Midterm 1 (Due 27 Feb)

Midterm 1 (Due 27 Feb)#

Spring 2026

Practicalities about Midterms

  1. You can work in up to a group of 3. If you work as a group, you can hand in one answer only if you wish. Remember to write your name(s)!

  2. How do I(we) hand in? You can hand in the paper and pencil exercises as a single scanned PDF document. Your jupyter notebook file should be converted to a PDF file, attached to the same PDF file as for the pencil and paper exercises. All files should be uploaded to Gradescope.

  3. Cite any resources you use for your midterm, and you can, of course, meet with any instructors or ask any questions in class about the midterm. We will not work any of the exercises on the midterm in help sessions as we have with homework exercises.

  4. Midterms are meant to be challenging and might cover topics and areas that we have not covered directly in class. They might also ask you to complete your own research or design your own model; there are plenty of resources available on the course website and you are welcome to ask any instructors for help or guidance.

Instructions for submitting to Gradescope.

Exercise 0 is marked Submit on D2L only, you must do submit this separately to D2L, not Gradescope. The work on Exercise 0 will be graded pass/fail and may be completed with a different group than your homework group.

You are submitting to D2L, so that Danny can give you feedback on your work in a timely manner to help with questions on your project and guide its development.

Reminder of AI Policy for our class

We can use AI for brainstorming, help, and editing.

  • AI cannot be used for direct answers or completion of assignments.

  • We expect documentation of AI use, but it can be informal.

  • Violations are discussed with Danny; the first violation requires a redo of the assignment, and repeated violations result in a failing grade.

What is informal documentation?

A summary of the use of AI and the kinds of questions asked, and what the AI replied with. Detailed prompts and replies are not needed but the extent to which AI was used to create products in the midterm is important to document. Of course, this is a minimal standard. You can include more detailed documentation such as screenshots of your interactions. The benefit of including screenshots, especially for project work, is that instructors can help you use AI tools more effectively.

Midterms in this class are also a learning opportunity. There are different ways to arrive at the solutions and you might have to learn something new to develop that solution. Using your resources and asking for help from instructors is very much encouraged, especially early on for the more open-ended exercises.

import numpy as np
from math import *
import matplotlib.pyplot as plt
import pandas as pd
%matplotlib inline
plt.style.use('seaborn-v0_8-colorblind')

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 4
      2 from math import *
      3 import matplotlib.pyplot as plt
----> 4 import pandas as pd
      5 get_ipython().run_line_magic('matplotlib', 'inline')
      6 plt.style.use('seaborn-v0_8-colorblind')

ModuleNotFoundError: No module named 'pandas'

Final Project Exercise (Submit on D2L only)#

Exercise 0, Planning your project (10 points)#

SUBMIT THIS ON D2L only - One submission per group

Your final project will be a computational essay of your own design. My colleague, Tor Ole Odden, and I have borrowed this idea from a proposal by Stephen Wolfram. In his original post, Wolfram talks about the importance of the computational medium as a way of communicating science.

Tor, myself, and others have started building this idea into a theory of computational learning in physics, using computational essays to argue for the importance of computing in physics, building the theory out, and trying to identify the ways making course work and materials to promote agency, creativity, and ownership.

Important Notes:

  • You can work with a different partner/group for this exercise than your homework group

  • Only one submission per group is required

  • This planning exercise will help guide your final project development

Understanding Computational Essays (4 points)#

To get started, you should read/skim the following articles; they are not very long:

  1. Wolfram’s What is a Computational Essay?

  2. Tor and my short paper: Computational Essays: An Avenue for Scientific Creativity in Physics

  3. Wolfram’s Steps to Writing a Computational Essay

If you are short on time, read Wolfram’s original post and the guidelines, and skim/skip the paper by Tor and myself.

  • 1a. (2 pts) Write a summary of your readings. What did you learn? What was important? What did you find interesting? What questions do you still have? Full credit will be given for a summary that is at least 200 words long.

Computational essays are a new way to communicate your science. It might be a good idea to look at some examples. Review the University of Oslo’s Computational Essay Showroom.

