Search: rotation

Kinesthetic

30 min.

Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

Quantum Fundamentals 2021

arms complex numbers phase rotation reflection math

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.

Homework

Undo Formulas for Center of Mass (Geometry)

Central Forces 2023 (3 years)

(Sketch limiting cases) Purpose: For two central force systems that share the same reduced mass system, discover how the motions of the original systems are the same and different.

The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).

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 \(m_1=m_2\).
Repeat, for \(m_2>m_1\).

Kinesthetic

10 min.

Using Arms to Represent Overall and Relative Phase in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years)

quantum states complex numbers arms Bloch sphere relative phase overall phase

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).

Homework

Centrifuge

Centrifugal potential Thermal and Statistical Physics 2020 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.

Small Group Activity

30 min.

Applying the equipartition theorem

Contemporary Challenges 2021 (4 years)

equipartition theorem

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature \(T\).

Small Group Activity

30 min.

Heat capacity of N₂

Contemporary Challenges 2021 (4 years)

equipartition quantum energy levels

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.”

Small Group Activity

10 min.

Sequential Stern-Gerlach Experiments

Quantum Fundamentals 2023 (3 years)

Small Group Activity

30 min.

Grey space capsule

Contemporary Challenges 2021 (4 years)

blackbody Stefan-Boltzmann Law

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

Kinesthetic

10 min.

Using Arms to Represent Time Dependence in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years)

Arms Representation quantum states time dependence Spin 1/2

Arms Sequence for Complex Numbers and Quantum States

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

Homework

Sun vs. Jupiter

Central Forces 2023 (3 years)

(Numbers and Units) Purpose: Gain experience with the relative sizes of objects and distances in the Solar System. Gain experience with realistic reduced masses.

Calculate the following quantities:

Find \({\vec r}_{\rm sun}-{\vec r}_{\rm cm}\) and \(\mu\) for the Sun-Earth system. Compare \({\vec r}_{\rm sun}-{\vec r}_{\rm cm}\) to the radius of the Sun and to the distance from the Sun to the Earth. Compare \(\mu\) to the mass of the Sun and the mass of the Earth.
Repeat the calculation for the Sun-Jupiter system.

Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)

Lie Groups and Lie Algebras 23 (4 years)

arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate math

Arms Sequence for Complex Numbers and Quantum States

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.

Small Group Activity

30 min.

Conic Sections

Central Forces 2023 (3 years) Students are asked to explore the parameters that affect orbit shape using the supplied Maple worksheet or Mathematica notebook.

Small Group Activity

10 min.

Using Tinker Toys to Represent Spin 1/2 Quantum Systems

spin 1/2 eigenstates quantum states

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

Homework

Visualization of Wave Functions on a Ring

Central Forces 2023 (3 years) Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)

Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

Computational Activity

120 min.

Electrostatic potential of four point charges

Computational Physics Lab II 2023 (2 years)

electrostatic potential python

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.

Small Group Activity

30 min.

The Hillside

Vector Calculus I 2022 (2 years)

Gradient Sequence

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

Small Group Activity

120 min.

Box Sliding Down Frictionless Wedge

Theoretical Mechanics (4 years)

Lagrangian Mechanics Generalized Coordinates Special Cases

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.

Mathematica Activity

30 min.

Visualization of Quantum Probabilities for a Particle Confined to a Ring

Central Forces 2023 (3 years)

central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

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: \(\vec{F}=q\vec{E}\)

Small Group Activity

60 min.