Circular Motion — Quick Summary

Mr Toogood's Physics · Periodic motion

AQA 3.6.1.1
ω = v/r = 2πf
Angular speed
v = ωr
Linear speed
a = v²/r = ω²r
Centripetal accel.
F = mv²/r = mω²r
Centripetal force

Key definitions

  • Period, T — time (s) to complete one full revolution.
  • Frequency, f — revolutions per second (Hz); f = 1/T.
  • Angular displacement, θ — angle in radians swept in time t.
  • Angular velocity, ω — rate of change of angular displacement (rad s⁻¹), ω = 2π/T = 2πf.

Linking angular & linear motion

For a point at radius r, the linear (tangential) speed is the circumference travelled per period:

v = 2πr/T

Combining this with ω = 2π/T gives the key link between angular and linear speed:

v = ωr
Radians, not degrees: all angular quantities in this topic use radians. One full revolution = 2π rad.

Centripetal acceleration & force

Velocity vector diagram used to derive centripetal acceleration

Velocity direction changes continuously — this change is the acceleration.

Even at constant speed, an object moving in a circle has changing velocity (its direction keeps changing), so it accelerates. Combining δθ=vδt/r with δθ=δv/v gives:

a = v²/r = ω²r

Applying Newton's 2nd law, F=ma, gives the resultant force needed to maintain the circular path:

F = mv²/r = mω²r
Not a new force: "centripetal force" is the name for whatever resultant force points toward the centre — gravity (satellites), tension (a whirled mass), friction (cornering car), electrostatic attraction (orbiting charges). Without it, Newton's 1st law means the object flies off tangentially.

Exam essentials

Equivalent forms

  • v = 2πr/T = ωr
  • ω = 2π/T = 2πf
  • a = v²/r = ω²r
  • F = mv²/r = mω²r

Direction

  • Centripetal acceleration & force point towards the centre of the circle at every instant.
  • Velocity is always tangential — perpendicular to the radius.

Common slips

  • Don't draw "centripetal force" as an extra arrow on a free-body diagram — label the real force providing it instead.
  • Keep ω in rad s⁻¹ throughout; don't mix with rev/min or degrees.
  • On F–r or a–r graphs, remember the r vs 1/r relationships depend on whether ω or v is held constant.