Energy Graphs: Difference between revisions

Revision as of 20:39, 24 November 2018

Claimed by Yuntao Liu Fall 2018

What is an Energy Graph?

Energy graphs are extremely useful for analyzing the interactions between two different objects. These graphs can also be confusing if not properly understood. How do all of the different pieces of an energy graph fit together? This page serves as a resource on how to construct and interpret energy graphs. Let's get started!

The Basics

Energy graphs are very useful for understanding different repulsive and attractive situations. These graphs consist of three main components: kinetic energy, potential energy, and kinetic plus potential energy. These three energies are plotted as a function of energy (y-axis) versus separation between two objects (x-axis). When it comes to spring potential and kinetic energy, energy is graphed as a function of time.

Let's review:

Kinetic Energy: the energy that a moving object has in addition to its Rest Mass Energy. Kinetic energy is K on an energy graph and has the formula [math]\displaystyle{ \frac{1}{2}mv^2 }[/math].

Potential Energy: the energy associated with multi-particle systems. This is the energy between interacting objects, and this energy can be either attractive or repulsive. Potential energy is U on an energy graph.

What is the sum? Simply the sum of the potential and kinetic energies displayed on the graph. We will discuss how to plot this value momentarily. K+U symbolizes the sum of the potential and kinetic energies. This generally constant when the surrounding are not performing any work on the system (aka when the surrounds are "The Nothing"). However, if there is energy being lost to the surroundings or work being done on the system, then the K+U can change.

Positive or Negative?

Kinetic energy can NEVER be negative! Why? Taking a look at the formula [math]\displaystyle{ \frac{1}{2}mv^2 }[/math], while velocity is a vector and can be negative, squaring the velocity results in a positive number. As such, if the object is moving, the kinetic energy is greater than 0, if it is at rest, the kinetic energy is 0.

Potential energy is negative if the interaction is attractive and positive if the interaction is repulsive.

For spring potential energy, the value is always positive as the spring is always working to counteract a stretch (repulsive). [math]\displaystyle{ U_s=\frac{1}{2}k_ss^2 }[/math] where s is the stretch and k_s is the spring constant.
For electric potential energy [math]\displaystyle{ U_{ele} = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\left|r\right|} }[/math]. Where [math]\displaystyle{ \frac{1}{4\pi\varepsilon_0} }[/math]is [math]\displaystyle{ 9\times10^9 \frac{N\cdot m^2}{C^2} }[/math], r is the distance between the charges and the q's are the charges. If the change of the two particles have the same sign (both positive or both negative), then the force is repulsive and results in a negative potential energy. If the change of the two particles have different signs (one positive and one negative), then the force is attractive and results in a postie potential energy.
- An electron has a change of -1.6 × 10⁻¹⁹ C and a proton has a change of 1.6 × 10⁻¹⁹ C
For gravitational potential energy [math]\displaystyle{ U_{grav} = -G\frac{M_1M_2}{\left|r\right|} }[/math]where G is [math]\displaystyle{ 6.7\times10^{-11} \frac{N\cdot m^2}{kg^2} }[/math] , M is the mass of the object and r is the distance between the objects. the force is always attractive and the potential energy is negative

The sum of the potential and kinetic energies depends on a few factors. If an object can "escape" from an attractive interaction, and, after having escaped, comes to rest some distance away, then K+U is equal to zero. If, however, the object escapes with some final velocity and continues to move, then K+U is greater than zero. Lastly, in a bound state, an object cannot escape the interaction it is involved in, and K+U is negative.

More Important Tidbits

In an attractive interaction, such as Gravitational Potential Energy, U (potential energy) increases (becomes less negative) and K decreases as two objects move farther away from one another (radius increases). In a repulsive interaction, U (potential energy) decreases and K (kinetic energy) increases as the two objects move farther away from one another. When a spring is at equalibrium length after being strched, kinetic energy is at a maximum.

How to Construct an Energy Graph

1. Draw U first, based upon whether or not the interaction is attractive or repulsive. See the examples below in order to visualize how U looks when an interaction is attractive versus repulsive.

2. Choose an r where you know K, and at this point, plot the point (r, K).

3. Add the K value from step 2 to the U value at that same r value.

4. Plot K+U at this r value that you know both K and U.

5. Draw a horizontal line through this K+U point.

6. Find a K value at a different r value that, when added to U, gives K+U (the horizontal line you just drew).

7. With these two points on the K graph, sketch K vs. r.

A Computational Model

Vpython is great for modeling energy graphs. Using Vpython, we can model many different systems that have kinetic and potential energies. For example, we can model a spacecraft orbiting the Earth, and we can create graphs to display the kinetic, potential, and kinetic+potential energies of this system. See this code for how to model these interactions using Vpython! This specific code shows the spacecraft + Earth system.

