Moving in a circle

Same speed, changing velocity

A cyclist rides round a circular track, never changing speed. Is she accelerating? Yes. Velocity has a direction, and her direction changes all the time. A change of velocity is an acceleration.

Uniform circular motion

Motion round a circle at a steady speed is called uniform circular motion.

• The velocity at any moment points along the tangent, the edge of the circle.
• The change of velocity, and so the acceleration, points towards the centre.

Let go of a stone whirled on a string, and it flies off along the tangent.

Speed, period and frequency

The time for one turn is the period, $T$. In one turn the object goes once round the circle, a distance $2\pi r$. So

The number of turns each second is the frequency, $\dfrac{1}{T}$.

You should find that the speed readout does not change as the object goes round, while the velocity arrow keeps turning. A bigger circle in the same time means a higher speed.

Watch out. 'Uniform' means the speed is steady. The velocity is not steady, because its direction changes.

A worked example

A ceiling fan blade is $0.6\ \text{m}$ long and turns $4$ times a second. A runner goes round a track of radius $50\ \text{m}$ in $80\ \text{s}$. Find each speed.

1. Fan tip. $T = 0.25\ \text{s}$, so $v = \dfrac{2\pi \times 0.6\ \text{m}}{0.25\ \text{s}} = 15.1\ \text{m/s}$.
2. Runner. $v = \dfrac{2\pi \times 50\ \text{m}}{80\ \text{s}} = 3.9\ \text{m/s}$.
3. So the fan's tip moves about four times as fast as the runner.

Practise with new numbers

Try it in Python

The program works out both speeds.

Remember these

• Steady speed round a circle is still an acceleration, because the direction changes.
• Velocity: along the tangent. Acceleration: towards the centre.
• $v = \dfrac{2\pi r}{T}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Why is uniform circular motion called accelerated motion?

2. A stone whirled on a string is let go. Which way does it fly?

3. A wheel of radius $0.35\ \text{m}$ turns once every $0.5\ \text{s}$. How fast does a point on its rim move?

4. What is the frequency of a fan that takes $0.2\ \text{s}$ for one turn?