How turning forces and mass distribution control rotation
Rotational motion has quantities that mirror those of straight-line motion. Angular displacement θ, angular velocity ω and angular acceleration α play the roles of distance, velocity and acceleration, with v = rω. The turning effect of a force is torque, τ = r × F, or rF sin θ, and a body is in rotational equilibrium when the net torque is zero. The centre of mass is the point that moves as if the whole mass were concentrated there.
Moment of inertia, I = Σmr², measures how hard it is to change a body’s rotation and depends on how mass is spread around the axis. For a ring it is MR², and for a solid disc ½MR². The parallel and perpendicular axis theorems help find I about other axes. Newton’s second law for rotation is τ = Iα, and angular momentum L = Iω is conserved when no external torque acts. A body rolling without slipping has kinetic energy ½mv² + ½Iω².
Why it matters
Spanners, door handles and see-saws use torque, while flywheels store energy using moment of inertia. Divers and figure skaters spin faster by pulling in their arms, conserving angular momentum. Engineers use these ideas to design wheels, gears, turbines and the stability of vehicles, and they are essential for understanding planetary motion and satellites.
Key terms
- Torque
- The turning effect of a force, equal to the force times its perpendicular distance from the axis.
- Moment of inertia
- A measure of a body’s resistance to changes in its rotational motion about an axis.
- Angular momentum
- The product of moment of inertia and angular velocity for a rotating body.
- Centre of mass
- The point where the entire mass of a system can be considered to be concentrated.