Homework 8 (Due 17 Apr)

Homework 8 (Due 17 Apr)#

Grading Breakdown

  • Individual Exercise (10 points) - Ex 0

  • Pencil and Paper Exercises (65 points) - Ex 1-4

  • Numerical Exercise (35 points) - Ex 5

Practicalities about homeworks

  1. Individual exercises. You must work alone on these exercises and hand in your own answers. This should be submitted on D2L only (Homework 8 Exercise 0). Individual exercises are marked with “Individual Exercise” in the title and are counted separately from the rest of the homework.

  2. 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.

  3. 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).

  4. Homeworks are available approximately ten days before the deadline. You should anticipate this work.

  5. 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.

  6. 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 (10pt): Looking Back, Looking Forward#

Over the course of this semester, Exercise 0 has asked you to examine physics from angles that are rarely foregrounded in physics courses: whose labor makes science possible, whose stories get told, who benefits, and who is left out. These exercises have asked you to consider the instrument makers of the 17th century and Global South data workers training AI, and to reflect on Nobel Prize controversies and the summit of Maunakea.

You may have engaged with all of these exercises, some of them, or only a few. That’s ok. This final exercise asks you to reflect on what has stayed with you, regardless of how much of the sequence you completed.

  • 0a (5pt) In ~200 words, pick one theme from the Exercise 0 sequence: labor, credit, capital, land, access, or another you identify, and trace how it appeared across this course. This can be in the Exercise 0 readings, in the physics that you learned this semester, or in both. You don’t need to reference every homework; focus on where this thread felt most important or most surprising to you.

  • 0b (5pt) In ~200 words, describe a question from this sequence that you are leaving the course without a good answer to. What makes it hard to answer? Is it an empirical question, an ethical one, or something else? What would you need (e.g., more knowledge, more experience, more conversation) to begin answering it?

Pencil and Paper Exercises (Submit on Gradescope only)#

Exercise 1 (10pt) Paths along a curved surface#

We have shown by minimizing the path length on a 2D flat surface that we have generated a linear equation, which illustrates the shortest distance between two points is a straight line. On a curved surface, the shortest distance will be a curve of some kind. In the exercise you will be asked to show that the shortest distance between two points on a sphere and a cylinder, both of fixed radii.

A step in spherical coordinates is given by:

\[d\vec{s} = dr \hat{r} + r d\theta \hat{\theta} + r \sin\theta d\phi \hat{\phi}.\]

A step in cylindrical coordinates is given by:

\[d\vec{s} = d\rho \hat{\rho} + \rho d\phi \hat{\phi} + dz \hat{z}.\]

  • 1a. (5 pt) Demonstrate the integral for the shortest distance between two points on a sphere of radius \(R\) is given by:

\[L = R \int_{\theta_1}^{\theta_2} \sqrt{1 + \left(\sin^2\theta\right) \left(\frac{d\phi}{d\theta}\right)^2} d\theta\]

where \(\theta\) is the polar angle and \(\phi\) is the azimuthal angle. Note that you want to travel from (\(\theta_1, \phi_1\)) to (\(\theta_2, \phi_2\)) at fixed \(R\).

  • 1b. (5 pt) Construct the integral for the shortest distance between two points on a cylinder of radius \(R\) and height \(h\). The coordinates are \((R, \phi_1, z_1)\) and \((R, \phi_2, z_2)\) in the cylindrical coordinate system, where \(\rho\) is the radial difference, \(\phi\) is the azimuthal angle, and \(z\) is the height.

Exercise 2 (15pt) Snell’s Law#

We can develop Snell’s Law by minimizing the time it takes for light to travel between two points in different media. We note that the speed of light in a media is given by \(v = c/n\), where \(c\) is the speed of light in a vacuum and \(n\) is the refractive index of the media. The time it takes for light to travel between two points is given by \(t = L/v\), where \(L\) is the distance between the two points.

Let a light ray travel in a medium with refractive index \(n_1\) into another medium with refractive index \(n_2\).

  • 2a. (5 pt) Sketch the problem and indicate the relevant angles and distances in your figure. Recall that the angle of incidence is \(\theta_1\) and the angle of refraction is \(\theta_2\). It is ok to use a 2D sketch.

  • 2b. (5 pt) Set up the “integral” for the time it takes for light to travel between two points in different media. Use the fact that the speed of light in a media is given by \(v = c/n\). Your answer needs to include the angles \(\theta_1\) and \(\theta_2\) and the refractive indices \(n_1\) and \(n_2\).

  • 2c. (5 pt) Minimize the time it takes for light to travel between two points in different media to derive Snell’s Law.

\[n_1 \sin\theta_1 = n_2 \sin\theta_2\]

Exercise 3 (15pt) Lagrangian Examples#

Constructing the Lagrangian for a system can be a bit tricky. In this exercise you will be asked to construct the Lagrangian for a few systems and quickly find equations of motion to illustrate the process.