  • 1b. (2 pts) Find at least one computational essay in the showroom that you find interesting. Write a summary of the computational essay and evaluate it based on your readings. What did you like? What did you not like? How well does it follow the guidelines for a good computational essay? What are its strengths and weaknesses? Full credit will be given for a summary that is at least 200 words long.

Planning Your Computational Essay (6 points)#

We hear from students that often the hardest part of developing a computational essay is finding a question worth investigating. So we are trying to help you with that. You are not expected to write the computational essay for this midterm. Instead, you are expected to plan it.

Your answers to this question do not commit you to completing this project on the exact idea you present here. You can change your mind later. This is a good exercise to get you thinking about what you might want to do.

For each of the questions below, you should write a paragraph or two (minimum 100 words, maximum 200 words per question) to answer the question. You may include equations, code, or figures in any of your answers.

  • 1c. (1 pt) Who is your team and how are you going to communicate regularly? What is the question(s) you want to investigate? What background information do you need to understand the question(s)? Where can you find that information?

  • 1d. (1 pt) What is the motivation for the question(s)? Why is it interesting to you? Why is it interesting to others?

  • 1e. (1.5 pts) What is the computational aspect of the question(s)? What will you calculate? What will you simulate? What will you visualize? Include at least one plot, figure, or image that relates to your topic.

  • 1f. (1.5 pts) Outline the structure of the setup and analysis that you think you need in order to answer the question(s) you have. What are the key steps you would need to take? What are the key challenges you anticipate?

  • 1g. (1 pt) How will you know if you have answered the question(s)? What are the key results you would be looking for?

Alternative Format: If you prefer, you can write up a single document that answers parts (c) through (g) in a style you prefer. That document should be roughly 400-800 words long and may include equations, code, or figures.

Submit your answers on D2L through the assignment titled “Midterm 1 Exercise 0”.

Pencil and Paper Exercises (Submit on Gradescope only)#

Exercise 1, Spin dependent forces (20 points)#

We’ve modeled objects falling, bouncing, and moving under different forms of fluid resistance. One key feature we have failed to account for are the forces that result from the lift force that occurs for objects spinning in the presence of fluid flow.

The Magnus Effect#

This is the Magnus Effect and it is a result of a broken symmetry in the fluid flow due to the spinning of the object. Fluid in the direction of the spin moves differently than the fluid that moves opposite the direction of flow. The figures below illustrate the effect:

Magnus Effect illustration showing the slower fluid below the spinning cylinder

ABOVE: A back-spinning cylinder (\(\mathbf{L}\) into of the page, fluid flow left to right) produces a Von Kármán vortex street that is directed downward. This is a signature of an upward lift.

Magnus Effect illustration showing the slower fluid below the spinning cylinder

ABOVE: A top-spinning cylinder (\(\mathbf{L}\) into the page, fluid flow right to left) will drive the fluid upward, the fluid on the bottom is accelerated in the direction of flow as it moves with the rotation. The causes a reaction force on the cylinder downward.

The Magnus Effect in Sports#

The Magnus Effect is important in nearly all sports where a ball is used. It produces trajectories that can curve in complex ways. This phenomena can be used to expert effect in tennis, volleyball, soccer, baseball, table tennis, golf, pickleball, jai lai, and so on.

Once place where the role of fluids is more complicated is bowling; the interaction with the spinning ball and the surface of the alley is a very different interaction. Here, lubrication theory helps, as the fluid underneath (the lane oil) plays a huge roll. In fact, there are many different oil lane patterns that expert bowlers learn to play on.

Mathematical Model of the Magnus Effect#

For this problem, you will add the spin dependent interaction to some model of a ball moving through air. The mathematical model of the of the force involves the cross (vector) product, so you will need to carefully set up your coordinate system.

The mathematical model of the Magnus Force for a spinning sphere is given by:

\[\mathbf{F}_{magnus} = S(v) \boldsymbol\omega \times \mathbf{v}.\]

  • Here, \(S(v)\) is a function of speed, but for our purposes we will treat as constant for all speeds.

  • The other two terms are the velocity \(\mathbf{v}\) of the ball and the angular velocity \(\boldsymbol\omega\) of the ball as it spins.

