Question

The centripetal force F acting on particles 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.
(A) $F=\dfrac{km{{v}^{2}}}{r}$
(B) $F=\dfrac{kmv}{r}$
(C) $F=\dfrac{km{{v}^{2}}}{{{r}^{2}}}$
(D) $F=\dfrac{k{{m}^{2}}{{v}^{2}}}{{{r}^{2}}}$

Answer 1

This problem of dimensional analysis, in this we can deduce the value from the dimensional formula of these individual elements by finding the degree of dependence of a physical quantity on another. The principle of consistency of two expressions is what guides us to find the equation relating these two quantities. It has its own limitations too.

Complete step by step answer:
Let us write the dependency of force on the given quantities $F\alpha {{m}^{a}}{{v}^{b}}{{r}^{c}}$, removing the proportionality sign we write a constant, $F=k{{m}^{a}}{{v}^{b}}{{r}^{c}}$ where k is the dimensionless constant of proportionality and a, b, c are the powers of m, v and r.
Equating the dimensions on both the sides we get; we need to keep in mind that constant is always dimensionless. Thus,
$[{{M}^{1}}{{L}^{1}}{{T}^{-2}}]={{[M]}^{a}}{{[L{{T}^{-1}}]}^{b}}{{[L]}^{c}}$
Applying the principle of homogeneity of dimensions, we get a = 1, b +c = 1 -b = -2 or, b = 2
Solving we get, c = 1 -b = 1 -2 = -1
Putting the values of a, b and c, in $F=k{{m}^{a}}{{v}^{b}}{{r}^{c}}$ we get $F=\dfrac{km{{v}^{2}}}{r}$.

Thus, the correct option is A.

Note: This method of finding out the relationship between two quantities or checking out whether the given equation is correct or not has some limitations. The quantities which are dimensionless get hollowed out during the use of this method, thus, this is not at all a 100% fool proof method. Also, many times it becomes quite complex. we need to keep in mind that constant is always dimensionless.