\[ \renewcommand{\C}{\mathbb{C}} \renewcommand{\R}{\mathbb{R}} \renewcommand{\Q}{\mathbb{Q}} \renewcommand{\N}{\mathbb{N}} \renewcommand{\Z}{\mathbb{Z}} \renewcommand{\bar}{\overline} \renewcommand{\Lin}{\mathrm{Lin}} \]

# Instructions

Submission is through GradeScope as a PDF.

Please be considerate of the grader:

- ensure your answers are legible;
- if you scan/photo your handwritten work adjust your settings so that your handwriting is higher visibility than the lines on your paper;
- when you submit to GradeScope, mark the answers for each problem you are submitting: this helps greatly.

Teamwork is encouraged but every person must turn in their own assignment. Please reference who you worked with and all external sources used. External help (not full solutions) or information is permitted but must be acknowledged. Lecture notes, OH help, course books, recorded lectures are not considered external and can be used without reference.

By submitting this HW assignment you pledge to the UVA honor code.

# 1 Circular motion

This sequence of exercises is meant to study in depth circular motion. Recall that uniform circular motion in \(\R^2\) around the point \(\Big[x_0;y_0\Big]\) with radius \(r\) and angular velocity \(\omega\) is given by

\begin{equation*} \vec{p}(t) = \Big[x_0;y_0\Big] + r \Big[\cos(\omega t);\sin(\omega t)\Big]= \Big[x_0+r\cos(\omega t);y_0+r\sin(\omega t)\Big] \end{equation*}## 1.1

Find:

- velocity vector
- acceleration vector
- tangent vector
- normal vector

at time \(t\) of uniform circular motion \(\R^2\) around the point \(\Big[x_0;y_0\Big]\) with radius \(r\) and angular velocity \(\omega\).

Your answer should be expressed in terms of \(\Big[x_0;y_0\Big]\), \(r\), \(\omega\), and \(t\)

## 1.2

Find the tangent line to uniform circular motion \(\R^2\) around the point \(\Big[x_0;y_0\Big]\) with radius \(r\) and angular velocity \(\omega\) at time \(t\). You may either write down the line in equation for or in parametric form.

Your answer should be expressed in terms of \(\Big[x_0;y_0\Big]\), \(r\), \(\omega\), and \(t\)

## 1.3

Find the normal line to uniform circular motion \(\R^2\) around the point \(\Big[x_0;y_0\Big]\) with radius \(r\) and angular velocity \(\omega\) at time \(t\). You may either write down the line in equation for or in parametric form.

Your answer should be expressed in terms of \(\Big[x_0;y_0\Big]\), \(r\), \(\omega\), and \(t\)

## 1.4

Just for this point let \(\Big[x_0,y_0\Big]=(-3,4)\), \(r=5\), \(\omega=2\). Draw the path of motion, the center, and

- the tangent line at \(t=\pi/8\)
- the normal line at \(t=\pi/8\)
- the tangent line at \(t=\pi/2\)
the normal line at \(t=\pi/2\)

Preferably use some software.

## 1.5

Show that for uniform circular motion \(\R^2\) around the point \(\Big[x_0;y_0\Big]\) with radius \(r\) and angular velocity \(\omega\) at time \(t\) the center of rotation \(x_0,y_0\) lies on the normal line.

You must show that this holds WHATEVER \(\Big[x_0;y_0\Big]\), \(r\), \(\omega\), and \(t\) are.

# 2 Circular motion

In this problem we do the opposite of the previous problem: given initial velocity and acceleration we FIND the center, radius, and velocity of the circle.

The next points will lead us to solving the following problem:

You are on \(\R^2\). Suppose that at time \(t=0\) you start at the point \(\Big[1;1\Big]\) your velocity is \(\dot{\vec{p}}(0)=\Big[1;2\Big]\) and your acceleration has always magnitude \(2\) and is directed 90 degrees to the left of your direction of motion. We will show that you are moving in a circle, we will find the center of the circle, its radius, your angular velocity, and conclude by writing down your position as a function of time.

## 2.1

Show that under the conditions above, your speed never changes.

(Compute the rate of change of speed and show that it is \(0\)).

## 2.2

Draw the initial position, initial velocity vector, acceleration vector.

## 2.3

We guess that we are moving in a circle, to show this we first need to find its center. Based on problem 1 we suppose that the center is on the normal line to motion at time \(t=0\).

Explain why the equation of the normal line to motion at time \(t=0\) is given by

\begin{equation*} L = \{\Big[1;1]+ s \Big[-2 ; 1 \Big] \} \end{equation*}i.e.

\begin{equation*} \begin{aligned}[t] &x=1-2s \\ & y= 1+s \end{aligned} \end{equation*}## 2.4

According to the equation above, the center is \(\Big[1-2s;1+s\Big]\) for some unknown \(s\), we are moving with an unknown radius \(r\) and with angular velocity \(\omega\).

- Using the formula for the velocity, acceleration, magnitude of acceleration, and position from P1 express these quantities at \(t=0\) with \(s,r,\omega\) as unknowns.
- Equate these quantities to the known position, velocity, and magnitude of acceleration and solve for \(s,r,\omega\)

## 2.5

Using the formula in P1 write the parameterization of motion. Compute velocity and acceleration, and check that the conditions at the beginning of P2 are satisfied.

# 3 Winding around a torus

A torus is a "doughnut" shape in \(\R^3\). We describe a torus by two quantities: inner radius \(r>0\) and outer radius \(R>r\).

You may use https://www.geogebra.org/3d/bswvfq9k to help you understand this problem.

Below is a picture (in red) of a torus of inner radius \(r=1\) and outer radius \(R=3\). In blue we have the center ring of the torus. In green we have a cross section circle of the torus.

We need to understand how to parameterize the torus above.

## 3.1

The blue line is a circle halfway between the inner and outer radius and it lies in the \(x,y\) plane.

- Parameterize the blue line \(\vec{p}\) with the variable \(a\in[0,2\pi)\). Write

- For any \(a\) find the
**unit**vector \(\vec{u}(a)\) in the direction of \(\vec{p}(a)\).

## 3.2

Now let us look at the cross section in \(\R^3\) in which the green circle lies. This corresponds to when the position of the blue point \(\vec{p}(a)\) at some time \(a\)

Let us now think of the above picture as if it were in \(\R^2\):

Explain why the parameterization of the green line in the above picture is given by

\begin{equation*} \Big[2+cos(b); sin(b) \Big] \qquad b\in[0,2 \pi) \end{equation*}## 3.3

Compare the two pictures

In the second picture the horizontal direction is given by \(\vec{u}(s)\) while the vertical direction is given by the \(z\) axis.

Explain why the \(x,y,z\) coordinates of the point on the torus at the cross section passing through \(\vec{p}(a)\) with angle \(b\) with respect to the \(\vec{u}(s)\) direction is given by

\begin{equation*} \Big[cos(a)\,(2+cos(b)),sin(a)\,(2+cos(b)); sin(b) \Big] \end{equation*}## 3.4

Parameterize the path \(\vec{q}(t)\) that as time \(t\) goes from \([0,2 \pi)\) starts at \(\Big[3,0,0\Big]\), loops once in the \(x,y\) plane and winds 5 times around the torus. Explain your work.

## 3.5

Compute the velocity of \(\vec{q}(t)\) and find the tangent line at time \(t=0\).

## 3.6

Write down the integral that represents the length of the path

## 3.7

Compute the length of the path to 2 digit precision.