## Activity: Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
What students learn
• The choice of where $U=0$ is arbitrary and doesn't affect the difference in potential energy between two locations.
• Gravitational potential energy is conventionally negative, $U<0$. “Increasing” potential energy approaches $U=0$.
• $U=-GMm/r$, is valid outside spherically symmetric Earth, but not inside it.
• The Earth-satellite system can be idealized as spherically symmetric. This prepares students for later discussions about equipotentials and about radial symmetry.
• Media

Your group has a plastic surface that represents the gravitational potential energy of a space station-Earth system as a function of the position of the space station relative to Earth. The height of the surface corresponds to the value of gravitational potential energy.

Interpret the Surface:

• Mark a point on the surface where the gravitational potential energy of the system is zero.

Answer: Any point is okay. Students might choose:

• the center of Earth,
• the surface of Earth,
• the blue circle,
• some arbitrary point,
• the far corner of the surface to indicate infinitely far away

• If a one inch difference of height corresponds to an energy difference of 1 TJ, what is the gravitational potential energy of the system when the space station is at the blue circle?

Answer: Students need to measure the difference in height between the blue circle and their zero. Some methods for doing this include:

• hold ruler vertically and align estimate vertical displacement between zero location and blue circle
• measure height from base to zero and from base to blue circle
• turn the surface over and measure depths

• What is the difference in gravitational potential energy of the system if the space station moves from the blue circle to the red star?

Answer: All groups should see the potential energy increase by +0.5 TJ

As a short class discussion, have the students help you build a table for the two measured values. One column will vary, the other column should be approximately +0.5 TJ for all groups.

Find Patterns: For each of the locations listed below, identify all other points on the plastic surface model with the same gravitational potential energy:

1. the blue circle
2. the red star
3. the green square
4. the point you marked in question 1.

What patterns are you noticing?

Hint: “If the space station was at the blue circle, how could it move so that the gravitational potential energy doesn't change?”

Discussion: Patterns “What patterns did your group notice?” (Give students time to discuss within their group then let groups report out)

Possible patterns:

• circles/arcs
• horizontal spacing for circle/star/square increasing
• vertical spacing for circle/star/square the same

Describe an Orbit: Draw a dot on your whiteboard to represent the center of Earth. Draw a closed, elliptical orbit of the space station. Using the information in the plastic surface model, what happens to the gravitational potential energy as it moves around the orbit?

If the space station does not use its engines during the orbit so that the total energy is constant, what happens to the speed of the space station?

Hint: To simplify, ask students about the earlier case of a circular orbit.

Answer: The speed will change. Conservation of energy says that if the space station is closer to earth, its speed must increase.

Extension: Draw an orbit so that the space station is always over the same spot on the Earth as it rotates (a geostationary orbit).

Generate a Graph: Using your whole whiteboard, sketch a large graph that shows how the gravitational potential energy of the system depends on the space station's distance from the center of Earth. Clearly label the axes of your graph. Be ready to show where you chose $U=0$ and to explain your reasoning to your classmates.

Extension: If a group gets done early, ask them to mark where the blue circle, orange star, and green square are on your plots.

Extension: Remind students that GPE is a function of three spatial variables. Ask them to describe the equipotentials (spherical shells) or the (spherical) symmetry involved.

What Students Learn:

• The choice of where $U=0$ is arbitrary and doesn't affect the difference in potential energy between two locations.
• The zero of gravitational potential energy is conventionally set at $r\rightarrow \infty$, making the potential energy negative, $U<0$. “Increasing” potential energy approaches $U=0$.
• $U=-GMm/r$, is valid outside spherically symmetric Earth, but not inside it.
• The Earth-satellite system can be idealized as spherically symmetric. This prepares students for later discussions about equipotentials and about radial symmetry.

Time Estimate: 30 minutes + 20 minute class discussion.

Equipment:

• Green “sphere” surface for each group
• Dry-erase markers & erasers
• Whiteboard for each group
• Student handout for each student

Prerequisites:

• This activity works particularly well when done before the Gravitational Force activity, even if you introduce force before gravitational potential energy, because students examine how the surface represents the physical system.
• Students should know about potential energy but do not need to know about universal gravitation before the activity. The instructor will introduce the equation $U=-GMm/r$ during this activity.

