Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

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 \(\text{km}\).

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.

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.

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.

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.

Determine the total mass of each of the slabs below.

1. (2pts) A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation*} \rho=A\pi\sin\left[\tfrac{\pi z}h\right]. \end{equation*}
2. (2pts) A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation*} \rho = 2A \Big[\Theta(z)-\Theta(z-h) \Big] \end{equation*}
3. (2pts) An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
4. (2pts) An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
5. (2pts) What are the dimensions of \(A\)?
6. (2pts) Write several sentences comparing your answers to the different cases above.

(Messy algebra) Convince yourself that the expressions for kinetic energy in original and center of mass coordinates are equivalent. The same for angular momentum.

Consider a system of two particles of mass \(m_1\) and \(m_2\).

1. Show that the total kinetic energy of the system is the same as that of two “fictitious” particles: one of mass \(M=m_1+m_2\) moving with the velocity of the center of mass and one of mass \(\mu\) (the reduced mass) moving with the velocity of the relative position.
2. Show that the total angular momentum of the system can similarly be decomposed into the angular momenta of these two fictitious particles.

The center-of-mass motion is determined by the net external force, even when the particles are not interacting. Practice with center-of-mass coordinates.

Consider two particles of equal mass \(m\). The forces on the particles are \(\vec F_1=0\) and \(\vec F_2=F_0\hat{x}\) (for this problem, ignore gravitational forces between the two particles). If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways,

1. solve for the motion of each of the particles, separately, then see what happens to the center of mass
2. solve directly for the center of mass motion
3. Write a short description comparing the two solutions.

Using your favorite graphing package, make a plot of the reduced mass \begin{equation} \mu=\frac{m_1\, m_2}{m_1+m_2} \end{equation} as a function of \(m_1\) and \(m_2\). What about the shape of this graph tells you something about the physical world that you would like to remember? You should be able to find at least three things. Hint: Think limiting cases.

(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.

The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).

1. Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
2. Repeat, for \(m_2>m_1\).

For systems of particles, we used the formulas \begin{align} \vec{R}_{cm}&=\frac{1}{M}\left(m_1\vec{r}_1+m_2\vec{r}_2\right) \nonumber\\ \vec{r}&=\vec{r}_2-\vec{r}_1 \label{cm} \end{align} to describe the system of two objects in terms of the center of mass and relative position instead of the positions of each object. After solving for the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system to examine the motion of each object.

Find the positions of the two objects in terms of the position of the center of mass and the relative position, i.e. solve for: \begin{align} \vec{r}_1&=\\ \vec{r}_2&= \end{align}

Hint: The system of equations (\ref{cm}) is linear, i.e. each variable is to the first power, even though the variables are vectors. In this case, you can use all of the methods you learned for solving systems of equations while keeping the variables vector valued, i.e. you can safely ignore the fact that the \(\vec{r}\)s are vectors while you are doing the algebra as long as you don't divide by a vector.

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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.