems.

===A Mathematical Model===

===Kinematic Equations===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

===Momentum Principle===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a string. You hold the toy such that the feathers hang suspended from the string when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

<math> d= d_0+ v_0t </math>

remember the definition of velocity in relation to momentum: <math> \vec{p} \over\ m </math> <math>= \vec{v} </math>

<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>

===Difficult===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{v}_{f}-\vec{v}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which demand an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay ''Essai sur la philosophie des sciences'' about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
In 1666, Newton formulated early versions of his three laws of motion, of which the firstl aw describes the momentum principle. Two decades later, he would publish ''Principia'' which is often cited as one of greatest scientific books ever written.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Newton.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-28T00:39:50Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016

Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and there are several conceptual models in physics such as kinematics or the momentum principle which describe motion. Because kinematics and the momentum principle are both vector quantities, it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. The kinematic equations are most useful when the object under observation is subject to a constant force <math> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\vec{F}_{net} \over\ m </math> is constant.

The use of the momentum principle is most applicable because we can apply it to any situation. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object.

For both models, an object in motion has properties along each axis which are independent of other axes allowing us to decompose three-dimensional motion into three one-dimensional problems.

===A Mathematical Model===

===Kinematic Equations===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

===Momentum Principle===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{v}_{f}-\vec{v}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a string. You hold the toy such that the feathers hang suspended from the string when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

<math> d= d_0+ v_0t </math>

remember the definition of velocity in relation to momentum: <math> \vec{p} \over\ m </math> <math>= \vec{v} </math>

<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which demand an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay ''Essai sur la philosophie des sciences'' about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
In 1666, Newton formulated early versions of his three laws of motion, of which the firstl aw describes the momentum principle. Two decades later, he would publish ''Principia'' which is often cited as one of greatest scientific books ever written.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Newton.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-27T23:02:57Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016

Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and there are several conceptual models in physics such as kinematics or the momentum principle which describe motion. Because kinematics and the momentum principle are both vector quantities, it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. The kinematic equations are most useful when the object under observation is subject to a constant force <math> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\vec{F}_{net} \over\ m </math> is constant.

The use of the momentum principle is most applicable because we can apply it to any situation. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object.

For both models, an object in motion has properties along each axis which are independent of other axes allowing us to decompose three-dimensional motion into three one-dimensional problems.

===A Mathematical Model===

===Kinematic Equations===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

===Momentum Principle===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a string. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

<math> d= d_0+ v_0t </math>

remember the definition of velocity in relation to momentum:

<math> \vec{p} \over\ m </math> <math>= \vec{v} </math>

<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which demand an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay ''Essai sur la philosophie des sciences'' about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
In 1666, Newton formulated early versions of his three laws of motion, of which the firstl aw describes the momentum principle. Two decades later, he would publish ''Principia'' which is often cited as one of greatest scientific books ever written.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Newton.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-27T22:42:38Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016

Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and there are several conceptual models in physics such as kinematics or the momentum principle which describe motion. Because kinematics and the momentum principle are both vector quantities, it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. The kinematic equations are most useful when the object under observation is subject to a constant force <math> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\vec{F}_{net} \over\ m </math> is constant.

The use of the momentum principle is most applicable because we can apply it to any situation. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object.

For both models, an object in motion has properties along each axis which are independent of other axes allowing us to decompose three-dimensional motion into three one-dimensional problems.

===A Mathematical Model===

===Kinematic Equations===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

===Momentum Principle===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-27T06:22:49Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of velocity:

<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of velocity:

<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin with a fundamental equation of motion

<math> {\vec{v}_{2}}= {\vec{v}_{1}+ \vec{a}\Delta t} </math>

<math> {\vec{v}_{2}-\vec{v}_{1} \over\ \Delta t}= {\vec{a}} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:
<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-27T06:09:26Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of velocity:

<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of velocity:

<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin with a fundamental equation of motion

<math> {\vec{v}_{2}}= {\vec{p}_{1}+ \vec{a}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{a}} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:
<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
http://kmoddl.library.cornell.edu/what.php

3-Dimensional Position and Motion

2016-11-27T06:09:02Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of velocity:

<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of velocity:

<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin with a fundamental equation of motion

