Uniform circular motion

In uniform circular motion a body goes round a circle at a constant speed. But its direction keeps changing (the velocity always points along the tangent), so its velocity is never constant, and a changing velocity means there is an acceleration.

This acceleration points to the centre of the circle, so it is called centripetal ('centre-seeking'). Its size is a_c = v²/R. It changes only the direction of motion, never the speed.

Why v²/R? Take the body from A to B, turning through a small angle Δθ. The two radii and the chord AB form a triangle, and the two velocity arrows and Δv form a similar triangle with the same angle. So Δv/v = Δs/R, giving Δv = vΔs/R. Divide by Δt: a = v × v/R = v²/R.

Angles in circular motion are measured in radians (θ = arc ÷ radius), which gives a set of simple links:

• Angular speed ω = Δθ/Δt, in rad s⁻¹.
• Period T: the time for one full turn, 2πR/v. Frequency ν = 1/T, in hertz (turns per second).
• One turn is 2π radians, so ω = 2π/T = 2πν.
• Distance along the arc is s = Rθ, so v = Rω, and a_c = v²/R = ω²R.

Example: a ceiling fan's blade tip is 0.5 m from the centre and turns at 300 revolutions per minute. ν = 300/60 = 5 Hz, ω = 2π × 5 ≈ 31.4 rad/s, v = 0.5 × 31.4 ≈ 15.7 m/s, and a_c = ω²R ≈ 986 × 0.5 ≈ 493 m/s², about fifty times g.

Angular speeds compare easily through the period: a watch's hour hand turns once in 12 h, the Earth once in 24 h, so the hand's ω is twice the Earth's.

The centripetal acceleration has a constant size but its direction keeps turning, so it is not a constant vector. If the speed also changes, there is a second part along the tangent, a_t, and the total is √(a_c² + a_t²), no longer pointing at the centre: a_c = 6 m/s² and a_t = 8 m/s² give 10 m/s².