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The centripetal force, F acting on a particle moving uniformly in a circle depend upon the mass (m), velocity (v) and radius (r) of the circle. The formula for F using the method of dimensions.
Let F = km a v b r c … (i)
Where, k is the dimensionless constant of proportionality, and a, b, c are the powers of m, v, r respectively.
On writing the dimensions of various quantities in (i), we get
[M 1 L 1 T −2 ] = M a [LT −1 ] b L c
= M a L b T −b L c
M 1 L 1 T −2 = M a L b + c T −b
On applying the principle of homogeneity of dimensions, we get
a = 1,
b= 2,
b + c = 1 …(ii)
From (ii), c = 1 − b = 1 − 2 = −1
On putting these values in (i), we get
F = km 1 v 2 r −1
OR
F = k mv^2 / r
F = k* mrw^2
This is the required relation for centripetal force.
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