<math> {\vec{v}_{2}}= {\vec{p}_{1}+ \vec{a}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{a}} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:
<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.
==History==
Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics.
== See also ==
===Further Reading===
http://kmoddl.library.cornell.edu/what.php
==References==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

3-Dimensional Position and Motion

2016-11-27T05:54:54Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of velocity:

<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of velocity:

<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin with a fundamental equation of motion

<math> {\vec{v}_{2}}= {\vec{p}_{1}+ \vec{a}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{a}} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:
<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T05:49:19Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of velocity:

<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of velocity:

<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin with a fundamental equation of motion

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T05:28:24Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average velocity in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{v}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{v}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T05:26:50Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==
http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T05:26:32Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.

===A Mathematical Model===

The fundamental equations of motion allow us to observe motion in three dimensions.

<math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>

<math> v= v_0+at </math>

<math> v^2= v_0^2 + 2a(d-d_0) </math>

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

<math> \vec{v}= \vec{v}_0+\vec{a}t </math>

<math> \vec{v}-\vec{v}_0 \over\ t </math> <math> = \vec{a} </math>

Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>

Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S
Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T04:15:37Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that the net force is provided, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>, <math> S = \left\vert \vec{L} \right\vert- L_0</math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

# We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.

#Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>
#Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T04:07:46Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that the net force is provided, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T04:07:07Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.

Now we can begin to solve this problem. As always, begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T04:06:06Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step <math> \Delta t </math> of 0.1 s?

Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

1. Identify a coordinate system. We choose <math> +\hat{x} </math> to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.

2. Choose a system and its surroundings. For this example, we choose the system to be the clump of feathers and the surroundings to be the spring and earth.

3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.
Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T03:53:34Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a spring with stiffness 0.915 N/m and relaxed length 0.265 m. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
and the moving ball is at location <math> \langle −0.2, −0.61, 0 \rangle </math> m relative to an origin located at the base of the spring.

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Yes! Every force is in three dimensions, as is every object and its movement.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T03:28:44Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 8.09, 6.18, -58.35 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 8.17, 6.37, -59.07 \rangle </math> m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

===Difficult===
A ball is kicked from a location <math> \langle 8,0,-7 \rangle </math> with initial velocity of <math> \langle -11, 15, 2 \rangle </math> m/s.

(a) What is velocity of the ball 0.2 seconds after being kicked?

(b) What is the net impulse during this time interval?

(c) What is the location of the ball at the end of this time interval?

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Yes! Every force is in three dimensions, as is every object and its movement.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T03:22:10Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.

==The Main Idea==

Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic? Position is determined by the net force <math> \vec{F}_{net} </math> so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.

Spring Force: <math> \vec{F}_{spring} = k_sS\hat{L} </math>

Gravity Force: <math> \vec{F}_{grav}= </math> <math> {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Electric Force: <math> \vec{F}_{elec}= </math> <math> {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} </math>

Momentum Principle: <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

The general workflow to solving position-update problems by hand would be as follows:

1. Calculate the current net force <math> \vec{F}_{net} </math> acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces.

2. Update the new momentum <math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

3. Update the new position <math> {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} </math>.

For this step, <math> \vec{v}_{avg} </math> can take many forms:

Constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{v}_{f}+ \vec{v}_{i} \over\ 2 </math>

Non-constant net force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>

===A Computational Model===
The following is code for a simple computational model showing the effects of a constant force on a mass:

https://trinket.io/embed/glowscript/75acfdd1c6

==Examples==

Here are a few examples:

===Simple===
At t = 10.0 seconds a mass of 3 kg has velocity of <math> \langle 12, 27, −8 \rangle </math> m/s. At t = 10.1 seconds its velocity was <math> \langle 24, 19, 22 \rangle </math> m/s. What was the average (vector) net force acting on the object?

Begin from a fundamental principle:

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

<math> {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} </math>

Substitute the provided values into the symbolic expression and you should arrive at your final answer:

<math> \left\langle 360,-240, 900\right\rangle N = \vec{F}_{net} </math>

===Middling===
An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

First Interval:

At t = 1.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 1.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

Second Interval:

At t = 3.56 s, the position was <math> \langle 4.22, 2.45, −9.63 \rangle </math> m.

