Rigid body rotation

The idea

Once a body’s size and shape matter (gears, cranks, linkages), it can’t be treated as a single point. In flat (planar) mechanisms, a rigid body can only move in three ways.

Type	What happens	In a slider-crank
Translation	Every line on the body stays parallel to itself. Straight paths (rectilinear) or curved ones (curvilinear)	The slider (piston)
Rotation about a fixed axis	Every point moves in a circle round the axis	The crank
General plane motion	Translation and rotation at once	The connecting rod

Translation

Every point has the same velocity and acceleration. Treat the whole body as a particle.

Rotation about a fixed axis

$\omega =\frac{d\theta }{dt}\text{(rad/s)}\alpha =\frac{d\omega }{dt}=\frac{d^2\theta }{d{t}^2}{\text{(rad/s}}^2\text{)}$ A point at radius $r$: $v=\omega r$, $a_t=\alpha r$, $a_n={\omega }^{2}r$ (see Curvilinear and circular motion).

• Constant α: use the angular constant-acceleration equations.
• α varies with time: integrate, $\int d\omega =\int \alpha dt$, then $\int d\theta =\int \omega dt$.
• α depends on angle: use $\omega d\omega =\alpha d\theta$.

A smooth pin support stops a body moving sideways or up and down, but lets it rotate freely.

Worked examples

1. A motor spins up uniformly from rest to 3000 rpm in 2 s. Find α and how many revolutions it makes.

Answer

$\omega =3000\times 2\pi /60=314.2$ rad/s, so $\alpha =314.2/2=157.1$ rad/s² $\theta =\frac{1}{2}\times 157.1\times 2^2=314.2$ rad $=50$ rev

2. A gear starts at 10 rad/s with $\alpha =6t$ rad/s². Find ω and θ after 2 s.

Answer

$\omega =10+3t^2=22$ rad/s $\theta =10t+t^3=28$ rad

Check yourself

1. Which type of motion does a lift car have?

Translation (rectilinear): every point moves the same way.

2. A flywheel slows from 50 to 20 rad/s in 6 s at a constant rate. α?

$\alpha =(20-50)/6=-5$ rad/s².