Question

why is a force needed to keep a body moving over a circular path find a formula for a force acting on a body moving with uniform speed over a circular path

Answer 1

I want to remind you Newton's law of motion , according to law " body changes the direction or move in accelerating motion when any external force act on it .

Now when a body moving over a circular path with uniform speed , body moves in accelerating motion due to change in direction of velocity of body every point of trajectory of motion , due to this , a centripetal force act on body towards centre of circular path.

Now you can say me , if a force act on you and how you have to balance this force , then what you have to do ?
Of course you have to apply same amount of force but in opposite direction .
Same here , a centrifugal force act on body for balancing its motion in circular path.

Let u is the uniform speed of object of mass m moving in circular path of radius r
Then, F = ma
But in circular path a = u²/r [ a is centripetal acceleration ]
so, F = mu²/r

Answer 2

When a body moves in circular path , it has uniform speed but its velocity changes continuously so that the motion is accelerated .

By Newton's First law and Second laws of motion we know that  unbalanced force produces acceleration.
Since velocity of a body is changing in circular motion , there must be a non zero acceleration causing the change in velocity.


Derivation:
Let us consider a body with mass m moving in circular path with uniform speed v.
As we already know that a force is needed  to keep the body accelerated   to produce circular motion.

⇒F=maThis force is called as Centripetal force which always acts towards centre of circleCentripetal force=F=mxa 

Fc=mxv²/r

where centripetal acceleration =a= v²/r