Rigid bodies, angular velocity and the kinematics of rotation

Rigid bodies

A rigid body does not bend or stretch: the distance between any two of its particles stays the same. A wheel, a door and a fan are good examples.

A rigid body can move in two ways: translation (every particle moves the same way) and rotation about an axis. Any motion is a mix of the two.

Angular velocity and acceleration

When a rigid body turns about a fixed axis, every particle turns through the same angle $\theta$ in the same time. So they all share

A particle at distance $r$ from the axis moves at $v = \omega r$, with a tangential acceleration $\alpha r$ and a centripetal acceleration $\omega^2 r$.

The equations of rotation

With a steady angular acceleration, the equations of motion carry over, letter for letter:

Watch a wheel speed up.

You should find that with a steady $\alpha$ the turns pile up faster and faster, because $\theta$ grows as $t^2$.

Watch out. Angles in these equations are in radians. One revolution is $2\pi$ radians, and $1\ \text{rpm} = \dfrac{2\pi\ \text{rad}}{60\ \text{s}}\ \text{rad/s}$.

A worked example

A CD starts from rest and reaches $50\ \text{rad/s}$ in $2.5\ \text{s}$, speeding up steadily. Its radius is $6.0\ \text{cm}$.

1. Angular acceleration. $\alpha = \dfrac{50\ \text{rad/s}}{2.5\ \text{s}} = 20\ \text{rad/s}^2$.
2. Angle turned. $\theta = \tfrac12 \times 20\ \text{rad/s}^2 \times (2.5\ \text{s})^2 = 62.5\ \text{rad}$, or $9.95$ turns.
3. Rim speed at the end. $v = \omega r = 50\ \text{rad/s} \times 0.060\ \text{m} = 3.0\ \text{m/s}$.

Think first. Write down what you know among $\omega_0$, $\omega$, $\alpha$, $\theta$ and $t$, and choose the equation that has the one you want.

Practise with new numbers

Try it in Python

The program follows a turntable as it speeds up from rest.

Remember these

• Every particle of a rigid body shares $\omega$ and $\alpha$.
• $v = \omega r$; tangential acceleration $\alpha r$; centripetal $\omega^2 r$.
• $\omega = \omega_0 + \alpha t$, $\theta = \omega_0t + \tfrac12\alpha t^2$, $\omega^2 = \omega_0^2 + 2\alpha\theta$, in radians.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A clock's second hand turns once a minute. What is its angular velocity?

2. A wheel slows steadily from $12\ \text{rad/s}$ to rest in $4.0\ \text{s}$. Through what angle does it turn meanwhile?

3. Two children sit on a merry-go-round, one near the edge and one near the middle. Which has the bigger angular velocity? The bigger speed?

4. A fan turning at $20\ \text{rad/s}$ speeds up at $2.0\ \text{rad/s}^2$. How fast is it turning after it has turned $96\ \text{rad}$?