Note: scroll down on the black display window on the right to see the energy graph for this system.

Sample Vpython code:https://trinket.io/glowscript/4010e21bc3

Examples

See end of examples for solutions to all examples.

Beginner

Label K, U, and K+U on the energy graphs for the following situations:

Example 1: Two electrons are held at rest some finite distance apart, and they move away from each other after they are released. Their initial velocities are zero.

Example 2: A proton and an electron start out far apart. Their initial velocities are nonzero, and their interaction is attractive.

Example 3: A daughter has just enough energy to escape from her controlling mother (they have an attractive relationship).

Intermediate

Label K, U, and K+U on the following energy graphs. THEN, label the following three energy graphs 1, 2, or 3 based on the following scenarios:

1: One of the components of this energy graph is incorrect.

2: A proton and a electron are at rest, and they start out infinitely far apart.

3: An astronaut orbits the moon.

Advanced

Create an energy graph for each of the following situations.

Situation 1: A spacecraft is orbiting a moon. The spacecraft is given an initial velocity that allows the spacecraft to leave the moon's orbit with a final velocity greater than zero.

Situation 2: A boy jumps onto a merry-go-round and is attracted to the merry-go-round's axle. The boy's initial velocity is not large enough for him to escape the merry-go-round, so he continues to "orbit" the merry-go-round.

Situation 3: Two people are repulsed by on another and are trying to fight. They are held at rest by two of their friends a finite distance apart, and they of course move away from one another as soon as they are released, since their friends will not let them approach one another.

Situation 4: A spring with a spring constant of k and has an inital

Situation 4a: The spring

Solutions to Beginner Examples

Example 1:

Example 2:

Example 3:

Solutions to Intermediate Examples

Example 1:

K+U is incorrect!

Example 2:

Example 3:

Solutions to Advanced Examples

Situation 1:

Situation 2:

Situation 3:

Connectedness

How is energy related to my future? How is energy related to the things I am passionate about? How are energy and physics related to biochemistry, medical school, and surgery? In order to get into medical school, I must first take the MCATs. Physics and math majors score the highest on the MCATs, so physics is extremely important to master in order to successfully master the MCATs. Energy is everywhere, so I need to understand energy in order to be successful in the future. Physics is a part of everyday life. Energy is a part of everyday life. Mastering both now will only aid me in my dream of becoming a surgeon. From medical school student, to intern, to surgical resident, to, one day, a surgeon, energy and physics will follow me. Energy graphs help to explain different energy interactions and make understanding energy simpler. Energy is a part of every scalpel I will hold, and kinetic energy will push me through the late nights and long shifts in my future. Is energy applicable to my future? Is energy applicable to medicine and surgery? Absolutely! Physics is more applicable to every aspect of my life than I ever could have imagined.Why is the discovery of energy so important? Energy and the conservation of energy drive our everyday lives. Everything we touch, eat, drink, and use has energy! Without energy, there would be no life as we know it. The laptop you are using right now? Potential energy. Your hand scrolling through this page? Kinetic energy. Everything has energy! And these energy graphs are a great way to help us learn how to interpret energy in different situations.

History

Gustave-Gaspard Coriolis first described "kinetic energy" in 1829, and William Rankine coined the term "potential energy" in 1853.

Energy was proven mathematically in 1918.

Emmy Noether (1882-1935) uncovered the fundamental justification for conservation laws.

External links

The Energy Principle

Conservation of Energy

Kinetic Energy

Potential Energy

Electric Potential

Gravitational Potential Energy

Rest Mass Energy

Escape Velocity

References

Chabay, R. W. & Sherwood, B. A. (2015). Matter and interactions. North Carolina State University: John Wiley & Sons, Inc.

