If a body moves in a circle there must be a resultant force acting towards the centre. This is because the change in direction implies that the body is accelerating.

Deriving the equation is not so simple though...

Key Concepts

Perpendicular force

A force acting at right angles to the direction of motion causes the body to travel in a circular path.

We can model this using a ramp made from a circular cut-out in which the normal reaction force is perpendicular to the velocity. When the force stops, the object moves along the tangent to the circle.

Here is another example:

Relationship between ω and v

ω is the angular velocity = \(2\pi \over T\) v is the speed = \(2\pi r \over T\)

From this we can deduce that \(v = ωr\)

Essentials

When deriving the equation for centripetal acceleration we start by considering a small part of the motion we can deduce that

Fill in the blanks of this justification that the force causing a body to move around a circle at constant speed must be towards the centre.

Word list: 1st, 2nd 3rd, direction , perpendicular, parallel, force, tangential, radial, kinetic, potential, work, acceleration

When a body travels in a circle its is always changing so it must have . According to Newtons law there must be an unbalanced force acting on it.

If the speed is constant then the energy must be constant which means no is being done on the body.

done = x distance moved in direction of so if no work is done the must be to the direction of motion, i.e. .

A mass on a string completes 3 revolutions in 2 seconds.

The angular velocity in radians s^{-1} is:

3 revs is 6π rads so ω = 6π/2

A mass travels in a circle on the end of a spring at constant speed

Which of the following statements is true

centripetal force is the name given to the force that acts towards the centre so doesn't pull the spring out, however the spring pulls the ball in so according to newt 3 the ball must pull the spring out.

A ball moves in a circle of radius 2 m with constant speed 2 ms^{-1}.

The time to complete 1 revolution is:

distance travelled in 1 revolution = 2πr

t = d/v = 2π x 2/2

10° in radians is about

10/360 x 2π approx 6/36 = 1/6

process of elimination, all the others are way too big

A ball on a string moves in a circle as shown

The magnitude of change in velocity is:

Must calculate change vectorially so use Pythagoras.

A mass travels in a circle of radius 2 m with constant speed 3 ms^{-1}.

The acceleration of the mass is:

a = v^{2}/r

If a = v^{2}/r and v = ωr then:

a = ω^{2}r^{2}/r =ω^{2}r

A 2 kg mass travels in a circle of radius 5m with time period π s.

Angular velocity = rads^{-1}

Centripetal acceleration = rads^{-2}

Centripetal force = N

ω = 2π/T

a = ω^{2}r

F = ma

Exam-style Questions

Online tutorials to help you solve original problems

Question 1

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