Circular motion

Objective. Resolve radial acceleration and forces.

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Learn: the key idea

Circular motion analysis resolves the net force towards the centre of a circular path as the centripetal force, F = mv²/r = mω²r, where ω is angular speed. Even at constant speed, direction constantly changes, so there is always a centre-directed acceleration a = v²/r. For a car of mass m rounding a flat curve of radius r at speed v, friction alone provides the centripetal force, giving the maximum safe speed v = √(μgr). On a banked curve, the normal force also contributes a horizontal component, allowing higher speeds without relying on friction.

Worked example

A 0.5 kg mass moves in a horizontal circle of radius 2 m at 4 m/s; use circular motion formulas to find the centripetal force required.

1 $Centripetal force formula: F = mv²/r$
2 $Substitute values: F = 0.5 \times 4²/2$
3 $Compute numerator: 0.5 \times 16 = 8$
4 $Divide by radius: F = 8/2 = 4$

Answer: 4 N

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Common mistake. Treating centripetal force as an extra outward force to add to a free-body diagram; it is not a separate force but the name for the net inward resultant of the real forces already acting.

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