At t = 3.59 s, the position was <math> \langle 4.26, 2.37, −10.35 \rangle </math> m.

(a) Begin with the definition of momentum:

<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,1}}= </math>

(b) Begin with the definition of momentum:

<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

<math> {\vec{p}_{avg,2}}= </math>

(c) Begin from a fundamental principle

<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

===Difficult===
A ball is kicked from a location <math> \langle 8,0,-7 \rangle </math> with initial velocity of <math> \langle -11, 15, 2 \rangle </math> m/s.

(a) What is velocity of the ball 0.2 seconds after being kicked?

(b) What is the net impulse during this time interval?

(c) What is the location of the ball at the end of this time interval?

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.
#How is it connected to your major?
I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.
#Is there an interesting industrial application?
Yes! Every force is in three dimensions, as is every object and its movement.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

3-Dimensional Position and Motion

2016-11-27T03:06:58Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T03:05:25Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T02:55:19Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T02:40:40Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T02:34:48Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T01:43:01Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T01:29:33Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T00:54:56Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T00:53:49Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T00:52:53Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T00:52:20Z

Bath3:

3-Dimensional Position and Motion

2016-11-27T00:45:42Z

Bath3:

Scattering: Collisions in 2D and 3D

2016-11-27T00:31:21Z

Bath3:

==The Main Idea==

Unlike normal collisions, atomic and nuclear collisions are far too small to observe the curving trajectories of the interacting particles. The only thing that can be noticed is the initial and final states of the interaction. Scattering experiments are incorporated in the world of collisions to be able to study the minute details (structure) of atoms, nuclei, and other tiny particles as the interact with one another.

==Impact Parameters==
Definition: The distance between centers perpendicular to the incoming velocity. Impact parameter is often denoted by the variable b.

A head-on collision has an impact parameter of zero and with equal masses fully transfers the momentum such as with Newton's Cradle. As the impact parameter gets smaller the collision has a larger effect, and an even large deflection angle (scattering).

Elastic collisions between two billiard balls (Double Click Image)
[[File:ImpactParameter.png]]

==Examples==
===Simple===
The collision of an alpha particle (helium nucleus) with the nucleus of a gold atom

==History==

In 1871 Lord Rayleigh published a paper on scattering. Rayleigh scattering is the dispersion of electromagnetic radiation by particles that have a minute radius less than approximately 1/10 the wavelength. It laid the foundation to research on scattering and information we have today.

== See also ==

[[Category:Collisions] (Main page)

===Further reading===

Matter and Interactions, Volume I: Modern Mechanics, 4th Edition. (Chapter 10.6)

===External links===
[http://physics-animations.com/Physics/English/par_txt.htm]
[http://www.britannica.com/science/Rayleigh-scattering]
[http://hypernews.slac.stanford.edu/slacsite/aux/HiPPP/scattering/]

==References==

Chabay, Ruth W., Bruce Sherwood. Matter and Interactions, Volume I: Modern Mechanics, 4th Edition. Wiley, 19/2014.

[[Category:Collisions]]

3-Dimensional Position and Motion

2016-11-27T00:31:15Z

Bath3:

Claimed by Benjamin Tasistro-Hart Fall 2016
In order to be able to calculate the effect of forces on an object, you need to first be able to describe its position and motion in three dimensional space. For locating entities, we have position vectors. The change in position over time creates the velocity vector, which describes motion in space. From here, we can apply three dimensional forces.

==The Main Idea==

Objects, exist, move and accelerate in three dimensions, so we have to describe them in three dimensions as well.

===A Mathematical Model===

What are the mathematical equations that allow us to model this topic. For example <math><{\frac{d\vec{x}}{dt}},{\frac{d\vec{y}}{dt}},{\frac{d\vec{z}}{dt}}></math> is the velocity and <math>{\frac{d\vec{(velocity)}}{dt}}</math> is the acceleration.

===A Computational Model===

To program the position in VPython for an object, obj, write obj.pos=(xp,yp,zp). Here xp, yp, and zp are the x, y, and z coordinates, respectively, of the object. Velocity and acceleration are programmed similarly with obj.velocity=(xv,yv,zv) and obj.acceleration=(xa,ya,za). The x, y, and z velocity and acceleration values are xv, yv, and zv and xa, ya, and za respectively.

