group Small Group Activity

10 min.

Angular Momentum in Polar Coordinates
Students learn how to express Angular Momentum as a vector quantity in polar coordinates, and then in Cylindrical and Spherical Coordinates

• Found in: Central Forces course(s)

None

Central Force Definition

(Quick) Purpose: Recognize the definition of a central force. Build experience about which common physical situations represent central forces and which don't.

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

1. The force on a test mass $m$ in a gravitational field $\vec{g~}$, i.e. $m\vec g$
2. The force on a test charge $q$ in an electric field $\vec E$, i.e. $q\vec E$
3. The force on a test charge $q$ moving at velocity $\vec{v~}$ in a magnetic field $\vec B$, i.e. $q\vec v \times \vec B$

• Found in: Central Forces course(s)

None

Solar Sail

The first spacecraft using a solar sail for propulsion was launched in 2010. Its name is IKAROS. It has a square sail with dimensions 14 m x 14 m. Assume that the sail's mass is 2 kg and it reflects 100% of incident photons. When IKAROS is loaded with other equipment, the total mass of the vehicle is 10 kg. The sail is orientated to receive maximum light from the sun.

1. Calculate the momentum of the photons that come from the sun and hit the solar sail in 1 second. Assume a solar intensity of 1300 J/(s.m2).
2. How much momentum will be transferred from solar photons to IKAROS in one day? Give a numerical answer in units of kg.m/s (assume a constant solar intensity).
3. What is the change in the solar sail's velocity in one day? (assume that acceleration is only caused by sunlight).

• Found in: Contemporary Challenges course(s)

group Small Group Activity

30 min.

Applying the equipartition theorem
Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.

• Found in: Contemporary Challenges course(s)

None

Center of Mass for Two Uncoupled Particles

(Straightforward) Purpose: Discover that a system of two masses can be a central force system even when they are not interacting at all. Practice with center-of-mass coordinates.

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

• Found in: Central Forces course(s)

group Small Group Activity

120 min.

Projectile with Linear Drag
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.

• Found in: Theoretical Mechanics course(s)

None

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.

None

Hockey

(Synthesis Problem: Brings together several different concepts from this unit.) Use effective potential diagrams for other than $1/r^2$ forces.

Consider the frictionless motion of a hockey puck of mass $m$ on a perfectly circular bowl-shaped ice rink with radius $a$. The central region of the bowl ($r < 0.8a$) is perfectly flat and the sides of the ice bowl smoothly rise to a height $h$ at $r = a$.

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance $\frac{a}{2}$ from the center of the rink, is hit in such a way that it's initial velocity $\vec v_0$ is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of $\vec v_0$ does the puck just escape the rink?

• Found in: Central Forces course(s)

None

Scattering

Consider a very light particle of mass $\mu$ scattering from a very heavy, stationary particle of mass $M$. The force between the two particles is a repulsive Coulomb force $\frac{k}{r^2}$. The impact parameter $b$ in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass $\mu$ is $\vec v_0$. Answer the following questions about this situation in terms of $k$, $M$, $\mu$, $\vec v_0$, and $b$. (It is not necessarily wise to answer these questions in order.)

1. What is the initial angular momentum of the system?
2. What is the initial total energy of the system?
3. What is the distance of closest approach $r_{\rm{min}}$ with the interaction?
4. Sketch the effective potential.
5. What is the angular momentum at $r_{\rm{min}}$?
6. What is the total energy of the system at $r_{\rm{min}}$?
7. What is the radial component of the velocity at $r_{\rm{min}}$?
8. What is the tangential component of the velocity at $r_{\rm{min}}$?
9. What is the value of the effective potential at $r_{\rm{min}}$?
10. For what values of the initial total energy are there bound orbits?
11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

• Found in: Central Forces course(s)

face Lecture

30 min.

Equipartition theorem
This lecture introduces the equipartition theorem.

• Found in: Contemporary Challenges course(s)

group Small Group Activity

10 min.

Velocity and Acceleration in Polar Coordinates
Use geometry to find formulas for velocity and acceleration in polar coordinates.
• Found in: Central Forces course(s)

None

Centrifuge
A circular cylinder of radius $R$ rotates about the long axis with angular velocity $\omega$. The cylinder contains an ideal gas of atoms of mass $M$ at temperature $T$. Find an expression for the dependence of the concentration $n(r)$ on the radial distance $r$ from the axis, in terms of $n(0)$ on the axis. Take $\mu$ as for an ideal gas.
• Found in: Thermal and Statistical Physics course(s)

accessibility_new Kinesthetic

10 min.

Acting Out Current Density
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear $\vec{I}$, surface $\vec{K}$, and volume $\vec{J}$ current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence, E&M Ring Cycle Sequence sequence(s)

group Small Group Activity

30 min.

A glass of water
Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

• Found in: Energy and Entropy course(s)

assignment_ind Small White Board Question

5 min.

Representations of Vectors
Students each recall a representation of vectors that they have seen before and record it on an individual whiteboard. The instructor uses these responses to generate a whole class discussion that compares and contrasts the features of the representations. If appropriate for the class, the instructor introduces bra/ket notation as a new, but valuable representation.

• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

groups Whole Class Activity

10 min.

Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
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.
• Found in: Central Forces course(s)

group Small Group Activity

30 min.

Wavefunctions on a Quantum Ring
This activity lets students explore translating a wavefunction that isn't obviously made up of eigenstates at first glance into ket and matrix form. Then students explore wave functions, probabilities in a region, expectation values, and what wavefunctions can tell you about measurements of $L_z$.

• Found in: Central Forces course(s)

group Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring

Students work in small groups to use the superposition principle $\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}$ to find an integral expression for the magnetic vector potential, $\vec{A}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{A}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

face Lecture

120 min.

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.