Homework 6 (Due 20 Mar)

Homework 6 (Due 20 Mar)#

Grading Breakdown

Individual Exercise (10 points) - Ex 0
Pencil and Paper Exercises (40 points) - Ex 1-4
Numerical Exercise (50 points) - Ex 5

Practicalities about homeworks

Individual exercises. You must work alone on these exercises and hand in your own answers. This should be submitted on D2L only (Homework 6 Exercise 0). Individual exercises are marked with “Individual Exercise” in the title and are counted separately from the rest of the homework.
For pencil and paper, or numerical exercises, you may work in groups of up to 3. If you work as a pair/group you may hand in one answer only if you wish. Remember to write your name(s)! These exercises are marked with “Pencil and Paper Exercises” or “Numerical Exercise” in the title, and are counted together for the homework grade.
Beyond the group you work on homework with, you may collaborate with others to discuss concepts and approaches, but you must write up your own answers (alone or as a group of 3).
Homeworks are available approximately ten days before the deadline. You should anticipate this work.
How do I(we) hand in? You can hand in the paper and pencil exercises as a single scanned PDF document. For this homework this applies to exercises 1-5. 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.
Make sure your work is legible. If we cannot read it, we cannot grade it.

Instructions for submitting to Gradescope.

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')

Individual Exercise (Submit on D2L only)#

Exercise 0: Algorithmic Science: Who Builds It, Who Benefits, and Who Disappears?#

Artificial intelligence has rapidly become a central tool in scientific research. From predicting protein structures to detecting gravitational waves to analyzing climate data, AI is reshaping how science is done. AI can be an incredible enabler of scientific progress.

But like all scientific work, AI-powered science happens within institutions, economies, and labor systems and those systems are rarely made visible. When we say a machine “learned” something, we should ask: who taught it, and at what cost?

The Nobel Prize Goes to… a Corporation?

Image credit: Wikimedia Commons, Public Domain

In 2024, the Nobel Prize in Chemistry was awarded to Demis Hassabis and John Jumper of Google DeepMind, along with David Baker of the University of Washington, for their work on protein structure prediction using AI. The centerpiece was AlphaFold, a system developed inside one of the world’s largest corporations that was trained on publicly funded biological databases built up over decades by thousands of researchers. The prize raised sharp questions: Can a corporate AI system win the Nobel Prize? Who really did this work? And what does it mean when scientific credit flows to a company’s executives rather than the broader community of researchers whose data and labor made the tool possible?

For this exercise, read the following two sources:

On AlphaFold and the Nobel: Chris Palmer, “AlphaFold Wins Nobel Prize, Gains Functionality, Drops Open Access”, Engineering, 2024. (Focus on the sections about open science controversy and the scientific credit debate.)
On the hidden labor of AI: Brookings Institution, “Reimagining the future of data and AI labor in the Global South”, 2025.

Why are we reading this?

AI tools are increasingly part of scientific workflows, including in physics. Understanding who builds these tools, and under what conditions, is part of understanding the political economy of modern science. Just as earlier exercises asked you to look at the hidden labor behind observatories and instruments, this exercise asks you to look at the hidden labor behind the algorithms. The workers who label data, moderate content, and annotate training sets are as structurally essential to AI as the ceramics makers and glass blowers of the 17th and 18th century, the technicians who polished telescope mirrors were to 19th century astronomy, and the human computers of the 20th century. They are rarely credited.

0a (3pt) In ~150 words, summarize the AlphaFold Nobel Prize controversy. The prize was awarded to corporate executives at Google DeepMind for a tool trained on publicly funded scientific databases. What does this episode reveal about how scientific credit is assigned when algorithms, corporations, and cumulative community labor all play a role? What would a more complete accounting of that credit look like?
0b (4pt) In ~150 words, AI systems — including those used in scientific research — are trained on data labeled by workers, often in the Global South, earning as little as $1–2 per hour under precarious conditions. This labor is structurally invisible: workers rarely know what systems they are building, and the companies using their work rarely disclose the supply chain. How does recognizing this labor chain change your understanding of what “automated” science actually is? How does it connect to the broader patterns of hidden labor we have discussed throughout this course?
0c (3pt) In ~150 words, as a future scientist or engineer, you will very likely use AI tools in your work. What practices would you adopt to be transparent about how those tools work, who built them, and whose labor made them possible? What structural changes — in how AI tools are credited, licensed, or governed — might better reflect the collective labor that produces them? You might draw on the principles of Open Science from Homework 4, Exercise 0 as a starting point.

