When something rotates in a circle, it moves through a *position angle* \begin{align*} \theta \end{align*} that runs from to \begin{align*}2\pi\end{align*} radians and starts over again at . The physical distance it moves is called the *path length.* If the radius of the circle is larger, the path length traveled is longer. According to the arc length formula \begin{align*} s = r\theta \end{align*}, the path length \begin{align*} \Delta s \end{align*} traveled by something at radius \begin{align*} r \end{align*} through an angle \begin{align*} \theta \end{align*} is:\begin{align*} \Delta s = r \Delta \theta \text{ [1]} \end{align*}Just like the linear velocity is the rate of change of distance, angular velocity, usually called \begin{align*} \omega \end{align*}, is the rate of change of \begin{align*} \theta \end{align*}. The direction of angular velocity is either clockwise or counterclockwise. Analogously, the rate of change of \begin{align*} \omega \end{align*} is the angular acceleration \begin{align*} \alpha \end{align*}.

For an object moving in a circle, the objects tangential speed is directly proportional to the distance it is from the rotation axis. the tangential speed (as shown in the key equations) equals this distance multiplied by the angular speed (in radians/sec).

\begin{align*}\omega = 2\pi / T = 2\pi f \end{align*} ; Relationship between period and angular frequency.

\begin{align*} \omega = \frac{\Delta \theta}{\Delta t} = \frac{\Delta s}{rt}= \frac{v}{r} \end{align*}

\begin{align*} v = \omega r \end{align*}

#### Example

A Merry Go Round is rotating once every 4 seconds. If you are on a horse that is 15 m from the rotation axis, how fast are you moving (i.e. what is your tangential speed)?

\begin{align*} v = r \omega \end{align*}

Now we need to convert the angular speed to units of radians per second.

\begin{align*} \omega = \frac{1}{4} \frac{\cancel{rotations} }{second} * \frac{2 \pi}{1} \frac{radians}{\cancel{rotation}} = \frac{\pi}{2} \frac{radians}{second} \end{align*}

\begin{align*} v = 15 m \times \frac{\pi}{2} = \frac {15}{2} \pi \approx 23.6 \frac{radians}{second} \end{align*}

### Interactive Simulations

### Review

- You are riding your bicycle and going 8 m/s. Your bicycle wheel is 0.25 m in radius.
- What is its angular speed in radians per second?
- What is the angular speed in rotations per minute?

- The angular speed of a record player is 33 rotations per minute. It has a diameter of 12 inches.
- What is the angular speed in radians per second?
- What is the tangential speed of the outer most part of the record?
- What is its tangential speed halfway out on the record?

**Review (Answers)**

- a. 32 rads/s b. 306 rot/min
- a.3.5 rads/s b. 0.53 m/s c. 0.26 m/s