Question

the centripetal force f acting on a particle moving uniformly in a circle may depend upon mass(m) velocity (v) and radius (r) of the circle . derive the formula for f using the method of dimensions

Answer 1

Given :

The quantities on which the centripetal force F acting on a particle moving uniformly in a circle may depends is :

mass  , velocity  and radius  

To Find :

What is the formula for F , the centripetal force using the method of dimensions = ?

Solutions :

Since from the given information we have :

$F \propto (mass)^a.(velocity)^b . (radius)^c$

Or, $F \propto m^a . v^b . r^c$

Or, $F \propto [M]^a [LT^{-1}]^b [L]^c$

Or, $[MLT^{-2}] \propto [M]^a[LT^{-1}]^b [L]^c$

Or, $[MLT^{-2}] \propto [M]^a[L^{b +c}] [T^{-b}]$

On comparing we get :

a = 2 , b + c = 1 and  -b = -2 or b = 2

Putting b = 2 in  b + c = 1 , we get c = -1

So, $F \propto m^1 . v^2. r^{-1}$

Or, $F \propto \frac{mv^2}{r}$

Or, $F = k \frac{mv^2}{r}$

Hence , the formula for F using dimensional method is F =$k \frac{mv^2}{r}$