## Undo Formulas for Reduced Mass (Algebra)

• 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 $m_2>m_1$.

• face Systems of Particles Lecture Notes

face Lecture

10 min.

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

assignment Homework

##### Reduced Mass
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.

• group Box Sliding Down Frictionless Wedge

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
Theoretical Mechanics (4 years)

Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance $d$ down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
• group Flux through a Cone

group Small Group Activity

30 min.

##### Flux through a Cone
Static Fields 2021 (4 years)

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2023 (4 years)

Integration Sequence

Students use known algebraic expressions for vector line elements $d\vec{r}$ to determine all simple vector area $d\vec{A}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

• assignment Cone Surface

assignment Homework

##### Cone Surface
Static Fields 2023 (6 years)

• Find $dA$ on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
• Using integration, find the surface area of an (open) cone with height $H$ and radius $R$. Do this problem in both cylindrical and spherical coordinates.

• assignment Flux through a Plane

assignment Homework

##### Flux through a Plane
Static Fields 2023 (4 years) Find the upward pointing flux of the vector field $\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}$ through the rectangle $R$ with one edge along the $y$ axis and the other in the $xz$-plane along the line $z=x$, with $0\le y\le2$ and $0\le x\le3$.
• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2021 (4 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• assignment Center of Mass for Two Uncoupled Particles

assignment Homework

##### Center of Mass for Two Uncoupled Particles
Central Forces 2023 (3 years)

Consider two particles of equal mass $m$. The forces on the particles are $\vec F_1=0$ and $\vec F_2=F_0\hat{x}$. 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, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

• 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.