  • 3a. (2 pt) Construct the Lagrangian in 3D for a projectile subject to no air resistance in a uniform gravitational field. The coordinates are \((x, y, z)\) in the Cartesian coordinate system. Find the 3 equations of motion.

  • 3b. (2 pt) Construct the Lagrangian for a one-dimensional simple harmonic oscillator (\(F=-kx\)). The coordinate is \(x\) in the Cartesian coordinate system. Find the equation of motion.

  • 3c. (3 pt) A mass moves in a potential well \(U(x,y) = \frac{1}{2}k(x^2 + y^2)\). Construct the Lagrangian in 2D. The coordinates are \((x, y)\) in the Cartesian coordinate system. Find the 2 equations of motion.

It’s common to use coordinates that are convenient for the problem. In these cases, you will need to write down the kinetic and potential energy in terms of the coordinates you choose. The single particle kinetic energy in these coordinates are commonly used.

  • 3d. (4 pt) Write or derive the velocity vector in spherical coordinates. The coordinates are \((r, \theta, \phi)\) in the spherical coordinate system. Show the kinetic energy is given by \(T = \frac{1}{2}m(\dot{r}^2 + r^2\dot{\theta}^2 + r^2\sin^2\theta\dot{\phi}^2)\).

  • 3e. (4 pt) Write or derive the velocity vector in cylindrical coordinates. The coordinates are \((\rho, \phi, z)\) in the cylindrical coordinate system. Show the kinetic energy is given by \(T = \frac{1}{2}m(\dot{\rho}^2 + \rho^2\dot{\phi}^2 + \dot{z}^2)\).

Exercise 4 (25pt) Changing Coordinates#

Consider two particles with masses \(m_1=m_2=m\) connected by a spring with spring constant \(k\). They sit on a frictionless surface but are constrained to the \(x\)-axis. The location of the first particle is \(x_1\) and the location of the second particle is \(x_2\). The spring is at equilibrium when the particles are a distance \(L\) apart. Assume the spring is massless and that the left mass never changes position with the right mass (i.e., the spring is always horizontal).

  • 4a. (3 pt) Write down the kinetic and potential energy for this system in terms of \(x_1\) and \(x_2\) and their associated velocities.

  • 4b. (4 pt) Write down the Lagrangian for this system in terms of \(x_1\) and \(x_2\) and their associated velocities \(\mathcal{L}(x_1, x_2, \dot{x}_1, \dot{x}_2)\). Find the 2 equations of motion.

  • 4c. (5 pt) Now consider a new coordinate system where the center of mass is at \(x_{cm} = \frac{1}{2}(x_1 + x_2)\) and the relative coordinate is \(x = x_1 - x_2\). Write down the kinetic and potential energy in terms of \(x_{cm}\) and \(x\) and their associated velocities.

  • 4d. (8 pt) Write down the Lagrangian for this system in terms of the center of mass coordinate (\(x_{cm}\)) and the spring’s extension (\(x\)) and the associated velocities \(\mathcal{L}(x_{cm}, x, \dot{x}_{cm}, \dot{x})\). Find the 2 equations of motion.

  • 4e. (5 pt) Find \(x_{cm}(t)\) and \(x(t)\) for the system for some generic initial conditions of your choosing. Describe the motion of the system.

In this problem we are changing the coordinates to explore a different aspect of the problem (and to make the math easier). This is a common technique that will not always be obvious. But practice will help us identify when it is useful.

Exercise 5 (35 pt) Motion inside a paraboloid#

Let’s figure out the motion of a particle stuck inside a paraboloid. The potential energy of the bead is given by \(U = mgz\), where \(m\) is the mass of the bead, \(g\) is the acceleration due to gravity. The coordinates are \((x, y, z)\) in the Cartesian coordinate system.

  • 5a. (5 pt) Write down the Lagrangian for the system in terms of \(x\), \(y\), and \(z\) and their associated velocities \(\mathcal{L}(x, y, z, \dot{x}, \dot{y}, \dot{z})\). Find the 3 equations of motion.

  • 5b. (15 pt) The bead is constrained by the paraboloid \(z = \frac{1}{2}k(x^2 + y^2)\). Find the constraint equation and use it to eliminate one of the coordinates from the Lagrangian. Find the 2 equations of motion. Here it can be useful to change to cylindrical coordinates \((\rho, \phi, z)\). Hint it should be that the Lagrangian is a function of \(\rho\) and \(\phi\) and their associated velocities, \(\mathcal{L}(\rho, \dot{\rho}, \phi, \dot{\phi})\).

  • 5c. (15 pt) For an initial condition where the bead is moving relative to the parabolic at some height above the bottom of the paraboloid, numerically solve the equations of motion (i.e., for \(\rho(t)\) and \(\phi(t)\)) and plot the the motion of the bead. Does it stick to the paraboloid?

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:

  1. Attend an MSU research talk (see approved clubs and seminars below).

  2. Write a summary of the talk (at least 150 words).

  3. 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.

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