==Examples==

Here are a few examples

===Simple===
obj. is at position (0,0,0) meters, moving at a velocity of (-1, 4, 9) meters per second for n seconds. What is obj.'s position now?
(0-n,0+4n,0+9n)=(-n,4n,9n)

===Middling===
obj. is at position (2,5,8) meters. Acceleration is (2, 9, 0) meters per second squared for 5 seconds. new position= (2,5,8)+(2,9,0)*1/2*5^2= (2,5,8)+(25,112.5,0)=(27,117.5,8)

===Difficult===
obj. starts at position <math>(1,2,1)</math> meters with initial velocity <math>(1,5,2)</math> and an acceleration of <math>(-1,4,-2)</math>.
After four seconds, what is the position? <math>position= (initial position) + (initial velocity)*(time) + (acceleration)*1/2(time)^2</math>.
<math>(1,2,1) + 4*(1,5,2) + 4^2/2*(-1,4,-2)= (1,2,1)+(4,20,8)+(-8,32,-16)=(1+4-8,2+20+32,1+8-16)=(-3,54,-7)</math>

==Connectedness==
#How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion. Any force acting upon an object is doing so in three dimensions. If I throw a football, the force I use to throw it is in three dimensions, as is its position and velocity.
#How is it connected to your major?
I am a mechanical engineering major. As mentioned before, whenever an object is acted upon by a force, this happens in three dimensions.
#Is there an interesting industrial application?
Yes! Every force is in three dimensions, as is every object and its movement.

== See also ==

===External links===
[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

Reciprocity

2016-11-27T00:20:32Z

Bath3:

This topic covers why forces on each other are equal in magnitude.

[[File:BookForce.JPG|thumb|This is an example of reciprocity. The book is exerting a contact force on the table and the table is exerting a contact force on the book.]]

==The Main Idea==

Reciprocity is the idea that the force object 1 exerts on object 2 is the same as the force object 2 exerts on object 1. This idea comes from Newton's Third Law of Motion. Forces are results of interactions. If i put my hand on a table, I am exerting a contact force on the table, but at the same time the table is using a exerting force on me. Though it seems like I am putting in more effort, the forces are the same. Forces come in pairs. The two forces are called "action" and "reaction" pairs. When forces are in these pairs, the magnitude of the two forces equal each other. However, in vector form, the two forces would be in opposite directions of each other, so one force would have a negative sign on it.

===A Mathematical Model===

Here is a formulaic representation of reciprocity.

F1on2=-F2on1.

The vector form of reciprocity is this:

<F,0,0> is one force.

<-F,0,0> is the exact same force but in the opposite direction.

This is the equation in vector format. When the vector of one force is in one direction. Usually, the vector is in the other direction

This equation on the left side. shows that object one is acting on object 2. The equation on the right side shows the reverse.

==Examples==

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

===Simple===

If you exert 20 N on the table, what would be the normal force of the table on you?

Since you are exerting 20 Newtons, due to reciprocity the table will be exerting a normal force of '''20''' Newtons.

===Middling===

A 60 kilogram man stands on the surface of the Earth. What is the force Earth exerts on the man? What is the force the man exerts on the Earth?

60x9.8=588N. Due to reciprocity 588 is the force on both Earth and the man.

===Difficult===

This problem is from our test.

Two blocks of mass m1(under rod) and m3(above rod) are connected by a rod of mass m2. A constant unknown force F pulls upward on the top block while both blocks and the rod move upward at a constant velocity v near the surface of the Earth. The direction of the gravitational force on each block points down. Find F1on2, the force exerted by the bottom block on the rod.

Fnet1=0 due to constant v

F2on1-m1gy=0

F2on1=m1gy

F1on2=-m1gy

==Connectedness==
# The first physics I ever learned was Newtons laws. Before heading into any science class, I always thought, every reaction gets an equal and opposite reaction. I didnt really understand it. That is a fundamental principle that we use in almost all physics problems. It has been test questions and homework questions. The thing that intrigues me the most is how an ant can be pushing against a rhino and though the rhino is so much bigger, they are still exerting the same force.
#I am an industrial engineering major and though there is very minimal use of physics in that field, I do believe it is something that will help us go about our days knowing that force isn't how much effort you put in but about the action reaction pairs.
#Forces are something we deal with everyday. Everything we touch, me typing this page right now is all the result of forces. An important industry that deals with this is the automobile industry. If we understand the forces of the wheels on the road, we will know how to make wheels that best suit an automobile.

