From Point Particles to Rigid Bodies
Everything you've learned about linear motion — Newton's laws, momentum, kinetic energy — applies to point particles or objects treated as if all their mass is concentrated at a single point. But the real world is full of extended rigid bodies: doors, wheels, spinning tops, and gymnasts.
An extended object in which all parts maintain fixed distances from one another — it does not deform under the forces applied to it.
When a force acts on a rigid body, two things can happen simultaneously:
- Translational motion — the centre of mass accelerates (linear physics applies)
- Rotational motion — the body spins about some axis (new physics is needed)
The key insight is that the effect of a force depends not just on its magnitude and direction, but also on where it is applied relative to the axis of rotation. Pushing a door near the hinge requires far more force to achieve the same rotation as pushing near the outer edge. This motivates the concept of torque.
Think of a rigid body as a team of point particles all rigidly connected. The centre of mass of the team obeys Newton's second law (), but the internal arrangement also determines how the team rotates — and that's where moment of inertia and torque come in.
Kinematics of Rotational Motion
When a rigid body rotates about a fixed axis, every point on it sweeps through the same angle in the same time — but points at different radii travel different arc lengths. We describe this motion using angular quantities.
The angle, measured in radians, swept by a reference line from the axis of rotation to a point on the body.
The rate of change of angular displacement: measured in rad s⁻¹.
The rate of change of angular velocity: measured in rad s⁻².
These angular quantities map directly onto their linear counterparts:
| Linear | Rotational |
|---|---|
| displacement | angular displacement |
| velocity | angular velocity |
| acceleration | angular acceleration |
| mass | moment of inertia |
| force | torque |
Equations of motion for constant angular acceleration (analogous to the SUVAT equations):
Linking linear and angular quantities for a point at radius from the axis:
Always work in radians, never degrees, when using these equations. To convert: , so rad.
Example: A flywheel starts from rest and reaches an angular velocity of rad s⁻¹ in s under constant angular acceleration. Find the angular acceleration and the total angle swept.
Step 1 — Angular acceleration:
Step 2 — Angle swept:
This corresponds to complete revolutions.