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Body Moving in a Circular Path at Constant Speed
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 A-O 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 B-O (which is parallel to A-O) is no longer equal to zero.  
It is now given by  
 
   
So, there has been a change in velocity along the direction A-O (or B-O) which means that there has been an acceleration in the direction A-O (or B-O).  
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 ARROWRIGHT 0 (Δθ ARROWRIGHT 0) then sinΔθ ARROWRIGHT Δθ (in radians) and so the acceleration is given by   
   
which, of course, gives   
   
also, as Δt ARROWRIGHT 0 (Δθ ARROWRIGHT 0) the line B-O approaches the line A-O, 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, electro-static 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.   
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