==History==

Isaac Newton was born in Woolsthorpe, England. When he was a child, one day he was resting under an apple tree when suddenly an apple fell on his head. He thought about why things fall down and not fall back up. He spent years figuring out the phenomenon. After all this, he came up with three laws of motion. This is when he discovered gravitation as a force. Where Newton's Law comes into play is that the Earth is exerting a force on us to stay with it since closer objects exert stronger forces on each other. We are also exerting a force on Earth so that we stay on the ground and don't go flying off. The date of this story is not known, and some even believe it to be a myth. However William Stukeley, author of ''Memoirs of Sir Isaac Newton's Life'' noted that he had a conversation with Newton and Newton talked about why an apple falls to the ground due to gravitational interaction.

== See also ==

===Further reading===

:'''1:'''http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''2:'''Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

===External links===

https://www.youtube.com/watch?v=NfuKfbpkIrQ

==References==

:'''1:''' http://www.mainlesson.com/display.php?author=baldwin&book=thirty&story=newton

:'''2:''' https://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.html

:'''3:''' http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''4:''' Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

[[Category:Which Category did you place this in?]]

Scattering: Collisions in 2D and 3D

2016-11-27T00:19:40Z

Bath3:

Claimed by: Benjamin Tasistro-Hart (Fall 2016)

==The Main Idea==

Unlike normal collisions, atomic and nuclear collisions are far too small to observe the curving trajectories of the interacting particles. The only thing that can be noticed is the initial and final states of the interaction. Scattering experiments are incorporated in the world of collisions to be able to study the minute details (structure) of atoms, nuclei, and other tiny particles as the interact with one another.

==Impact Parameters==
Definition: The distance between centers perpendicular to the incoming velocity. Impact parameter is often denoted by the variable b.

A head-on collision has an impact parameter of zero and with equal masses fully transfers the momentum such as with Newton's Cradle. As the impact parameter gets smaller the collision has a larger effect, and an even large deflection angle (scattering).

Elastic collisions between two billiard balls (Double Click Image)
[[File:ImpactParameter.png]]

==Examples==
===Simple===
The collision of an alpha particle (helium nucleus) with the nucleus of a gold atom

==History==

In 1871 Lord Rayleigh published a paper on scattering. Rayleigh scattering is the dispersion of electromagnetic radiation by particles that have a minute radius less than approximately 1/10 the wavelength. It laid the foundation to research on scattering and information we have today.

== See also ==

[[Category:Collisions] (Main page)

===Further reading===

Matter and Interactions, Volume I: Modern Mechanics, 4th Edition. (Chapter 10.6)

===External links===
[http://physics-animations.com/Physics/English/par_txt.htm]
[http://www.britannica.com/science/Rayleigh-scattering]
[http://hypernews.slac.stanford.edu/slacsite/aux/HiPPP/scattering/]

==References==

Chabay, Ruth W., Bruce Sherwood. Matter and Interactions, Volume I: Modern Mechanics, 4th Edition. Wiley, 19/2014.

[[Category:Collisions]]

Power (Mechanical)

2016-11-12T14:22:13Z

Bath3:

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]

Reciprocity

2016-11-12T14:21:48Z

Bath3:

[['''Claimed by Benjamin Tasistro-Hart Fall 2016''']]
This topic covers why forces on each other are equal in magnitude.

[[File:BookForce.JPG|thumb|This is an example of reciprocity. The book is exerting a contact force on the table and the table is exerting a contact force on the book.]]

==The Main Idea==

Reciprocity is the idea that the force object 1 exerts on object 2 is the same as the force object 2 exerts on object 1. This idea comes from Newton's Third Law of Motion. Forces are results of interactions. If i put my hand on a table, I am exerting a contact force on the table, but at the same time the table is using a exerting force on me. Though it seems like I am putting in more effort, the forces are the same. Forces come in pairs. The two forces are called "action" and "reaction" pairs. When forces are in these pairs, the magnitude of the two forces equal each other. However, in vector form, the two forces would be in opposite directions of each other, so one force would have a negative sign on it.