Typically, the spin-drag interaction will cause the spinning of the ball to slow, this is what \(S(v)\) can capture. For extra credit, you can add that (see below).

  • 1a. (4 pts) Set up the situation that you intend to model; explain it and indicate the coordinate system.

    • Find all the necessary constants and properties that you need.

    • Make sure things have the correct SI units.

    • Make a table of these values, so we can follow your work. This will also help with your computational model.

  • 1b. (4 pts) For a model that includes that gravitational interaction, the turbulent drag model, and the spin dependent interaction, write out the differential equation that includes each aspect.

    • You should write this as both a vector differential equation, make sure you are careful about the signs and directions, and a set of scalar ODEs that have the correct signs, these are needed for the next part.

    • These scalar ODEs are second order, how else do we characterize them?

    • Do you notice something about how many ODEs you must have to model this system compared to what we’ve done with 1D and 2D drag models?

  • 1c. (4 pts) The ODEs that you developed in part (b), which are second order, need to be made into sets of first order ODEs. Remember that you are going to need to develop twice as many first order equations as you have second order in part (b).

    • Write them both as analytical equations and discrete equations that you would use in a computational model, that is, how does each time step get computed in a Euler-Cromer set up?

  • 1d. (4 pts) Using prior codes that have been posted (e.g., Euler integration resource and Integration Methods, or homework solutions on D2L), or written from scratch, write a computational model that integrates these equations of motion to model a trajectory.

    • Choose appropriate initial conditions for your model and attempt to be as close to realistic choices as possible.

    • Show the trajectory in 3 dimensions and make plots of the velocities and the magnitudes of the different forces acting on the ball.

  • 1e. (4 pts) What do you notice in your graphs about the motion of your ball?

    • How accurate is your work? How are you judging that?

    • From the graphs you’ve made, is anything surprising or explanatory for the “goodness” or “badness” of your model? What might improve it?

Extra Credit (10 pts)#

  • 1f. (10 pts) The spin dependent interaction is “coupled” to the drag force both through the cross product of the angular velocity and the linear velocity, and the tunable parameter \(S(v)\). We’ve treated it as constant so far. How should it be treated? Should \(S\) go up or down as the speed goes up or down? Is it monotonic (i.e., always increasing or always decreasing)?

    • Research or develop a simplified model of this coupling constant \(S(v)\) and describe what you find. Implement this approach for the same trajectory you computed in part (d). How do they compare?

Exercise 2, Particle in a one-dimensional potential (40 points)#

We consider a particle (for example an atom) of mass \(m\) moving in a one-dimensional potential,

\[ V(x)=\frac{V_0}{d^4}\left(x^4-2x^2d^2+d^4\right). \]

We will assume all other forces on the particle are small in comparison, and neglect them in our model. The parameters \(V_0\) and \(d\) are known constants.

  • 2a. (3 pts) Sketch or plot the potential and find the equilibrium points (stable and unstable) by requiring that the first derivative of the potential is zero. Make an energy diagram (see for example Malthe-Sørenssen chapter 11.3) and mark the equilibrium points on the diagram and characterize their stability. The position of the particle is \(x\).

  • 2b. (2 pts) Choose two different energies that give two distinct types of motions, draw them on the energy diagram, and describe the motion in each case.

  • 2c. (5 pts) If the particle starts at rest at \(x=2d\), what is the velocity of the particle at the point \(x=d\)?

  • 2d. (5 pts) If the particle starts at \(x=d\) with velocity \(v_0\), how large must \(v_0\) be for the particle to reach the point \(x=−d\)?

  • 2e. (5 pts) Use the above potential to find the net force acting on the particle. Find the acceleration acting on the particle. Is this net force conservative? Calculate the curl of the force \(\boldsymbol{\nabla}\times \boldsymbol{F}\) in order to validate your conclusion.

  • 2f. (5 pts) Are linear momentum and angular momentum conserved? You need to show this by calculating these quantities.

  • 2g. (5 pts) Write a numerical algorithm to find the position and velocity of the particle at a time \(t+\Delta t\) given the position and velocity at a time \(t\). Here you can use either the standard forward Euler, or the Euler-Cromer or the Velocity Verlet algorithms. You need to justify your choice here (hint: consider energy conservation). You might find this numerical integration resource helpful.

