Consider a small mass, on the end of a light rigid rod, moving in a circular
path at constant speed. 

The magnitude
of the velocity of the body is constant but the direction is
constantly changing. 

This means that, even though the speed is not changing, the
velocity is changing. 

We conclude that a body moving in a circular (or, in fact, any curved path)
is accelerating. 






At any instant, the direction of the velocity is a tangent
to the circular path. 



At time t = 0, the body is at A (an arbitrarily chosen point), and the
magnitude the component of its velocity in the direction AO is zero. 

(By definition, the tangent is at 90° to the radius.) 














At time Δt later, the body has moved to B and
the angular displacement is
Δθ. 



When at B, the
magnitude of the component of the velocity in the direction
BO’ (which is parallel to AO) is no longer equal to zero. 

It is now given by 





So, there has been a change in velocity along
the direction AO (or BO’) which means that there
has been an acceleration in the direction AO (or BO’). 

The magnitude of the acceleration is equal to the
change in velocity divided by the time taken for the change. 

Therefore 



For small
angles, the sine of the angle is very nearly equal the
angle in radians. See here for proof. 



In order
to find the instantaneous value of the acceleration, consider B getting
closer and closer to A. 

This is equivalent to considering smaller and
smaller time intervals. 

As Δt 0
(Δθ 0)
then sinΔθ
Δθ (in radians) and so the acceleration
is given by 



which, of course, gives 



also, as
Δt
0
(Δθ 0) the line
BO’ approaches the line AO, so the direction of the
acceleration is directed towards the centre of the circular path. 

Remembering
that v=rω this gives us two useful equations
to calculate the magnitude of this acceleration: 





Conclusions 

A body
moving at constant speed in a circular path
experiences an acceleration directed towards the
centre of the circular path. 



This acceleration is called a centripetal
acceleration and is provided by a centripetal
force. 



The force might be due to gravity, electrostatic
attraction, the tension is a string etc. 



A centripetal
force does not change the K.E. of the body because it
acts at
90° to the direction of motion. 
