Uniform circular motion and centripetal force: WASSCE Physics

A body moving in a circle at steady speed is accelerating, even though its speed never changes. That surprising fact explains why you slide across the seat when a vehicle turns, how a spin dryer works and what holds the Moon in its orbit.

Learning objectives

By the end of this lesson you should be able to:

define angular displacement, angular velocity, period and frequency
relate linear speed to angular velocity, $v = \omega r$
explain why a body in uniform circular motion is accelerating
calculate centripetal acceleration and centripetal force

Describing circular motion

Definition

Uniform circular motion is motion in a circle at constant speed.

Quantity	Symbol	Meaning	Unit
Angular displacement	θ	angle turned through	radian (rad)
Angular velocity	ω	rate of change of angle, $\omega = \dfrac{\theta}{t}$	rad s⁻¹
Period	$T$	time for one complete revolution	s
Frequency	$f$	revolutions per second, $f = \dfrac{1}{T}$	Hz

One complete revolution is $2\pi$ radians (360°), so

\omega = \dfrac{2\pi}{T} = 2\pi f

Definition

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. $2\pi\ \text{rad} = 360°$ , so $1\ \text{rad} \approx 57.3°$ .

Linear speed and angular velocity

In one revolution a body travels the circumference $2\pi r$ in time $T$ :

v = \dfrac{2\pi r}{T} = \omega r

Points further from the centre move faster for the same ω: the outside of a turning wheel moves faster than the hub.

Why circular motion is accelerated

A body moving round a circle of radius r: its velocity points along the tangent while the centripetal force and acceleration point towards the centre O

Figure 1. Velocity is along the tangent; acceleration and force point to the centre. WASSCE Tutor diagram.

Velocity is a vector. Although the speed is constant, the direction of motion changes continuously, so the velocity changes. A changing velocity means an acceleration. This acceleration is directed towards the centre of the circle and is called centripetal ("centre-seeking") acceleration:

a = \dfrac{v^2}{r} = \omega^2 r

By Newton's second law, a resultant force must cause it. This centripetal force also acts towards the centre:

F = \dfrac{mv^2}{r} = m\omega^2 r

Common mistake

Centripetal force is not a new kind of force. It is the name for whatever resultant force points to the centre: tension in a string, friction between tyres and road, gravity on a satellite, or the electric force on an electron in an atom. If that force disappears, the body moves off along the tangent in a straight line (Newton's first law), not outwards along the radius.

Worked examples

Worked example

1. A stone of mass 0.2 kg is whirled in a horizontal circle of radius 0.5 m at 3 revolutions per second. Find (a) the angular velocity, (b) the linear speed, (c) the tension in the string.

(a) $\omega = 2\pi f = 2\pi \times 3 = 18.85$ rad s⁻¹.

(b) $v = \omega r = 18.85 \times 0.5 = 9.42$ m s⁻¹.

Worked example

2. A car of mass 800 kg takes a bend of radius 50 m on a level road at 15 m s⁻¹. Find the friction force needed. If the maximum friction is 4000 N, what is the greatest safe speed?

F = \dfrac{mv^2}{r} = \dfrac{800 \times 15^2}{50} = 3600\ \text{N}

\begin{aligned}F &= \dfrac{mv^2}{r} \\ &= \dfrac{800 \times 15^2}{50} \\ &= 3600\ \text{N}\end{aligned}

Greatest speed: $4000 = \dfrac{800v^2}{50} \Rightarrow v^2 = 250 \Rightarrow v = 15.8$ m s⁻¹.

Above this the tyres cannot supply enough centripetal force, and the car skids outwards along a tangent.

In Sierra Leone

A spin dryer or the spinning drum of a washing machine uses this idea: the drum wall pushes the clothes towards the centre, but the water passes through the holes, which supply no force. With nothing to keep it moving in a circle, the water flies off along tangents and leaves the clothes.

Applications

Vehicles on bends: friction provides the centripetal force; roads are banked to help.
Spin dryers and centrifuges: separate water from clothes, cream from milk, blood cells from plasma.
Satellites and the Moon: gravity provides the centripetal force.
Cyclists lean inwards when turning.
"Wall of death" riders and fairground rides.

Key points

Uniform circular motion: constant speed, changing velocity, so it is accelerated.
$\omega = \dfrac{2\pi}{T} = 2\pi f$ , $v = \omega r$ .
Centripetal acceleration $a = \dfrac{v^2}{r} = \omega^2 r$ , towards the centre.
Centripetal force $F = \dfrac{mv^2}{r}$ , provided by tension, friction, gravity or another real force.
If the force stops, the body moves off along the tangent.

WASSCE examination focus

WAEC tests this

Expect to explain why a body moving in a circle at constant speed is accelerating, to define angular velocity and the radian, and to calculate centripetal force or tension for a stone on a string or a car on a bend. Objective items often ask for the direction of centripetal acceleration (towards the centre) or what happens when a string breaks (moves along the tangent).

Quick check

In which direction does the centripetal acceleration act?
A wheel makes 120 revolutions per minute. Find its angular velocity.
What provides the centripetal force for a car turning on a flat road?

Answers: 1. Towards the centre of the circle. 2. $f = 2$ Hz, $\omega = 2\pi \times 2 = 12.6$ rad s⁻¹. 3. Friction between the tyres and the road.

Uniform circular motion and centripetal force