Uniform circular motion

Uniform circular motion is motion on a circle at constant speed. The velocity is along the tangent and keeps changing direction, so the body is accelerated.

The acceleration points towards the centre (centripetal acceleration) and has magnitude a_c = v²/R.

a_c has constant magnitude but its direction keeps turning, always towards the centre, so it is not a constant vector; the constant-acceleration equations do not apply.

Angular speed ω = Δθ/Δt (rad/s); v = R ω and a_c = ω² R.

The time period T is the time for one revolution and the frequency ν = 1/T is revolutions per second; v = 2πRν, ω = 2πν and a_c = 4π²ν²R.

Jogging at 5.0 m/s on a circle of radius 20 m: ω = 0.25 rad/s, one lap takes 2π/0.25 ≈ 25 s, and a_c = 25/20 = 1.25 m/s² towards the centre.

If the speed is changing as well, the acceleration also has a component along the tangent and no longer points at the centre.