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Homework 4

Total points: 100.

Introduction to Homework 4

This homework’s sets of classical pen and paper and computational exercises deal with some motion problems and conservation of energy. We also have a preparation exercise for the upcoming midterms and final project.

The relevant reading background is

  1. chapters 3, 4.1, 4.2 and 4.3 of Taylor (there are many good examples there)

  2. chapters 10-13 of Malthe-Sørenssen.

  3. for the numerical exercise see Malthe-Sørenssen section 7.5

In both textbooks there are many nice worked out examples. Malthe-Sørenssen’s text contains also several coding examples you may find useful.

The numerical homework focuses on another motion problem where you can use the code you developed in homework 3, almost entirely. Please take a look at the posted solution (jupyter-notebook) for homework 3 (POSTED AFTER HW3 DUE). You need only to change the forces at play. The numerical problem this time is based on your code from homework 3 and we will try to make the motion of a falling object in two dimensions more realistic by allowing to bounce up again due to a normal force from the floor.

Practicalities about homeworks and projects

  1. You can work in groups (optimal groups are often 2-3 people) or by yourself. If you work as a group you can hand in one answer only if you wish. Remember to write your name(s)!

  2. Homeworks are available ten days before the deadline.

  3. Submission instructions: Submit the paper-and-pencil exercises as a single scanned PDF document. For this homework, this applies to exercises 1–5. Convert the Jupyter notebook to a PDF and attach it to the same PDF document.

Exercise 1 (15 pts), Is this a conservative force?

Consider a particle of mass mm moving in two dimensions. The particle moves from (0,0)(0,0) to (1,1)(1,1) along three different paths, aa, bb and cc as shown in the figure below.

Paths

In this space, the particle experiences a force:

F⃗=⟨x2,2xy⟩=x2i^+2xyj^\vec{F} = \langle x^2, 2xy \rangle = x^2\hat{i} + 2xy \hat{j}
(1)

Exercise 2 (10 pt), Sliding puck

A small puck rests on a fixed sphere of radius RR. The puck is given a tiny nudge and it slides down the sphere. Using conservation of energy, we can determine the point at which the puck leaves the sphere.

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn-v0_8-colorblind')

R = 1.0
theta = np.radians(35)  # an arbitrary illustrative angle, not the answer

fig, ax = plt.subplots(figsize=(5, 5))
phi = np.linspace(0, 2 * np.pi, 200)
ax.plot(R * np.cos(phi), R * np.sin(phi), 'C0')

px, py = R * np.sin(theta), R * np.cos(theta)
ax.plot(px, py, 'C1o', markersize=10, label='Puck')
ax.plot([0, px], [0, py], 'k--', lw=1)
ax.plot([0, 0], [0, R], 'k:', lw=1)
ax.annotate(r'$\theta$', xy=(0.15, 0.75), fontsize=13)
ax.plot(0, 0, 'ks', markersize=5)

ax.set_xlim(-1.3, 1.3)
ax.set_ylim(-1.3, 1.3)
ax.set_aspect('equal')
ax.set_title('Puck sliding on a sphere (setup sketch)')
ax.set_xlabel('x')
ax.set_ylabel('y')
ax.grid(True)
plt.tight_layout()
plt.show()
<Figure size 500x500 with 1 Axes>

Exercise 3 (10pt), Example of potential

Consider a particle of mass mm moving according to the potential:

V(x,y,z)=Aexp⁡{−x2+z22a2}.V(x,y,z)=A\exp\left\{-\frac{x^2+z^2}{2a^2}\right\}.
(2)

We can think of this potential as the energy landscape of a particle in three dimensions. That is, you can imagine a particle moving around this potential like a ball rolling around a landscape. That analogy is not perfect, but it is a good way to help us think about stability and equilibrium.

Exercise 4 (15pt), forces and potentials

A particle of mass mm has velocity v=α/xv=\alpha/x, where xx is its displacement.

A particle is thereafter under the influence of a force F=−kx+kx3/α2F=-kx+kx^3/\alpha^2, where kk and α\alpha are constants and kk is positive.

Exercise 5 (10pt), Midterms and Final Project Preparation

Your final project will be a computational essay of your own design. The idea is informed by a proposal by Stephen Wolfram. In his original post, Wolfram talks about the importance of the computational medium as a way of communicating science.

In this homework question, we are going to start building your plan for your computational essay. I ask that you complete this particular homework problem by yourself because it is important for each of you to do this planning.

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

  1. Wolfram’s What is a Computational Essay?

  2. The short paper Computational Essays: An Avenue for Scientific Creativity in Physics

  3. Wolfram’s Steps to Writing a Computational Essay

You are, of course, welcome to read more, but these are the three that I would like you to read.

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.

These essays were made by students who were taking a course at the University of Oslo. The essays are not meant to be perfect, they are meant to be representative of the work that students can do.

Now, let’s move to your future plans.

Exercise 6 (40pt), Bouncing object

This exercise builds on the code you wrote for solving homework 3. We recommend strongly that you study the text of Malthe-Sørenssen, section 7.5.

In homework 3 we introduced gravity and air resistance and studied their effects via a constant acceleration due to gravity and the force arising from air resistance. But what happens when the ball hits the floor? What if we would like to simulate the normal force from the floor acting on the ball? This exercise shows how we can include more complicated forces with no pain! And the force we include here is an example of a case where analytical solutions may either be difficult to find or we cannot find an analytical solution at all.

We need then to include a force model for the normal force from the floor on the ball. The simplest approach to such a system is to introduce a contact force model represented by a spring model. We model the interaction between the floor and the ball as a single spring. But the normal force is zero when there is no contact. Here we define a simple model that allows us to include such effects in our models.

Assume that yy increases upward. The normal force from the floor on the ball is represented by a spring force. This is a strong simplification of the actual deformation process occurring at the contact between the ball and the floor due to the deformation of both the ball and the floor.

The deformed region corresponds roughly to the region of overlap between the ball and the floor. The depth of this region is Δy=R−y(t)\Delta y = R-y(t), where RR is the radius of the ball. This is supposed to represent the compression of the spring. Our model for the normal force acting on the ball is then

N⃗=k(R−y(t))e⃗y.\vec{N} = k (R-y(t)) \vec{e}_y.
(3)

The normal force must act upward when y<Ry < R, hence the sign is positive for an upward-pointing e⃗y\vec e_y. However, we must also ensure that the normal force only acts when the ball is in contact with the floor, otherwise the normal force is zero. The full form of the normal force is therefore

N⃗=k(R−y(t))e⃗y,\vec{N} = k (R-y(t)) \vec{e}_y,
(4)

when y(t)<Ry(t) < R and zero when y(t)≥Ry(t) \ge R. In the numerical calculations you can choose R=0.1R=0.1 m and the spring constant k=1000k=1000 N/m.

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn-v0_8-colorblind')

R = 0.1
k = 1000.0
y = np.linspace(0, 0.2, 400)
N = np.where(y < R, k * (R - y), 0.0)

fig, ax = plt.subplots(figsize=(6, 4.5))
ax.plot(y, N, 'C0')
ax.axvline(R, color='k', ls='--', lw=1, label=f'$y=R={R}$ m')
ax.set_xlabel('Height of ball center, $y$ (m)')
ax.set_ylabel('Normal force magnitude, $N$ (N)')
ax.set_title('Spring Model of Floor Normal Force vs. Height')
ax.legend()
ax.grid(True)
plt.tight_layout()
plt.show()
<Figure size 600x450 with 1 Axes>
Homework
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