§ 12.1 Kinematics of uniform circular motion
Key ideas
- One radian is the angle subtended at the centre by an arc equal in length to the radius; a full circle is 2π rad.
- Angular displacement θ = s/r. Angular speed ω is the rate of change of angle, the same for every point on a rigid rotating body.
- The linear speed of a point at radius r is v = rω.
Equations
ω = 2π / T = 2πfangular speed from period or frequencyrad s⁻¹
v = r ωlinear speed = radius × angular speedm s⁻¹
Watch out: ω must be in rad s⁻¹, not revolutions per second or degrees. Convert revs to radians (× 2π) before substituting into v = rω or any later formula.
§ 12.2 Centripetal acceleration
Key ideas
- An object moving in a circle at constant speed is still accelerating: its velocity direction changes, pointing the acceleration towards the centre.
- The centripetal force is the resultant force causing this; it is supplied by tension, gravity, friction or a normal force, never a separate "outward" force.
- Because the force is perpendicular to the velocity, it does no work and the speed stays constant.
Equations
a = r ω² = v² / rcentripetal acceleration, directed to the centrem s⁻²
F = m r ω² = m v² / rcentripetal force, directed to the centreN
Fig. 1 · Uniform circular motion: the velocity is always tangential, while the centripetal force and acceleration point inwards along the radius, turning the velocity without changing its size.
Watch out: there is no outward "centrifugal force" on the object. The only real force acts inwards; the outward feeling is the object's inertia resisting the change of direction.