## Reduced Mass

• assignment Undo Formulas for Reduced Mass (Geometry)

assignment Homework

##### Undo Formulas for Reduced Mass (Geometry)
Central Forces 2023 (3 years)

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 $m_2>m_1$.

• assignment Undo Formulas for Reduced Mass (Algebra)

assignment Homework

##### Undo Formulas for Reduced Mass (Algebra)
Central Forces 2023 (2 years) 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 switch from a rectangular coordinate system that is unrelated to the system to coordinates adapted to the center-of-mass. After you have solved the equations of motion in the center-of-mass coordinates, you may want to transform back to the original coordinate system. Find the inverse transformation, 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.
• face Systems of Particles Lecture Notes

face Lecture

10 min.

##### Systems of Particles Lecture Notes
Central Forces 2023 (3 years)
• assignment Frequency

assignment Homework

##### Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2022 (2 years) Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ where $c$ is real and positive. Let the initial state of the system be $\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle$, where $\left|{m_1}\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? This frequency is the Bohr frequency.
• 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.
• assignment Effective Potentials: Graphical Version

assignment Homework

##### Effective Potentials: Graphical Version
Central Forces 2023 (2 years)

Consider a mass $\mu$ in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is $\ell\ne 0$ for a given fixed value of $\ell$.

1. Give the definition of a central force system and briefly explain why this situation qualifies.
2. Make a sketch of the graph of the effective potential for this situation.
3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

• 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 Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• group Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics (4 years)

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?

• 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 $\text{km}$.

• Central Forces 2023 (3 years)

Using your favorite graphing package, make a plot of the reduced mass $$\mu=\frac{m_1\, m_2}{m_1+m_2}$$ 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.