A.4 Rigid body mechanics (HL only)

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.

Rigid Body: 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.

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.

Angular Displacement (Δθ): The angle, measured in radians, swept by a reference line from the axis of rotation to a point on the body.

Angular Velocity (ω): The rate of change of angular displacement: $\omega =\frac{\Delta \theta }{\Delta t}$ measured in rad s⁻¹.

Angular Acceleration (α): The rate of change of angular velocity: $\alpha =\frac{\Delta \omega }{\Delta t}$ measured in rad s⁻².

These angular quantities map directly onto their linear counterparts:

Equations of motion for constant angular acceleration (analogous to the SUVAT equations):
${\omega }_f={\omega }_i+\alpha t$
$\theta ={\omega }_{i}t+\frac{1}{2}\alpha t^2$
${\omega }_f^2={\omega }_i^2+2\alpha \Delta \theta$

Linking linear and angular quantities for a point at radius $r$ from the axis:
$v=\omega r,a=\alpha r$