Question

For a particle to move in a circular orbit uniformly, centripetal force is required which infact depends upon mass (m), velocity (v), and radius (r) of the circle. Express centripetal force in terms of these quantities.

Answer 1

According to the provided information, $F\propto m^a{v}^b{r}^c\Rightarrow F=k{m}^a{v}^b{r}^c$
where k is the dimensionless constant of proportionality and a, b, c are the constant powers of m, v, r, respectively.
Now using the principle of homogeneity, comparing the power of like quantities on both the sides, we have
$a=1\left(ii\right)b+c=1\left(iii\right)andb=2$ (iv)
Using (ii), (iii) and (iv), we have $a=1,b=2andc=-1$.
Using these values in (i), $F=k{m}^1{v}^2{r}^{-2}$$\Rightarrow F=K\frac{m{v}^2}{r}$ which is the desired relation.