Physics 2211 Test 3

https://en.wikipedia.org/wiki/History_of_energy

Physics 2211 lecture notes

Physcs 2211 labs via Webassign

@@ Line 1: / Line 1: @@
 '''Claimed by Yuntao Liu Fall 2018'''
-==What is an Energy Graph?==
+== What is an Energy Graph? ==
-[[File:Mainideapic.JPG|center|300px|Example 2]]
 Energy graphs are extremely useful for analyzing the interactions between two different objects. These graphs can also be confusing if not properly understood. How do all of the different pieces of an energy graph fit together? This page serves as a resource on how to construct and interpret energy graphs. Let's get started!
-===The Basics===
+=== The Basics ===
-Energy graphs are very useful for understanding different repulsive and attractive situations. These graphs consist of three main components: kinetic energy, potential energy, and kinetic plus potential energy. These three energies are plotted as a function of energy versus separation between two objects.
+Energy graphs are very useful for understanding different repulsive and attractive situations. These graphs consist of three main components: kinetic energy, potential energy, and kinetic plus potential energy. These three energies are plotted as a function of energy (y-axis) versus separation between two objects (x-axis). When it comes to spring potential and kinetic energy, energy is graphed as a function of time.
 Let's review:
-[[Kinetic Energy]]: the energy that a moving object has in addition to its [[Rest Mass Energy]]. Kinetic energy is '''K''' on an energy graph.
+[[Kinetic Energy]]: the energy that a moving object has in addition to its [[Rest Mass Energy]]. Kinetic energy is '''K''' on an energy graph and has the formula <math>\frac{1}{2}mv^2</math>.
 [[Potential Energy]]: the energy associated with multi-particle systems. This is the energy between interacting objects, and this energy can be either attractive or repulsive. Potential energy is '''U''' on an energy graph.
-'''What is the sum?''' Simply the sum of the potential and kinetic energies displayed on the graph. We will discuss how to plot this value momentarily. '''K+U''' symbolizes the sum of the potential and kinetic energies.
+'''What is the sum?''' Simply the sum of the potential and kinetic energies displayed on the graph. We will discuss how to plot this value momentarily. '''K+U''' symbolizes the sum of the potential and kinetic energies. This generally constant when the surrounding are not performing any work on the system (aka when the surrounds are "The Nothing"). However, if there is energy being lost to the surroundings or work being done on the system, then the K+U can change.
+=== Positive or Negative? ===
+Kinetic energy can NEVER be negative! Why? Taking a look at the formula <math>\frac{1}{2}mv^2</math>, while velocity is a vector and can be negative, squaring the velocity results in a positive number. As such, if the object is moving, the kinetic energy is greater than 0, if it is at rest, the kinetic energy is 0.
+Potential energy is negative if the interaction is attractive and positive if the interaction is repulsive.
-===Positive or Negative?===
+* For spring potential energy, the value is always positive as the spring is always working to counteract a stretch (repulsive). <math>U_s=\frac{1}{2}k_ss^2</math> where s is the stretch and k<sub>s</sub> is the spring constant.
-Kinetic energy can NEVER be negative!
+* For electric potential energy <math>U_{ele} = \frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{\left|r\right|}</math>. Where <math>\frac{1}{4\pi\varepsilon_0}</math>is <math>9\times10^9  \frac{N\cdot m^2}{C^2}</math>, r is the distance between the charges and the q's are the charges. If the change of the two particles have the same sign (both positive or both negative), then the force is repulsive and results in a negative potential energy. If the change of the two particles have different signs (one positive and one negative), then the force is attractive and results in a postie potential energy.
-Potential energy is negative if the interaction is attractive and positive if the interaction is repulsive.
+** An electron has a change of -1.6 × 10<sup>−19</sup> C and a proton has a change of 1.6 × 10<sup>−19</sup> C
-The sum of the potential and kinetic energies depends on a few factors. If an object can "escape" from an attractive interaction, and, after having escaped, comes to rest, then K+U is equal to zero. If, however, the object escapes with some final velocity, then K+U is greater than zero. Lastly, in a '''bound state,''' an object cannot escape the interaction it is involved in, and K+U is negative.
+* For gravitational potential energy <math>U_{grav} = -G\frac{M_1M_2}{\left|r\right|}</math>where G is <math>6.7\times10^{-11}   \frac{N\cdot m^2}{kg^2}</math> , M is the mass of the object and r is the distance between the objects. the force is always attractive and the potential energy is negative
-===More Important Tidbits===
