The centripetal force 'F' acting on a particle moving uniformly in a circle depends on the mass 'm' of the particle, its velocity, and the radius of the circle. Derive dimensionally the formula for the centripetal force 'F'. 


Solution:

As given the centripetal force ' F ' depends upon the following factors.

1. mass ' m ' of the body moving in a circle 
2. velocity ' v '
3. Radius  ' r ' of the circle

By expressing the above quantities in relation with F we have

F ∝ `\m^a` , F ∝ `\v^b` , F ∝ `\r^c` 

by combining these relations

F ∝ `\m^a` `\v^b` `\r^c` 

F = K `\m^a` `\v^b` `\r^c` ----------(1)

where K is constant and dimensionless

Now expressing the equation (1) in terms of dimension

[M¹ L¹ T⁻²] = `\[M]^a` `\[L¹ T⁻¹]^b` `\[L¹]^c`

M¹ L¹ T⁻² = `\M^a` `\L^b T^{-b}` `\L^c`

M¹ L¹ T⁻² = `\M^a` `\L^{b+c} T^{-b}` 

by comparing the power 

a = 1  ----------(i)
b + c = 1   ------------(ii)
-2 = -b 
or
b = 2  -----------(iii)
putting b=2 in (i) 
2 + c  = 1 
or
c = -1-----------(iv)


putting the above value of a, b, and c in equation (1)


F = K `\m^1` `\v^2` `\r^{-1}`

F = `\frac {mv^2}{r}` (where K=1)

Hence the required formula for Centripetal Force F.


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