Question

Prove centripetal force(F)=mv^2/r

Answer 1

Since F=ma , what you have to show is that for circular motion ,a = v^2/r .

Here is a straight forward derivation which uses calculus.

The position of the particle at any time, as it moves with constant angular velocity "w" and constant radius "r" has the rectangular components;

x = rCos(wt)

y = rSin(wt)

The first derivative gives the velocity components;

Vx = - rwSin(wt)

Vy = rwCos(wt)

The magnitude of the velocity is then;

v = SR[Vx^2 + Vy^2] = rw

The second derivative gives the acceleration components;

ax = - rw^2Cos(wt)

ay = - rw^2Sin(wt)

The magnitude of the acceleration is then;

a = SR[ax^2 + ay^2] = rw^2

You combine this with the velocity eq to elliminate "w" to finally get;

a = v^2/r