  • 2h. (5 pts) Use now your program to find the position of the particle as function of time from \(t=0\) to \(t=30\) s using a mass \(m=1.0\) kg, the parameter \(V_0=1\) J and \(d=0.1\) m. Make a plot of three distinct positions with initial conditions \(x_0=d\) and \(v_0=0.5\) m/s, \(x_0=d\) and \(v_0=1.5\) m/s, and \(x_0=d\) and \(v_0=2.5\) m/s. Plot the velocity. Perform these analyses with and without the term \(x^4\) in the potential. Do you see a difference? What do you notice?

  • 2i. (5 pts) Describe the behavior of the particle for the three initial conditions and sketch the motion in an energy diagram. Is energy conserved in your simulations?

Exercise 3, Model your own system (30 points)#

In this problem, you will choose a one dimensional system of your own. You may choose a known potential, or you may invent your own. Your potential must:

  1. Have at least one stable equilibrium point.

  2. Have at least one unstable equilibrium point, or some other interesting feature (e.g., asymptotic behavior).

  3. For some choice of total energy, it should have oscillatory motion (i.e., classical turning points).

  4. Produce a non-linear and conservative force.

  5. Be continuous and differentiable over the range of interest.

For this problem, you will need to perform the following tasks:

  • 3a. (5 pts) Write down the potential and start to demonstrate that it meets the above criteria. Make sure it is conservative and that the force is nonlinear before proceeding.

  • 3b. (5 pts) Sketch or plot the potential and find the equilibrium points. You need to show you can compute the equilibrium points and characterize their stability. For some choices of potential, you may need to use a numerical method to find the equilibrium points and conceptual arguments to determine their stability.

  • 3c. (5 pts) Pick a total energy that gives rise to oscillatory motion. Show this by sketching or plotting the energy diagram and describing the motion. Are there any other kinds of motion that can occur for other choices of total energy?

  • 3d. (10 pts) Write a numerical algorithm to find the position and velocity of the particle (it’s trajectories) for the choice of total energy where the motion is oscillatory. Here you must use two methods: (1) the standard forward Euler, and either (2) the Euler-Cromer or the Velocity Verlet algorithms. You will need to pick the time step \(\Delta t\) and the total time \(t_{\rm max}\) for your simulation. Compare the results of the two algorithms. Which one is better? Justify your answer. You might find this numerical integration resource helpful.

  • 3e. (5 pts) Use your program to plot the energy of the particle (T), the potential energy (V), and the total energy (E) as a function of time. Discuss the behavior of the energy between each choice of algorithm. Is energy conserved in your simulations?

Extra Credit: Professional Development Planning (5% on Midterm Grade)#

This extra credit opportunity allows you to earn up to 5% additional points on your midterm grade by engaging in structured professional development planning. This is an individual opportunity that requires personal reflection and planning for your career goals.

Requirements#

Before the Midterm Due Date:

  • Schedule and complete a meeting with Danny to discuss your career goals, interests, and potential pathways

  • Use this meeting to present, brainstorm, and/or outline your professional development plan

By the Midterm Due Date:

  • Submit a written professional development plan (approximately 500 words) via email to Danny

  • Your plan should address the following components:

    • Your current career goals and interests (both short-term and long-term)

    • An honest assessment of your current strengths and skills

    • Areas where you want to grow or develop new competencies

    • Specific steps or actions you plan to take to work toward your goals

    • How this physics course connects to your broader career aspirations

Submission Instructions#

  • Email your written plan directly to Danny (caball14@msu.edu) by the midterm late due date (March 1st)

  • Use the subject line: “PHY321 Midterm 1 Extra Credit - [Your Name]”

  • This is an individual assignment - each student must complete their own plan and reflection

Grading#

Full extra credit (5%) will be awarded for:

  • Completing the meeting with Danny before the deadline

  • Submitting a thoughtful, well-written plan that addresses all required components

  • Demonstrating genuine reflection on your goals and development needs

This extra credit can exceed 100% total on your midterm.

Note: Students who complete this extra credit for Midterm 1 will have the opportunity to continue this professional development work for additional extra credit on Midterm 2.