+The sum of the potential and kinetic energies depends on a few factors. If an object can "escape" from an attractive interaction, and, after having escaped, comes to rest some distance away, then K+U is equal to zero. If, however, the object escapes with some final velocity and continues to move, then K+U is greater than zero. Lastly, in a '''bound state,''' an object cannot escape the interaction it is involved in, and K+U is negative.
-K (kinetic energy) is zero or nonzero depending on the motion of the objects. At rest, kinetic energy is zero. In motion, kinetic energy is greater than zero.
-In an '''attractive''' interaction, such as [[Gravitational Potential Energy]], U (potential energy) increases (becomes less negative) and K decreases as two objects move farther away from one another.
-In a '''repulsive''' interaction, U(potential energy) decreases and K (kinetic energy) increases as the two objects move farther away from one another.
-The '''x-axis''' of an energy graph is separation, denoted r, and the '''y-axis''' is energy.
+=== More Important Tidbits ===
+In an '''attractive''' interaction, such as [[Gravitational Potential Energy]], U (potential energy) increases (becomes less negative) and K decreases as two objects move farther away from one another (radius increases).
+In a '''repulsive''' interaction, U (potential energy) decreases and K (kinetic energy) increases as the two objects move farther away from one another. When a spring is at equalibrium length after being strched, kinetic energy is at a maximum.
-===How to Construct an Energy Graph===
+=== How to Construct an Energy Graph ===
-Now that you know the basics, practice constructing and analyzing energy graphs is the best way to fully grasp this concept.
-Here are some helpful steps for how to draw an energy graph:
 . Draw U first, based upon whether or not the interaction is attractive or repulsive. See the examples below in order to visualize how U looks when an interaction is attractive versus repulsive.
@@ Line 46: / Line 48: @@
 . With these two points on the K graph, sketch K vs. r.
-===A Computational Model===
+=== A Computational Model ===
 Vpython is great for modeling energy graphs. Using Vpython, we can model many different systems that have kinetic and potential energies. For example, we can model a spacecraft orbiting the Earth, and we can create graphs to display the kinetic, potential, and kinetic+potential energies of this system. See this code for how to model these interactions using Vpython! This specific code shows the spacecraft + Earth system.
-*Note: scroll down on the black display window on the right to see the energy graph for this system.
+* Note: scroll down on the black display window on the right to see the energy graph for this system.
 Sample Vpython code:https://trinket.io/glowscript/4010e21bc3
-==Examples==
+== Examples ==
-'''See end of examples for solutions to all examples.
+'''See end of examples for solutions to all examples.'''
-'''
-===Beginner===
+=== Beginner ===
 Label K, U, and K+U on the energy graphs for the following situations:
 Example 1:
 Two electrons are held at rest some finite distance apart, and they move away from each other after they are released. Their initial velocities are zero.
-[[File:109.jpg|300px|center|Example 1]]
+[[File:109.jpg|center|300x300px|Example 1]]
 Example 2:
 A proton and an electron start out far apart. Their initial velocities are nonzero, and their interaction is attractive.
-[[File:111.jpg|300px|center|Example 2]]
+[[File:111.jpg|center|300x300px|Example 2]]
 Example 3:
 A daughter has just enough energy to escape from her controlling mother (they have an attractive relationship).
-[[File:JDO78.JPG|300px|center|Example 3]]
+[[File:JDO78.JPG|center|300x300px|Example 3]]
-===Intermediate===
+=== Intermediate ===
 Label K, U, and K+U on the following energy graphs. THEN, label the following three energy graphs 1, 2, or 3 based on the following scenarios:
@@ Line 84: / Line 86: @@
 : An astronaut orbits the moon.
-[[File:JDO78.JPG|300px|Example 1]]
+[[File:JDO78.JPG|300x300px|Example 1]]
-[[File:JDO100.JPG|300px|Example 2]]
+[[File:JDO100.JPG|300x300px|Example 2]]
-[[File:Jdo77.jpg|300px|Example 3]]
+[[File:Jdo77.jpg|300x300px|Example 3]]
-===Advanced===
+=== Advanced ===
 Create an energy graph for each of the following situations.
@@ Line 97: / Line 99: @@
 Situation 3: Two people are repulsed by on another and are trying to fight. They are held at rest by two of their friends a finite distance apart, and they of course move away from one another as soon as they are released, since their friends will not let them approach one another.
-===Solutions to Beginner Examples===
+Situation 4: A spring with a spring constant of k and has an inital
+Situation 4a: The spring
+=== Solutions to Beginner Examples ===
 Example 1:
-[[File:Simpleex1solution.JPG|300px|Example 1]]
+[[File:Simpleex1solution.JPG|300x300px|Example 1]]
 Example 2:
-[[File:Simpleex2solution.JPG|300px|Example 2]]
+[[File:Simpleex2solution.JPG|300x300px|Example 2]]
 Example 3:
-[[File:Simpleex3solution.JPG|300px|Example 3]]
+[[File:Simpleex3solution.JPG|300x300px|Example 3]]
-===Solutions to Intermediate Examples===
