## Tetrahedron

• assignment Divergence through a Prism

assignment Homework

##### Divergence through a Prism
Static Fields 2023 (6 years)

Consider the vector field $\vec F=(x+2)\hat{x} +(z+2)\hat{z}$.

1. Calculate the divergence of $\vec F$.
2. In which direction does the vector field $\vec F$ point on the plane $z=x$? What is the value of $\vec F\cdot \hat n$ on this plane where $\hat n$ is the unit normal to the plane?
3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

• assignment Cross Triangle

assignment Homework

##### Cross Triangle
Static Fields 2023 (6 years)

Use the cross product to find the components of the unit vector $\mathbf{\boldsymbol{\hat n}}$ perpendicular to the plane shown in the figure below, i.e.  the plane joining the points $\{(1,0,0),(0,1,0),(0,0,1)\}$.

assignment Homework

Consider the fields at a point $\vec{r}$ due to a point charge located at $\vec{r}'$.

1. Write down an expression for the electrostatic potential $V(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
2. Write down an expression for the electric field $\vec{E}(\vec{r})$ at a point $\vec{r}$ due to a point charge located at $\vec{r}'$. (There is nothing to calculate here.)
3. Working in rectangular coordinates, compute the gradient of $V$.
4. Write several sentences comparing your answers to the last two questions.

• groups Air Hockey

groups Whole Class Activity

10 min.

##### Air Hockey
Central Forces 2023 (3 years)

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
• assignment Lines in Polar Coordinates

assignment Homework

##### Lines in Polar Coordinates
Central Forces 2023 (3 years)

The general equation for a straight line in polar coordinates is given by: $$r(\phi)=\frac{r_0}{\cos(\phi-\delta)}$$ where $r_0$ and $\delta$ are constant parameters. Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.

1. $y=3$
2. $x=3$
3. $y=-3x+2$

• group Projectile with Linear Drag

group Small Group Activity

120 min.

##### Projectile with Linear Drag
Theoretical Mechanics (4 years)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
• assignment Building the PDM: Instructions

assignment Homework

##### Building the PDM: Instructions
PDM Energy and Entropy 2021 (2 years) In your kits for the Portable Partial Derivative Machine should be the following:
• A 1ft by 1ft board with 5 holes and measuring tapes (the measuring tapes will be on the top side)
• 2 S-hooks
• A spring with 3 strings attached
• 2 small cloth bags
• 4 large ball bearings
• 8 small ball bearings
• 2 vertical clamp pulleys
• A ziploc bag containing
• 5 screws
• 5 hex nuts
• 5 washers
• 5 wing nuts
• 2 horizontal pulleys
To assemble the Portable PDM, start by placing the PDM on a table surface with the measuring tapes perpendicular to the table's edge and the board edge with 3 holes closest to you.
1. one screw should be put through each hole so that the threads stick out through the top side of the board. Next use a hex nut to secure each screw in place. It is not critical that they be screwed on any more than you can comfortably manage by hand.
2. After securing all 5 screws in place with a hex nut, put a washer on each screw.
3. Slide a horizontal pulley onto screws 1 and 2 (as labeled above).
4. On all 5 screws, add a wing nut to secure the other pieces. Again, it does not need to be tightened all the way as long as it is secure enough that nothing will fall off.
5. Using the middle wingnut/washer/screw (Screw 4), clamp the shortest of the strings tied to the spring.
6. Loop the remaining 2 looped-ends of string around the horizontal pulleys and along the measuring tape.
7. Using the string as a guide, clamp the vertical pulleys into place on the edge of the board.
8. Through the looped-end of each string, place 1 S-hook.
9. Put the other end of each s-hook through the hole in the small cloth bag.
Here is a poor photo of the final result, which doesn't show the two vertical pulleys. If you would like, you could view a video of the building process.
• assignment Rubber Sheet

assignment Homework

##### Rubber Sheet
Energy and Entropy 2021 (2 years)

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call $y$, and the distance of horizontal stretch we will call $x$.

If I pull the bottom down by a small distance $\Delta y$, with no horizontal force, what is the resulting change in width $\Delta x$? Express your answer in terms of partial derivatives of the potential energy $U(x,y)$.

• 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.

• computer Visualizing Flux through a Cube

computer Computer Simulation

30 min.

##### Visualizing Flux through a Cube
Static Fields 2023 (6 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.
• Static Fields 2023 (6 years)

Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play with it to see how this works!)