Introduction:

• Explain that the
• center of Earth is in the lowest corner,
• the etched line represents the surface of Earth,
• the base is the xy-plane, and
• the height of the surface represents the gravitational potential energy for locations in the xy-plane.
Whole Class Discussion / Wrap Up:
• There are various strategies for measuring heights on the surface
• Differences in energy are what matter, not the values
• Spherical symmetry means we can use $r$ to fully describe the energy

Interpret the Surface
Purpose: We want students to physically manipulate the surface so that they kinesthetically understand the surface height as representing gravitational potential energy. We also want students to learn that although the choice of zero changes the value of the gravitational potential energy at any point, the difference in energy between two points will always be the same. In this situation, the difference is the physically meaningful quantity.
Discussion: Students must be told that the height of the surface represents GPE, and that one inch of height is a change of 1 TJ of energy. The goal of this task is for students to reinforce the idea of height difference as the meaningful measurement. For students who are stuck on making the measurement, you can guide them with a question like “If the space station is initially at the blue dot, how could the space station move so that the gravitational potential energy doesn't change?” Once they recognize that the equipotentials are circles, they may see that there is a point on the edge where they can measure the value of GPE. A discussion of the variety of measurement methods can happen in the whole class discussion: Students might place the ruler at the base or highest point and visually see where the blue dot falls, or they might place the ruler on the dot (perhaps turning the surface upside down) and visually see where the edge, or the highest/lowest points fall. They may draw on the surface to extend to one or both edges, using a straight line (incorrect) or the circular equipotentials. One note in all of this is that the lowest point on the plastic graph is higher than the base.

Find Patterns

Purpose: Students are being prompted here to draw equipotentials, which in this case appear as quarter circles on the surface. The prompt says “points” which may be confusing to some students. You are welcome to clarify that there may be many points.

Goal: The goal is for students to discover patterns and foreshadow concepts for later discussions. There is a lot to talk about regarding the relationship between the equipotentials and the equation, but keep in mind that the equation will be introduced later. So right now we just want students to identify the qualitative features of the equipotentials. Possible patterns:

• circles/arcs [spherical symmetry - what do these circles have to do with symmetry?]
• horizontal spacing for circle/star/square increasing [foreshadowing inverse law - what would the next point be? (Don't force it if students don't seem ready)]
• vertical spacing for circle/star/square the same [equal differences in potential. This is the convention for equipotentials.].

Describe an Orbit Students may need to be told that elliptical orbits are possible. The goal of this task is for students to explore the physics of elliptical orbits and discover that objects in an elliptical orbit will change speed.

Plot The whole class discussion is an interactive opportunity for students to explore the limitations of the mathematical model and investigate the meaning and conventions around the zero of gravitational potential.

• Why choose $U(r\rightarrow \infty) = 0$? To make the equation simpler - a different choice is just as correct! The conventional choice means that the gravitational potential energy is negative, which has no special physical meaning. Be careful about language - when potential energy “decreases”, it gets more negative (like temperature, not like velocity where “decrease” means “smaller magnitude”).
• When can you use the equation $U=-GMm/r$? This equation describes GPE well for regions outside the surface of Earth, but inside the GPE is a different function of $r$. $U=-GMm/r$ is the gravitational potential energy between two point masses. We can treat Earth as a point mass (but only for regions outside the Earth's surface) because of the spherical symmetry of how the mass is distributed.

Following is a detailed description of how this discussion might be facilitated.

