Motion in a circle at constant speed, angular speed, centripetal acceleration and the centripetal force that keeps an object on a circular path.

What this dot point is asking

AQA specification point 3.6.1.1 wants you to describe motion in a circle at constant speed, define and use angular speed, link it to linear speed and frequency, and explain that a resultant centripetal force directed towards the centre produces a centripetal acceleration without changing the speed.

Angular speed

Angular speed $\omega$ is the rate at which an object sweeps out angle, measured in radians per second. One complete revolution is $2\pi$ radians, covered in one period $T$.

The relation $v = \omega r$ shows that for a rigid rotating object all points have the same angular speed, but points further from the centre have a larger linear speed. This is why the rim of a wheel moves faster than a point near the axle.

Centripetal acceleration

Even at constant speed the velocity changes direction, so there is an acceleration. By considering the change in the velocity vector over a small time interval, this acceleration points towards the centre of the circle:

The acceleration grows with the square of the speed, which is why rounding a bend twice as fast needs four times the force. The two equivalent forms $\dfrac{v^2}{r}$ and $\omega^2 r$ are related through $v = \omega r$: substitute $v = \omega r$ into $\dfrac{v^2}{r}$ to get $\dfrac{\omega^2 r^2}{r} = \omega^2 r$. Choose whichever form matches the quantity you are given, the linear speed or the angular speed.

Centripetal force

The centripetal force is not a new kind of force. It is provided by an existing force such as tension (a mass on a string), friction (a car on a bend), gravity (a satellite in orbit) or the normal contact force, depending on the situation. If that force is removed (for example the string snaps), the object flies off along a tangent, in a straight line as required by Newton's first law. Because the centripetal force is always perpendicular to the velocity, it does no work and the speed is unchanged; the force only redirects the motion. A worked feel for the size of these forces helps: a passenger of mass $70 \text{ kg}$ on a fairground ride of radius $5.0 \text{ m}$ moving at $10 \text{ m s}^{-1}$ experiences a centripetal force of $\dfrac{70 \times 10^2}{5.0} = 1400 \text{ N}$, about twice their weight, which is felt as a strong inward push from the seat.

Try this

Q1. State the direction of the centripetal acceleration of an object moving in a circle. [1 mark]

• Cue. Towards the centre of the circle.

Q2. A mass on a string is whirled in a horizontal circle of radius $0.80 \text{ m}$ at $3.0 \text{ rad s}^{-1}$. Calculate its linear speed. [2 marks]

• Cue. $v = \omega r = 3.0 \times 0.80 = 2.4 \text{ m s}^{-1}$.

Q3. State what happens to a whirling mass if the string suddenly snaps. [1 mark]

• Cue. It flies off along a tangent in a straight line.