===A Mathematical Model===

Here is a formulaic representation of reciprocity.

F1on2=-F2on1.

The vector form of reciprocity is this:

<F,0,0> is one force.

<-F,0,0> is the exact same force but in the opposite direction.

This is the equation in vector format. When the vector of one force is in one direction. Usually, the vector is in the other direction

This equation on the left side. shows that object one is acting on object 2. The equation on the right side shows the reverse.

==Examples==

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

===Simple===

If you exert 20 N on the table, what would be the normal force of the table on you?

Since you are exerting 20 Newtons, due to reciprocity the table will be exerting a normal force of '''20''' Newtons.

===Middling===

A 60 kilogram man stands on the surface of the Earth. What is the force Earth exerts on the man? What is the force the man exerts on the Earth?

60x9.8=588N. Due to reciprocity 588 is the force on both Earth and the man.

===Difficult===

This problem is from our test.

Two blocks of mass m1(under rod) and m3(above rod) are connected by a rod of mass m2. A constant unknown force F pulls upward on the top block while both blocks and the rod move upward at a constant velocity v near the surface of the Earth. The direction of the gravitational force on each block points down. Find F1on2, the force exerted by the bottom block on the rod.

Fnet1=0 due to constant v

F2on1-m1gy=0

F2on1=m1gy

F1on2=-m1gy

==Connectedness==
# The first physics I ever learned was Newtons laws. Before heading into any science class, I always thought, every reaction gets an equal and opposite reaction. I didnt really understand it. That is a fundamental principle that we use in almost all physics problems. It has been test questions and homework questions. The thing that intrigues me the most is how an ant can be pushing against a rhino and though the rhino is so much bigger, they are still exerting the same force.
#I am an industrial engineering major and though there is very minimal use of physics in that field, I do believe it is something that will help us go about our days knowing that force isn't how much effort you put in but about the action reaction pairs.
#Forces are something we deal with everyday. Everything we touch, me typing this page right now is all the result of forces. An important industry that deals with this is the automobile industry. If we understand the forces of the wheels on the road, we will know how to make wheels that best suit an automobile.

==History==

Isaac Newton was born in Woolsthorpe, England. When he was a child, one day he was resting under an apple tree when suddenly an apple fell on his head. He thought about why things fall down and not fall back up. He spent years figuring out the phenomenon. After all this, he came up with three laws of motion. This is when he discovered gravitation as a force. Where Newton's Law comes into play is that the Earth is exerting a force on us to stay with it since closer objects exert stronger forces on each other. We are also exerting a force on Earth so that we stay on the ground and don't go flying off. The date of this story is not known, and some even believe it to be a myth. However William Stukeley, author of ''Memoirs of Sir Isaac Newton's Life'' noted that he had a conversation with Newton and Newton talked about why an apple falls to the ground due to gravitational interaction.

== See also ==

===Further reading===

:'''1:'''http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''2:'''Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

===External links===

https://www.youtube.com/watch?v=NfuKfbpkIrQ

==References==

:'''1:''' http://www.mainlesson.com/display.php?author=baldwin&book=thirty&story=newton

:'''2:''' https://www.newscientist.com/blogs/culturelab/2010/01/newtons-apple-the-real-story.html

:'''3:''' http://www.physicsclassroom.com/class/newtlaws/Lesson-4/Newton-s-Third-Law

:'''4:''' Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3.4

[[Category:Which Category did you place this in?]]

Power (Mechanical)

2016-11-12T14:18:49Z

Bath3:

==The Main Idea==
'''Claimed for Fall 2016 by Benjamin Tasistro-Hart'''

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]

Power (Mechanical)

2016-11-12T14:18:32Z

Bath3:

==The Main Idea==
'''Claimed for Fall 2016 by Benjamin Tasistro-Hart'''''Italic text''

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]

Power (Mechanical)

2016-11-12T14:18:09Z

Bath3:

==The Main Idea==
Claimed for Fall 2016 by Benjamin Tasistro-Hart
Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]