Describing rotation
Rotation uses angular versions of the linear quantities. Angular position θ is measured in radians, where one full turn is 2π rad (about 6.28 rad). Angular velocity ω is the rate of change of angle, and angular acceleration α is the rate of change of ω. To convert revolutions per minute to rad/s, multiply by 2π and divide by 60. The constant-acceleration equations carry over: ω = ω₀ + α t and θ = ω₀ t + ½ α t².
Linking to linear motion
A point at distance r from the axis moves with speed v = ω r and tangential acceleration a = α r. If it moves in a circle at constant speed it still accelerates toward the centre: a_c = v² / r = ω² r. For rolling without slipping, the centre of the wheel moves at v = ω r.
Torque
Torque is the turning effect of a force: τ = r F sin θ, where r is the distance from the pivot to the point of application and θ is the angle between the force and the arm. Torque has units N·m. A force along the line through the pivot gives zero torque. A longer wrench makes the same push turn a bolt more easily. For static balance (no rotation), the clockwise torques equal the anticlockwise torques about any chosen pivot, as when two children balance on a seesaw.
Moment of inertia and Newton's second law for rotation
The moment of inertia I measures resistance to a change in rotation. It depends on how mass is spread out: mass far from the axis counts more. The rotational second law is Στ = I α. Rotational kinetic energy is KE = ½ I ω².
Angular momentum
Angular momentum is L = I ω. With no net external torque it is conserved. When a spinning skater pulls in her arms, I falls and so ω rises in proportion. Note that kinetic energy is not conserved here, because she does work pulling her arms in.
Worked example
A wheel of radius 0.30 m starts from rest with α = 2.0 rad/s² for 5.0 s.
- Final angular speed: ω = 2.0 × 5.0 = 10.0 rad/s.
- Rim speed: v = ω r = 3.0 m/s.
- Angle turned: θ = ½ × 2.0 × 5.0² = 25 rad, which is 3.98 revolutions.
- Centripetal acceleration at the rim: ω² r = 30.0 m/s².
Common mistakes
- Using degrees or revolutions in formulas that need radians.
- Forgetting the sine of the angle in the torque formula.
- Using the total mass rather than the distribution of mass for I.
- Mixing linear and angular quantities in one equation without the factor r.