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.
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.
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 PAGEWhat 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 Small Group Activity
60 min.
Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics
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 Homework
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:
assignment Homework
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.
assignment Homework
assignment Homework
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: \begin{equation} \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} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
assignment Homework
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.
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 \(-\frac53\frac{GM^2}{R}\)).
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.
Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), \(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).
If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what is the density of the white dwarf?
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 Mathematica Activity
30 min.
groups Whole Class Activity
10 min.
group Small Group Activity
10 min.