Question
The centripetal force F acting on a particle moving uniformly in a circle may depend upon mass (m), velocity (u) and radius (r) of the circle. Derive the formula for F using the method of dimensions.
Solution
Let F = K ma vb rc ... (i)
where,
K is a dimensionless constant of proportionality and
a,b c are the powers of m,v and r respectively to represent F.
Now, writing the dimensions of various quantities in (i), we get
[M1 L 1 T-2] = Ma [LT-1]b Lc
= Ma Lb T--b Lc
= Ma Lb+c T-b
On applying the principle of homogeneity of dimensions, we get
a = 1 ,
b+c = 1 ... (ii)
-b = -2 or b = 2
From equation (ii),
c = 1-b = 1-2 = -1
Putting these values in (i), we get
F = K m1 v2 r-1 
This is the required relation.