+=== Solutions to Intermediate Examples ===
 Example 1:
 K+U is incorrect!
-[[File:Intermediateex1solution.JPG|300px|Example 1]]
+[[File:Intermediateex1solution.JPG|300x300px|Example 1]]
 Example 2:
-[[File:Intermediateex2solution.JPG|300px|Example 2]]
+[[File:Intermediateex2solution.JPG|300x300px|Example 2]]
 Example 3:
-[[File:Intermediateex3solution.JPG|300px|Example 3]]
+[[File:Intermediateex3solution.JPG|300x300px|Example 3]]
-===Solutions to Advanced Examples===
+=== Solutions to Advanced Examples ===
 Situation 1:
-[[File:Simpleex2solution.JPG|300px|Situation 1]]
+[[File:Simpleex2solution.JPG|300x300px|Situation 1]]
 Situation 2:
-[[File:Advancedex2solution.JPG|300px|Situation 2]]
+[[File:Advancedex2solution.JPG|300x300px|Situation 2]]
 Situation 3:
-[[File:Simpleex1solution.JPG|300px|Situation 3]]
+[[File:Simpleex1solution.JPG|300x300px|Situation 3]]
-==Energy and My Future==
+== Connectedness ==
-How is energy related to my future? How is energy related to the things I am passionate about? How are energy and physics related to biochemistry, medical school, and surgery? In order to get into medical school, I must first take the MCATs. Physics and math majors score the highest on the MCATs, so physics is extremely important to master in order to successfully master the MCATs. Energy is everywhere, so I need to understand energy in order to be successful in the future. Physics is a part of everyday life. Energy is a part of everyday life. Mastering both now will only aid me in my dream of becoming a surgeon. From medical school student, to intern, to surgical resident, to, one day, a surgeon, energy and physics will follow me. Energy graphs help to explain different energy interactions and make understanding energy simpler. Energy is a part of every scalpel I will hold, and kinetic energy will push me through the late nights and long shifts in my future. Is energy applicable to my future? Is energy applicable to medicine and surgery? Absolutely! Physics is more applicable to every aspect of my life than I ever could have imagined.
+How is energy related to my future? How is energy related to the things I am passionate about? How are energy and physics related to biochemistry, medical school, and surgery? In order to get into medical school, I must first take the MCATs. Physics and math majors score the highest on the MCATs, so physics is extremely important to master in order to successfully master the MCATs. Energy is everywhere, so I need to understand energy in order to be successful in the future. Physics is a part of everyday life. Energy is a part of everyday life. Mastering both now will only aid me in my dream of becoming a surgeon. From medical school student, to intern, to surgical resident, to, one day, a surgeon, energy and physics will follow me. Energy graphs help to explain different energy interactions and make understanding energy simpler. Energy is a part of every scalpel I will hold, and kinetic energy will push me through the late nights and long shifts in my future. Is energy applicable to my future? Is energy applicable to medicine and surgery? Absolutely! Physics is more applicable to every aspect of my life than I ever could have imagined.Why is the discovery of energy so important?
+Energy and the conservation of energy drive our everyday lives. Everything we touch, eat, drink, and use has energy! Without energy, there would be no life as we know it. The laptop you are using right now? Potential energy. Your hand scrolling through this page? Kinetic energy. Everything has energy! And these energy graphs are a great way to help us learn how to interpret energy in different situations.
-[[File:Syringe.jpg|200px|center]]
+== History ==
-[[File:Scalpel123.jpg|200px|center]]
-==History==
 Gustave-Gaspard Coriolis first described "kinetic energy" in 1829, and William Rankine coined the term "potential energy" in 1853.
@@ Line 156: / Line 159: @@
 Emmy Noether (1882-1935) uncovered the fundamental justification for conservation laws.
-Why is the discovery of energy so important?
+=== Further reading ===
-Energy and the conservation of energy drive our everyday lives. Everything we touch, eat, drink, and use has energy! Without energy, there would be no life as we know it. The laptop you are using right now? Potential energy. Your hand scrolling through this page? Kinetic energy. Everything has energy! And these energy graphs are a great way to help us learn how to interpret energy in different situations.
-===Further reading===
+Matter and Interactions 4th Edition by Ruth W. Chabay and Bruce A. Sherwood
-Matter and Interactions by Ruth W. Chabay and Bruce A. Sherwood
+* Section 6.10
-===External links===
+=== External links ===
 [[The Energy Principle]]
@@ Line 178: / Line 180: @@
 [[Rest Mass Energy]]
-==References==
+Escape Velocity
+== References ==
 Chabay, R. W. & Sherwood, B. A. (2015). Matter and interactions. North Carolina State University: John Wiley & Sons, Inc.
@@ Line 190: / Line 194: @@
 Physcs 2211 labs via Webassign
-[[Category: Energy]]
+[[Category:Energy]]