# Instructions

Submission is through GradeScope as a PDF.

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# 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 $$R>r$$.

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
\begin{equation*} \vec{p}(a)= \ldots \end{equation*}
• For any $$a$$ find the unit vector $$\vec{u}(a)$$ in the direction of $$\vec{p}(a)$$.
\begin{equation*} \vec{u}(a)= \ldots \end{equation*}

## 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.