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.
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:
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 Small Group Activity
30 min.
Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics
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 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
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.
face Lecture
5 min.
assignment Homework
(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.
assignment Homework
assignment Homework
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 \(\text{km}\).
assignment Homework
(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: