group Small Group Activity

30 min.

##### $|\pm\rangle$ Forms an Orthonormal Basis
Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.

• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

face Lecture

30 min.

##### A coarse-grained model for transportation
A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

• Found in: Contemporary Challenges course(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)

group Small Group Activity

30 min.

##### Acceleration
• Found in: Vector Calculus II course(s)

accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and we demonstrate the answers with coins under a doc cam.

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

5 min.

##### Acting Out Flux
Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

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

accessibility_new Kinesthetic

10 min.

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

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

groups Whole Class Activity

10 min.

##### Air Hockey
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.

• Found in: Central Forces course(s)

face Lecture

10 min.

##### Angular Momentum Commutation Relations: Lecture

• Found in: Central Forces course(s)

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)

assignment_ind Small White Board Question

5 min.

##### Angular Momentum SWBQ
• Found in: Central Forces 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)

face Lecture

10 min.

##### Basics of heat engines
This brief lecture covers the basics of heat engines.
• Found in: Contemporary Challenges course(s)

group Small Group Activity

30 min.

##### Black space capsule
In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

• Found in: Contemporary Challenges course(s)

computer Computer Simulation

30 min.

##### Blackbody PhET
Students use a PhET to explore properties of the Planck distribution.

• Found in: Contemporary Challenges course(s)

face Lecture

120 min.

##### Boltzmann probabilities and Helmholtz
These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.

• Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

120 min.

##### Box Sliding Down Frictionless Wedge
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.

• Found in: Theoretical Mechanics course(s)

group Small Group Activity

5 min.

##### Calculating a Total Differential
Students are placed into small groups and asked to calculate the total differential of a function of two variables, each of which is in turn expressed in terms of two other variables.

group Small Group Activity

30 min.

##### Calculating Coefficients for a Power Series

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the power series coefficients.

$c_n={1\over n!}\, f^{(n)}(z_0)$

Students use this formula to compute the power series coefficients for a $\sin\theta$ (around both the origin and (if time allows) $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up and in the follow-up activity: Visualization of Power Series Approximations.

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

face Lecture

5 min.

##### Central Forces Introduction Lecture Notes
• Found in: Central Forces course(s)

group Small Group Activity

30 min.

##### Chain Rule
This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.
• Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

##### Chain Rule Measurement
This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the $r$-direction and derivatives in $x$- and $y$-directions using the chain rule.
• Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

##### Change of Variables
• Found in: Vector Calculus II course(s)

group Small Group Activity

30 min.

##### Changes in Internal Energy
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

• Found in: Warm-Up sequence(s)

group Small Group Activity

10 min.

##### Changing Spin Bases with a Completeness Relation
Students work in small groups to use completeness relations to change the basis of quantum states.

• Found in: Quantum Fundamentals course(s) Found in: Completeness Relations sequence(s)

group Small Group Activity

30 min.

##### Charged Sphere
Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.

• Found in: Thermal and Statistical Physics course(s)

face Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.

• Found in: Completeness Relations sequence(s)

group Small Group Activity

30 min.

##### Conic Sections
Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.
• Found in: Central Forces course(s)

group Small Group Activity

5 min.

##### Constant Lines in the $u$, $v$-Plane

Students are asked to draw lines of constant $u$ and $v$ in a $u,v$ coordinate system. Then, in the same coordinate system, students must draw lines of constant $x$ and constant $y$ when

$x(u,v)=u$ and $y(u,v)=\frac{1}{2}u+3v.$

group Small Group Activity

30 min.

##### Covariation in Thermal Systems
Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

group Small Group Activity

10 min.

##### Cross Product
This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.

• Found in: Vector Calculus I course(s)

accessibility_new Kinesthetic

10 min.

##### Curvilinear Basis Vectors
Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).

• Found in: Static Fields, Central Forces, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Curvilinear Coordinate Sequence sequence(s)

groups Whole Class Activity

10 min.

##### Curvilinear Coordinates Introduction
First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles $\theta$ and $\phi$. Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.

• Found in: Static Fields, Central Forces, AIMS Maxwell, Vector Calculus I, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Curvilinear Coordinate Sequence sequence(s)

group Small Group Activity

30 min.

##### Curvilinear Volume Elements
Students construct the volume element in cylindrical and spherical coordinates.
• Found in: Vector Calculus I course(s)

group Small Group Activity

30 min.

##### de Broglie wavelength after freefall
In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.

• Found in: Contemporary Challenges course(s)

group Small Group Activity

30 min.

##### DELETE Navigating a Hill
In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.
• Found in: Static Fields, AIMS Maxwell course(s)

assignment_ind Small White Board Question

10 min.

##### Derivatives SWBQ
• Found in: Surfaces/Bridge Workshop course(s)

face Lecture

30 min.

##### Determining $|\pm_x\rangle$ and $|\pm_y\rangle$ in the $S_z$ basis

• Found in: Quantum Fundamentals course(s)

face Lecture

30 min.

##### Differentials
• Found in: Static Fields, Surfaces/Bridge Workshop course(s)

group Small Group Activity

30 min.

##### Directional Derivatives
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

• Found in: Vector Calculus I course(s) Found in: Gradient Sequence sequence(s)

group Small Group Activity

30 min.

##### Divergence and Curl
• Found in: Vector Calculus II course(s)

assignment_ind Small White Board Question

10 min.

##### Dot Product Review
This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.

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

group Small Group Activity

30 min.

##### Earthquake waves
In this activity students use the known speed of earthquake waves to estimate the Young's modulus of the Earth's crust.

• Found in: Contemporary Challenges course(s)

computer Mathematica Activity

30 min.

##### Effective Potentials
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.

• Found in: Central Forces course(s)

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge

Students work in small groups to use Coulomb's Law $\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the electric field, $\vec{E}(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $\vec{E}(\vec{r})$ 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)

keyboard Computational Activity

120 min.

##### Electric field for a waffle cone of charge
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.

• Found in: Computational Physics Lab II course(s)

group Small Group Activity

30 min.

##### Electric Field of Two Charged Plates
• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
1. objects with like charge repel and opposite charge attract,
2. object tend to move toward lower energy configurations
3. The potential energy of a charged particle is related to its charge: $U=qV$
4. The force on a charged particle is related to its charge: $\vec{F}=q\vec{E}$

group Small Group Activity

30 min.

##### Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.