Pencil and Paper Exercises (Submit on Gradescope only)#

Exercise 1 (10pt) Morse Potential as an SHO#

If the potential has a local minimum, we can often find SHO approximation for that potential near the local minimum.

The Morse potential is a convenient model for the potential energy of a diatomic molecule. The potential is a radial one and thus one-dimensional. It is given by,

\[U(r) = A\left[ \left(e^{(R-r)/S}-1\right)^2-1\right]\]

where the distance between the centers of the two atoms is $r$, and the constants $A$, $R$, and $S$ are all positive. Here $S<<R$.

1a (2pt) Sketch (or plot) the potential as a function of $r$.
1b (3pt) Find the equilibrium position of the potential, i.e. the position where the potential is at a minimum. We will call this $r_e$.
1c (3pt) Rewrite the potential in terms of the displacement from equilibrium, $r = r_e + x$. Expand the potential to second order in $x$.
1d (2pt) Find the effective spring constant, $k$, for the potential near the minimum. What is the frequency of small oscillations about the minimum?

Exercise 2 (10pt), Time Averaging and the SHO#

Time Averaging is a common tool to use with periodic systems. It also us to discuss what happens to different properties of the system over one period.

An SHO has a period $\tau$. We can find the time average of a variable $f(t)$ over one period, $\langle f \rangle$, by averaging over the period,

\[\langle f \rangle = \frac{1}{\tau}\int_0^\tau f(t)dt\]

2a (2pt) Show that the time average of the position of the SHO is zero.
2b (2pt) Show that the time average of the velocity of the SHO is zero.
2c (6pt) Show that the time average of the kinetic energy of the SHO is equal to the time average of the potential energy (and importantly, non-zero). If the total energy is $E$, these time averages are equal to $E/2$. You might need to show that the very useful trigonometric identity,

\[\langle \sin^2(\omega t-\delta)\rangle = \langle \cos^2(\omega t-\delta)\rangle = \frac{1}{2}\]

Exercise 3 (10pt), Toy Potential#

Consider a toy potential of the form,

\[U(r) = U_0\left(\dfrac{r}{R}+\lambda^2\frac{R}{r}\right)\]

where $U_0$, $R$, and $\lambda$ are all positive constants and the domain of the potential is $0<r<\infty$.

3a (2pt) Sketch (or plot) the potential as a function of $r$.
3b (3pt) Find the equilibrium position of the potential, i.e. the position where the potential is at a minimum. We will call this $r_e$.
3c (5pt) Rewrite the potential in terms of the displacement from equilibrium, $r = r_e + x$. Expand the potential to second order in $x$. What is the effective spring constant, $k$, for the potential near the minimum? What is the frequency of small oscillations about the minimum?

Exercise 4 (10pt), Defining Periodicity#

A common issue with oscillators is determining their periodicity. For the SHO, we can show that the period is $2\pi/\omega_0$ where $\omega_0$ is the natural frequency of the SHO. How might we define a periodicity more generally? Let’s start with the damped harmonic oscillator. Consider a weakly damped oscillator ($\beta < \omega_0$). The motion of the oscillator is given by,

\[x(t) = A e^{-\beta t}\cos(\omega_1 t - \delta)\]

where $A$ is the amplitude, $\beta$ is the damping constant, $\omega_1 = \sqrt{\omega_0^2 - \beta^2}$ and $\delta$ is the phase.

The motion decays with time, but we can still define a periodicity, $\tau_1$, which is the time between peaks in the motion.

4a (4pt) Sketch (or plot) the motion of the oscillator as a function of time. Show that you can find $\tau_1 = 2\pi/\omega_1$ from looking at successive maxima.
4b (3pt) Show that an equivalent definition of $\tau_1$ is twice the time between zero crossings of the motion.
4c (3pt) If the damping $\beta$ is half the natural frequency ($\omega_0$), how does the amplitude of the motion decay in one period?

