## Activity: 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.
What students learn
• Recall intuitions about gravitational force and potential energy.
• The magnitude of the gravitational force is given by the derivative of (the change in) the gravitational potential energy.
• The direction of the gravitational force is opposite the sign of the slope of the potential energy.
• Media

Your group has a plastic surface and a contour map that represent the gravitational potential energy of a space station-Earth system as a function of the position of the space station relative to Earth. Solve the following problems together and discuss the results.

You may need to remind students that the gravitational potential energy is zero infinitely far away from Earth.

Compare Potentials: Rank the three points marked on the surface by gravitational potential energy from highest to lowest.

Answer: This answer is relatively intuitive using the surface, especially if students have recently completed the Gravitational Potential Energy activity. The higher a point, the higher the gravitational potential energy.

Identify Forces: What direction is the gravitational force at each of the marked points? Indicate the direction of each force with a vector on the contour map.

Hint: Students can rely on their intuition that the gravitational force points toward Earth.

Potential Pitfall: It is common for students to think that the force vectors are tangent to the surface. In fact, all three points exist in a horizontal plane, and their force vectors also only have horizontal components.

Extension: If you were to draw the vectors on the surface, how would those vectors be the same or different from the vectors on the contour map?

Rank the three points by the magnitude of the gravitational force.

As with the previous questions, this question is relatively intuitive.

Plot: Sketch a graph of the gravitational force vs. distance from the center of the Earth. Use the convention that positive forces point away from the center of the Earth.

Goal: Students to visualize the function $F(r)$.

Note: If students make the force positive, they are plotting the magnitude of the force. The component of the force is negative since it points opposite to increasing r.

Examine Changes: At each point, imagine that the space station moves a small distance (about 10 m) directly toward the center of the Earth.

1. At each point, is the resulting change in gravitational potential energy positive, negative, or zero?
2. Rank the three points by the magnitude of the change in gravitational potential energy.

This question can be less intuitive than those above. Pay close attention to students' answers, as the conclusion in this question (that the change is larger closer to the Earth) is used in the conclusion of the activity.

Relate the Surfaces: The yellow surface represents the gravitational potential energy of an object close to the surface of Earth. How it is possible for both surfaces to correctly represent the gravitational potential energy when the object is near the Earth's surface?

This question is intended to get students to recognize that the force near the surface of the Earth, which is uniform, is simply a linear approximation to the actual force, which varies with distance. This idea can be especially difficult for students, but is a valuable connection to systems that students have already studied.

Generate Graphs: Draw graphs of gravitational potential energy vs. distance and gravitational force vs. distance for the yellow surface.

Find Patterns: Examine the graphs of gravitational potential energy and force for both the green and yellow surfaces.

What patterns do you see between gravitational force and gravitational potential energy?

The big takeaway of the activity is that gravitational force is related to the change (derivative) in the potential energy, not to the value of the potential energy. This can be an especially persistent idea because the graphs of $U$ and $F$ for universal gravitation are hard to distinguish for the green surface. The difference is much more clear for the yellow surface.

SUMMARY PAGE
What Students Learn:
• Recall intuitions about gravitational force and potential energy.
• The magnitude of the gravitational force is given by the derivative of (the change in) the gravitational potential energy.
• The direction of the gravitational force is opposite the sign of the slope of the potential energy.

Time Estimate: 20 minutes

Equipment

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

Introduction

• Students should be familiar with gravitational force and potential energy near the surface of the Earth.
• It can be helpful for students to be familiar with equations and graphs of universal gravitation, but it is not essential.
• Optional: We find it useful for the instructor to explain the context at the start of the activity.
• The surface of the Earth is marked by an indent in the surface near the lowest corner. The height of the surface corresponds to the value of gravitational potential energy.

Whole Class Discussion / Wrap Up:

• This activity is especially effective when given following the Gravitational Potential Energy activity.

• If students have done both activities, they will have graphs of both $U(r)$ and $F(r)$ (if not, you might encourage students to sketch $U(r)$ as well). Ask students to compare the two graphs directly, and especially to think about how the graphs can help them identify patterns. Many students will think that because the graphs look nearly identical, $U$ and $F$ must be directly proportional.

• The meaning of the negative sign for $U$ and $F$ is especially challenging. Asking students to describe the meaning of the negative sign in each case (it indicates the value of $U$ but the direction of $F$) and calling attention to the difference can help them develop more sophisticated reasoning about negative signs.

• group Gravitational Potential Energy

group Small Group Activity

60 min.

##### 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.
• 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 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. $q\vec v \times \vec B$

• face Central Forces Introduction: Lecture Notes

face Lecture

5 min.

##### Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)
• 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 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 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 $\text{km}$.

• computer Effective Potentials

computer Mathematica Activity

30 min.

##### Effective Potentials
Central Forces 2023 (3 years) Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
• groups Air Hockey

groups Whole Class Activity

10 min.

##### Air Hockey
Central Forces 2023 (3 years)

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
• group Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass

group Small Group Activity

10 min.

##### Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Central Forces 2023 (3 years) A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.

Keywords
Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics
Learning Outcomes