What is circular motion?
When an object moves along a circle, we say it is in circular motion. Straight-line motion is easy: the object keeps one direction. In a circle, the direction changes all the time.
If the object covers equal arcs in equal times, the motion is uniform circular motion. The speed (how fast) is constant. But the velocity (speed with direction) is not constant, because the direction keeps changing.
Everyday examples: a fan blade, a bicycle wheel, a Ferris wheel, the Moon going round the Earth (nearly a circle).
Period, frequency and units
Period (T) is the time for one full round. Unit: second (s).
Frequency (f) is how many rounds happen in one second. Unit: hertz (Hz), and 1 Hz = 1 round per second.
They are linked: f = 1 / T. If one round takes 0.5 s, then f = 2 Hz.
Machines often use rpm (rounds per minute). To change rpm into Hz, divide by 60. So 120 rpm = 2 Hz.
Angle in radians: a full circle is 2π radians (about 6.28 rad). One radian is the angle whose arc is as long as the radius.
Speed, angular speed and v = ωr
In one period the object covers the whole circumference, 2πr. So the linear speed is
v = 2πr / T
The angular speed ω tells how fast the angle grows: ω = 2π / T = 2πf. Unit: radian per second (rad/s).
Put them together: v = ω × r. Every part of a turning wheel has the same ω, but parts farther from the centre have bigger v.
The velocity arrow is always along the tangent, a line that just touches the circle at the ball.
Acceleration towards the centre
Acceleration means a change in velocity. In circular motion the speed is steady but the direction changes, so the velocity changes. So there IS an acceleration.
It points towards the centre and is called centripetal acceleration (centripetal means centre-seeking):
a = v² / r = ω² r
For the same speed, a tighter turn (smaller r) needs a bigger acceleration. Something must supply the push or pull towards the centre: a string, friction of the road, or gravity. That force is the next lesson.
Try it: measure your own circle
Tie a small soft ball to a string, about 1 m long. Hold the end and swing the ball gently in a flat circle (do this outside, away from people). Ask a friend to count 10 rounds and time them with a phone. Then T = total time ÷ 10, and v = 2πr / T. Now swing faster and see how T gets smaller and v bigger.
Key formulas and definitions
- f = 1 / T (Hz) and T = 1 / f (s)
- v = 2πr / T
- ω = 2π / T = 2πf (rad/s)
- v = ω r
- a = v² / r = ω² r (towards the centre)
Worked examples
1. A ball goes round a circle of radius 2 m in 4 s. Find its speed.
v = 2πr / T = 2 × 3.14 × 2 / 4 = 3.14 m/s.
2. A slow fan blade turns once every 12 s. Find ω.
ω = 2π / T = 6.28 / 12 = 0.52 rad/s.
3. A drum spins at 120 rpm. Find f and T.
f = 120 / 60 = 2 Hz. T = 1 / f = 0.5 s.
4. A car goes round a bend at 5 m/s. The bend has radius 2.5 m. Find the acceleration.
a = v² / r = 25 / 2.5 = 10 m/s², directed towards the centre of the bend.
5. A bicycle wheel has radius 0.35 m. The cycle moves at 7 m/s. Find ω and f of the wheel.
The rim moves at the cycle speed, so v = 7 m/s. ω = v / r = 7 / 0.35 = 20 rad/s. f = ω / 2π = 20 / 6.28 = 3.2 Hz.
6. Two horses on a merry-go-round take 8 s per round. One is at r = 4 m and the other at r = 2 m. Compare speed and acceleration.
Outer horse: v = 2π × 4 / 8 = 3.14 m/s and a = v² / r = 9.86 / 4 = 2.47 m/s². Inner horse: v = 2π × 2 / 8 = 1.57 m/s and a = 2.47 / 2 = 1.23 m/s². Same T, but the outer horse has twice the speed and twice the acceleration.
Common mistakes
- Saying 'constant speed means no acceleration'. Direction changes, so velocity changes and a = v²/r is not zero.
- Drawing the velocity arrow towards the centre. Velocity is along the tangent; only the acceleration points to the centre.
- Using rpm in the formulas without converting. Change to Hz (divide by 60) or rad/s (multiply by 2π/60).
- Mixing up T and f. T is seconds per round; f is rounds per second; f = 1/T.