Circular motion

Learning objectives

Relate angular speed ω to linear speed by v = rω, and state that a particle moving in a circle of radius r has acceleration rω² = v²/r directed towards the centre

Apply Newton's second law radially and vertically to horizontal circular motion, finding tensions, normal reactions or frictional forces for a conical pendulum, a particle on a rotating turntable and a vehicle on a banked track

Derive the condition tan θ = v²/(rg) for a vehicle to round a banked bend at speed v with no sideways friction, and extend the analysis to include limiting friction to find the maximum and minimum safe speeds

Analyse motion in a vertical circle by combining conservation of energy, which links speed to height, with Newton's second law along the radius, which gives the tension or normal reaction at any angle

State and use the condition for a particle on the inside of a circular track or on the end of a light inextensible string to complete a full vertical circle, namely that the speed at the top satisfies v² ≥ rg, and contrast this with a light rod, for which the condition is only v ≥ 0 at the top

Find the point at which a particle sliding on the outside of a smooth sphere or circular arc leaves the surface, by setting the normal reaction to zero and solving the resulting equation for the angle