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 ∝ ma , F ∝ vb , F ∝ rc 

by combining these relations

F ∝ ma vb rc 

F = K ma vb rc ----------(1)

where K is constant and dimensionless

Now expressing the equation (1) in terms of dimension

[M¹ L¹ T⁻²] = [M]a [L¹T⁻¹]b [L¹]c

M¹ L¹ T⁻² = Ma LbT-b Lc

M¹ L¹ T⁻² = Ma Lb+cT-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 m1 v2 r-1

F = mv2r (where K=1)

Hence the required formula for Centripetal Force F.


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