The force F acting on a body moving in a circular path depends on mass of the body (m) velocity(v) and radius (r) of the circular path. Obtain the expression for the force by dimensional analysis method `(k=1)`

Explanation

`F prop m^(a)v^(b)r^(c )`, `F=km^(a)v^(b)r^(c )`
where `k` is a dimensionless constant of proportionality. Rewriting the above equation in terms of dimensions and taking `k=1`, we have
`[MLT^(-2)]=[M]^(a)[LT^(-1)]^(b)[L]^(c )=[M^(a)L^(b)T^(c )]`
`[MLT^(-2)]=[M^(a)][L^(b+c)][T^(-b)]`
Comparing the powers of `M,L` and `T` on both sides
`a=1, b+c=1-b=-2`, `2+c=1`, `b=2`
`a=1`, `b=2` and `c=-1`
From the above equation we get `F=m^(a)v^(b)r^(c )`
`F=m^(1)v^(2)r^(-1)` or `F=(mv^(2))/(r )`