A2 Circular Motion

What these notes cover

These notes cover circular motion for Cambridge International A Level Physics (9702): the radian and angular displacement, angular velocity and its link to linear speed and period, centripetal acceleration and centripetal force, and examples where a force provides the centripetal force: a car on a level road and on a banked road, a banking aircraft, a mass on a turntable and the conical pendulum. They include worked examples and past-paper questions.

Key formulas and definitions

• Radian: the angle subtended at the centre of a circle by an arc equal in length to the radius. Arc length s = rθ, and 360° = 2π rad.
• Angular velocity is the rate of change of angular displacement: ω = Δθ / Δt = 2π / T = 2πf. Unit: rad s⁻¹.
• Linear speed: v = rω.
• An object moving in a circle at constant speed is accelerating, because its velocity is changing direction. This centripetal acceleration is directed towards the centre: a = v² / r = rω².
• Centripetal force: the resultant force towards the centre needed to keep an object moving in a circle: F = mv² / r = mrω². The object is not in equilibrium.
• The centripetal force is perpendicular to the velocity, so it does no work and the speed does not change.
• If the available force is less than mv²/r, the object moves outwards from the circular path. Friction provides the centripetal force for a car cornering on a level road; on a banked road (or for a banking aircraft) the horizontal component of the normal reaction (or lift) provides it: N sin θ = mv² / r and N cos θ = mg.
• Conical pendulum: T sin θ = mv² / r and T cos θ = mg, so tan θ = v² / rg.