## Activity: 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.
What students learn
• Lagrangian mechanics will work with non-orthonormal coordinate systems
• To use the dot product to find speed squared for the kinetic energy
• To make sense of their answer by checking dimensions, functional behavior, and special cases
• Media

Block Sliding Down a Frictionless Wedge

(Taylor Example 7.5) Consider a block with mass $m$ sliding frictionlessly down an wedge with mass $M$ that makes an angle $\alpha$. The wedge itself slides frictionlessly across a horizontal floor near the surface of Earth. The block is released from the top of the wedge, with both objects initially at rest.

If length of the sloping face is $d$, how long does the block take to reach the bottom? ## Structure of the Activity

This is a small group activity where everyone is solving the same problem. The problem is hard, so I break the problem up for the students as we go along. Students are asked to share their reasoning with the whole class.

1. Describe the problem situation, emphasizing the choice of coordinates. Distribute the handout. Tell students they are going to first solve for the acceleration in order to then calculate the time.

2. Ask students to work in groups for 5-10 min. to anticipate what the accelerations will look like:

• Dimensions
• Functional Behavior
• Special Cases (demonstrate one, like the case where the wedge is flat, i.e., $\alpha = 0$).
They will use these anticipations to make sense of their answers.

3. Bring the whole class together and make a list of the anticipations. Ask students to justify their anticipations.

4. Ask students to work in groups to find the accelerations. They will get stuck writing down the kinetic energy. Let them think about it for a little bit but bring the whole class together to talk it through.

5. Once you've written the Lagrangian down as a class, ask students to work in groups to solve for the accelerations.

6. Solve for the accelerations as a whole class and then talk thought comparing the anticipations with the calculations.

## Student Conversations

• Using a Non-Orthogonal Coordinate System Some students believe that coordinate systems have to be orthogonal. The question “How can you use a coordinate system like this?” can be responded to with, “Why would you not be able to?” Let them think for a minute, then mention: you do have to be careful with dot products - they're not zero.

Some students are also bothered with coordinate systems where the zeroes don't line up.

Some students are further still bothered by the fact that the zero of $q_1$ moves in space because it is attached to the top of the wedge. Relatedly, some students are bothered that the speed of the wedge is independent of $q_1$. Some students might notice that this is a non-inertial coordinate system. If you've discussed relativity already, it can help to talk about different reference frames.

It is helpful to tell students that it's ok for this to feel uncomfortable because its unfamiliar - they haven't really seen anything like this! But this coordinate system is perfectly reasonable. Examples they've previously worked with: they've rotated a Cartesian coordinate system to be parallel and perpendicular to an incline. For problems with rotational and translational motion, they might use Cartesian to describe the position of the center of mass and polar to describe the rotational motion about the center of mass - two coordinate systems with different origins for the same problem.

• Labeling Different Speeds as Different: The box and the wedge will have different speeds. They should be labeled differently.

• Special Cases: Students have a really hard time anticipating the special cases and then interpreting the cases once they get the answer, particularly the case where $m>>M$. Plan on spending a good amount of class time on this discussion.
• 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 Find Force Law: Logarithmic Spiral Orbit

assignment Homework

##### Find Force Law: Logarithmic Spiral Orbit
Central Forces 2023 (3 years)

In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a $1/r^2$ force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

Find the force law for a mass $\mu$, under the influence of a central-force field, that moves in a logarithmic spiral orbit given by $r = ke^{\alpha \phi}$, where $k$ and $\alpha$ are constants.

• assignment Extensive Internal Energy

assignment Homework

##### Extensive Internal Energy
Energy and Entropy 2021 (2 years)

Consider a system which has an internal energy $U$ defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?

• assignment Yukawa

assignment Homework

##### Yukawa
Central Forces 2023 (3 years)

In a solid, a free electron doesn't see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be screened” from far away. A common model for such problems is described by the Yukawa or screened potential: \begin{equation} U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}} \end{equation}

1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter $\alpha$.
2. Draw the effective potential for the two choices $\alpha=10$ and $\alpha=0.1$ with $k=1$ and $\ell=1$. For which value(s) of $\alpha$ is there the possibility of stable circular orbits?

• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
Static Fields 2022 (3 years) Consider the magnetic field $\vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases}$
1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
2. Find a formula for the current density that creates this magnetic field.
3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (5 years)

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

• 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.
• assignment Cube Charge

assignment Homework

##### Cube Charge
charge density

Integration Sequence

Static Fields 2022 (5 years)
1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
2. In a new physical situation: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube.
• assignment Current from a Spinning Cylinder

assignment Homework

##### Current from a Spinning Cylinder
A solid cylinder with radius $R$ and height $H$ has its base on the $x,y$-plane and is symmetric around the $z$-axis. There is a fixed volume charge density on the cylinder $\rho=\alpha z$. If the cylinder is spinning with period $T$:
1. Find the volume current density.
2. Find the total current.
• assignment Current in a Wire

assignment Homework

##### Current in a Wire
Static Fields 2022 (3 years) The current density in a cylindrical wire of radius $R$ is given by $\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}$. Find the total current in the wire.

Learning Outcomes