Exercise 5 (50 pt), Find your own 1D Oscillator#

We have built all the tools to study 1D unforced oscillators. Now you get to pick your own potential and study it. You can pick any 1D potential you like, but it should have a local minimum. Make sure it is not a driven oscillator (i.e., no explicit time dependence in the equations of motion). To earn full credit for this exercise, you must:

5a (5pt) Present the potential and describe its origin, why it is interesting, where it comes from, etc. Educate us about it.
5b (5pt) Sketch (or plot) the potential as a function of it’s argument (and chosen variables) and find the equilibrium position of the potential, i.e. the position where the potential is at a minimum.
5c (10pt) Rewrite the potential in terms of the displacement from equilibrium. Expand the potential to second order to find the effective spring constant, $k$, for the potential near the minimum. What is the frequency of small oscillations about the minimum?
5d (10pt) Construct the equations of motion for the potential and solve them numerically. Choose initial conditions and parameters that give oscillatory motion. Note it doesn’t have to be SHO (In fact, it probably won’t be). Plot the position as a function of time. Make sure we can see the oscillations.
5e (10pt) Plot the phase diagram of the trajectory (you don’t have to produce a phase diagram, but just plot the trajectory in phase space). What does the phase diagram tell you about the motion?
5f (10pt) Find the period of your motion. Here you might have to make some definitions of what periodicity means for your potential.

Note: this might seem similar to your midterm, but notice we expect you to do some research on the potential in 6a, and we are going into more depth with questions 6e and 6f. This is also a scaffold for your final project in terms of practicing aspects that should appear.

Examples of 1D potentials#

Simple Pendulum (Nonlinear Small Angle Approximation):
- Potential: $ V(\theta) = mgh(1 - \cos(\theta)) $, where $m$ is the mass, $g$ is the acceleration due to gravity, $h$ is the length of the pendulum, and $\theta$ is the angular displacement.
Nonlinear Spring (Hardening or Softening):
- Potential: $ V(x) = \frac{k}{2} x^2 + \frac{\beta}{3} x^3 $, where $k$ and $\beta$ are constants. Depending on the sign of $\beta$, the spring can exhibit hardening ($\beta > 0$) or softening ($\beta < 0$) nonlinearity.
Lennard-Jones Potential Oscillator (for a diatomic molecule model):
- Potential: $ V(r) = 4\epsilon \left[ \left(\frac{\sigma}{r}\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] $, where $\epsilon$ is the depth of the potential well, $\sigma$ is the finite distance at which the inter-particle potential is zero, and $r$ is the distance between particles.
Morse Potential (for molecular vibrations):
- Potential: $ V(x) = D_e \left(1 - e^{-a(x - x_0)}\right)^2 $, where $D_e$ is the depth of the potential well, $a$ is a constant related to the width of the well, $x$ is the displacement from equilibrium, and $x_0$ is the equilibrium bond length. This potential models the energy of a diatomic molecule as a function of the distance between atoms, showing oscillatory behavior that represents molecular vibrations.
Double Well Potential:
- Potential: $ V(x) = -\frac{\mu}{2} x^2 + \frac{\lambda}{4} x^4 $, where $\mu$ and $\lambda$ are positive constants. This system exhibits bistability with two stable equilibria, leading to interesting nonlinear dynamics and potential oscillations between the wells under certain conditions.

Extra Credit — Integrating Research#

Earning and Submitting Your Summary

Earn up to 5 extra credit points per homework by engaging with MSU research activities. These points can boost your grade above 100% or help offset missed exercises.

Send via email to Danny caball14@msu.edu

Earn up to 5 extra credit points per homework by engaging with MSU research activities. These points can boost your grade above 100% or help offset missed exercises.

To receive full credit:

Attend an MSU research talk (see approved clubs and seminars below).
Write a summary of the talk (at least 150 words).
Submit your summary with your homework (email to caball14@msu.edu).

Approved talks include:

Society for Physics Students (SPS): Meets Monday nights (alternates with Astronomy Club)
Astronomy Club: Meets Monday nights (alternates with SPS)
Any physics and astronomy seminar of interest
Any MSU research seminar/workshop relevant to physics (get approval if unsure)
Any other physics-related event approved in advance

If you have questions, please contact Danny.

Note: You can earn 5% extra credit on each homework by attending a seminar, workshop, or other physics-related event and submitting a short reflection (about 150 words) on your experience.