assignment Homework

Circle Trig Complex
Complex Numbers Exponential Form Rectangular Form Polar Form Unit Circle Quantum Fundamentals 2022

Find the rectangular coordinates of the point where the angle \(\frac{5\pi}{3}\) meets the unit circle. If this were a point in the complex plane, what would be the rectangular and exponential forms of the complex number? (See figure.)

assignment Homework

Circle Trigonometry
trigonometry cosine sine math circle Quantum Fundamentals 2022 (2 years)

On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)

assignment Homework

Circle Vector, Version 2
Static Fields 2022 (6 years)

Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

  1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
  2. Make a sketch of the graph \begin{equation} \vert \vec{r} - \vec{a} \vert = 2 \end{equation}

    for each of the following values of \(\vec a\): \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

  3. Derive a more familiar equation equivalent to \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation} for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
  4. Write a brief description of the geometric meaning of the equation \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation}

groups Whole Class Activity

10 min.

Pineapples and Pumpkins
Static Fields 2022 (6 years)

Integration Sequence

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

group Small Group Activity

60 min.

Gravitational Potential Energy

Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

assignment Homework

Using Gradescope (AIMS)
AIMS Maxwell 2021 (2 years)

Task: Draw a right triangle. Put a circle around the right angle, that is, the angle that is \(\frac\pi2\) radians.

Preparing your submission:

  • Complete the assignment using your choice of technology. You may write your answers on paper, write them electronically (for instance using xournal), or typeset them (for instance using LaTeX).
  • If using software, please export to PDF. If writing by hand, please scan your work using the AIMS scanner if possible. You can also use a scanning app; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF; photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.)
  • Please make sure that your file name includes your own name and the number of the assignment, such as "Tevian2.pdf."

Using Gradescope: We will arrange for you to have a Gradescope account, after which you should receive access instructions directly from them. To submit an assignment:

  1. Navigate to https://paradigms.oregonstate.eduhttps://www.gradescope.com and login
  2. Select the appropriate course, such as "AIMS F21". (There will likely be only one course listed.)
  3. Select the assignment called "Sample Assignment"
  4. Follow the instructions to upload your assignment. (The preferred format is PDF.)
  5. You will then be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. (In this assignment, there is only one of each.) You may associate multiple problems with the same page if appropriate.
  6. When you are finished, click "Submit"
  7. After the assignments have been marked, you can log back in to see instructor comments.

group Small Group Activity

30 min.

Charged Sphere

E&M Introductory Physics Electric Potential Electric Field

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.

accessibility_new Kinesthetic

10 min.

Using Arms to Visualize Complex Numbers (MathBits)
Quantum Fundamentals 2022 (2 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.

group Small Group Activity

30 min.

Grey space capsule
Contemporary Challenges 2022 (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.

group Small Group Activity

30 min.

Vector Differential--Curvilinear
Vector Calculus II 2022 (9 years)

vector calculus coordinate systems curvilinear coordinates

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

assignment 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.

accessibility_new Kinesthetic

10 min.

Acting Out Current Density
Static Fields 2022 (6 years)

Steady current current density magnetic field idealization

Ring Cycle Sequence

Integration Sequence

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.

assignment Homework

Ring Function
Central Forces 2023 (3 years) Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
  1. Find \(N\).

  2. Plot this wave function.
  3. Plot the probability density.
  4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
  5. What is the expectation value of \(L_z\) in this state?

assignment Homework

Effective Potentials: Graphical Version
Central Forces 2023 (2 years)

Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).

  1. Give the definition of a central force system and briefly explain why this situation qualifies.
  2. Make a sketch of the graph of the effective potential for this situation.
  3. How should you push the puck to establish a circular orbit? (i.e. Characterize the initial position, direction of push, and strength of the push. You do NOT need to solve any equations.)
  4. BRIEFLY discuss the possible orbit shapes that can arise from this effective potential. Include a discussion of whether the orbits are open or closed, bound or unbound, etc. Make sure that you refer to your sketch of the effective potential in your discussions, mark any points of physical significance on the sketch, and describe the range of parameters relevant to each type of orbit. Include a discussion of the role of the total energy of the orbit.

assignment Homework

Flux through a Paraboloid
Static Fields 2022 (6 years)

Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.

group Small Group Activity

30 min.

Flux through a Cone
Static Fields 2022 (5 years)

Integration Sequence

Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .

assignment Homework

Helix

Integration Sequence

Static Fields 2022 (6 years)

A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the helix?

group Small Group Activity

30 min.

Covariation in Thermal Systems

Thermo Multivariable Functions

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.

assignment Homework

Cone Surface
Static Fields 2022 (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.

group Small Group Activity

120 min.

Equipotential Surfaces

E&M Quadrupole Scalar Fields

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.