## Student handout: 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.

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.

• 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?

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

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?

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?

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.

• 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 2023 (6 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 2023 (5 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.

• face Unit Learning Outcomes: Classical Mechanics Orbits

face Lecture

5 min.

##### Unit Learning Outcomes: Classical Mechanics Orbits
Central Forces 2023 This handout lists Motivating Questions, Key Activities/Problems, Unit Learning Outcomes, and an Equation Sheet for a Unit on Classical Mechanics Orbits. It can be used both to introduce the unit and, even better, for review.
• assignment Central Force Definition

assignment Homework

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

(Quick) Purpose: Recognize the definition of a central force. Build experience about which common physical situations represent central forces and which don't.

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.
• face Central Forces Introduction Lecture Notes

face Lecture

5 min.

##### Central Forces Introduction Lecture Notes
Central Forces 2022
• 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 $1.5\times 10^{13}\text{cm}$.
• 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 Sun vs. Jupiter

assignment Homework

##### Sun vs. Jupiter
Central Forces 2023 (3 years)

(Numbers and Units) Purpose: Gain experience with the relative sizes of objects and distances in the Solar System. Gain experience with realistic reduced masses.

Calculate the following quantities:

1. Find ${\vec r}_{\rm sun}-{\vec r}_{\rm cm}$ and $\mu$ for the Sun-Earth system. Compare ${\vec r}_{\rm sun}-{\vec r}_{\rm cm}$ to the radius of the Sun and to the distance from the Sun to the Earth. Compare $\mu$ to the mass of the Sun and the mass of the Earth.
2. Repeat the calculation for the Sun-Jupiter system.

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