1. Begin by having students arrange themselves so they can all see each others' whiteboards. Give them a full 30 seconds to study the graphs.
2. Students were prompted to attend to where zero is on their graph, and ensure their graph is consistent with where they picked zero at the beginning of the activity. Call on a few representative groups to describe their graphs. Different groups may have chosen different locations as the “zero” for GPE---having groups share what they chose and why can help emphasize the freedom to choose where zero is located. We suggest calling on group that used the conventional zero at $r=\infty$ somewhere in the middle of these presentations, and encouraging the other choices, so students know their choice is equally valid to any other. (e.g.“I think it makes a lot of sense to place zero at the blue dot, since that is where the space station is located”) This will help students to not feel “wrong” when discussing the merits of the conventional choice.
3. Students are not expected to know or discover the equation for universal gravitation. They must be told this information. You can work with the students to determine the $-\frac{1}{r}$ dependence of the graph OR just cut to the chase and display the following equation to the students: “$U_{grav} = -G\frac{Mm}{r}$”. Tell students “Without erasing the graph on your whiteboard, add a graph of this equation.” Prompt the students to return to their groups and work on this.
4. Spend a few minutes observing the groups. If a group set their $U=0$ somewhere other than $r=\infty$ ask, “How could you adjust this equation so it fits your graph better?” (Answer: By adding a constant shift to the equation: $U_{grav} = -G\frac{Mm}{r} +B$.) Some groups may neglect the negative sign, and should be reminded that this flips the graph.
5. Gather the groups together again. Ask a few groups (that may have not shared out yet) what they did to adjust the fit. Goal of discussion: With the addition of a constant shift up or down, the equation fits well for $r>R_{earth}$, and but is still not a good description of the potential for $r<R_{earth}$.
6. Show the students the attached graph. This can be concluded in a variety of ways, depending on student interest:
• Highlight the fact that, although we don't have an equation for this system, the graph of $U$ can be determined from the surface itself.
• Ask the class, “Why is it reasonable to represent the system as a graph with only 2 axes (U and r)?”
• Students can be told what the conventional choice for $U=0$ is and why this convention is chosen.
• Have students identify where the zero of this graph is found on the surface.
• Ask the class, “What does it mean to say that a system has negative potential energy?”
• Compare the two types of questions that you can ask about potential energy: (1) questions where you must know where zero is (“What is U at this point?”) and (2) questions where you don't need to know the offset (“What is the change in potential between these two points? What is the derivative at this point?”)
• Ask students to speculate about why the $1/r$ model does not work inside the earth. (The equation can be found by integrated sources or by integrating the gravitational field found by Gauss' Law.)
• A discussion of energy diagrams follows easily from this activity.
• Initiate a discussion about the rings they drew, and define what an equipotential is.
• This model only considers the space station-earth system. What would happen if we wanted to also consider the sun?
• Despite the choice of where $U=0$ affecting the sign and value of the GPE at any particular point, the idea of “increasing” and “decreasing” still have reasonable meaning.

• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
• assignment Potential vs. Potential Energy

assignment Homework

##### Potential vs. Potential Energy
Static Fields 2022 (5 years)

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• assignment Gravitational Field and Mass

assignment Homework

##### Gravitational Field and Mass
Static Fields 2022 (4 years)

The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: $$\vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases}$$

This problem explores the consequences of the divergence theorem for this shell.

1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: $r<a$, $a<r<b$, and $r>b$.
2. Briefly discuss the physical meaning of the divergence in this particular example.
3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius $Q$, where $a<Q<b$. ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

• assignment Central Force Definition

assignment Homework

##### Central Force Definition
Central Forces 2023 (3 years)

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

1. The force on a test mass $m$ in a gravitational field $\vec{g~}$, i.e. $m\vec g$
2. The force on a test charge $q$ in an electric field $\vec E$, i.e. $q\vec E$
3. The force on a test charge $q$ moving at velocity $\vec{v~}$ in a magnetic field $\vec B$, i.e. $q\vec v \times \vec B$

• assignment Potential energy of gas in gravitational field

assignment Homework

##### Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal.
• assignment Surface temperature of the Earth

assignment Homework

##### Surface temperature of the Earth
Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature $T_{\odot}=5800\text{K}$; and the sun's radius $R_{\odot}=7\times 10^{10}\text{cm}$; and the Earth-Sun distance of $1.5\times 10^{13}\text{cm}$.
• face Central Forces Introduction: Lecture Notes

face Lecture

5 min.

##### Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)
• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• assignment Mass-radius relationship for white dwarfs

assignment Homework

##### Mass-radius relationship for white dwarfs
White dwarf Mass Density Energy Thermal and Statistical Physics 2020

Consider a white dwarf of mass $M$ and radius $R$. The dwarf consists of ionized hydrogen, thus a bunch of free electrons and protons, each of which are fermions. Let the electrons be degenerate but nonrelativistic; the protons are nondegenerate.

1. Show that the order of magnitude of the gravitational self-energy is $-\frac{GM^2}{R}$, where $G$ is the gravitational constant. (If the mass density is constant within the sphere of radius $R$, the exact potential energy is $-\frac53\frac{GM^2}{R}$).

2. Show that the order of magnitude of the kinetic energy of the electrons in the ground state is \begin{align} \frac{\hbar^2N^{\frac53}}{mR^2} \approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2} \end{align} where $m$ is the mass of an electron and $M_H$ is the mas of a proton.

3. Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), $M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}$.

4. If the mass is equal to that of the Sun ($2\times 10^{33}g$), what is the density of the white dwarf?

5. It is believed that pulsars are stars composed of a cold degenerate gas of neutrons (i.e. neutron stars). Show that for a neutron star $M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}$. What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in $\text{km}$.

• assignment Circle Trig Complex

assignment Homework

##### Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2022

Find the rectangular coordinates of the point where the angle $\frac{5\pi}{3}$ meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

Keywords
Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics
Learning Outcomes