If an object of mass m moves in uniform circular motion, a formce F acts on it, whose direction is always towards the centre of the circular path. If F depends on m, speed v of the object and radius r of the circular path, find an expression for F.

Explanation

Let the required expression be `F=am^x v^y r^z`
where a is a dimensionless constant .
x,y, z are powers on m,v and r respectively.
According to the principle of dimensional homogeneity.
`[F]=[m]^x [v]^y [r]^z`
`rArr [M^1 L^1 T^(-2)]=[M]^x [LT^(-1)]^y [L]^z`
`[M^1L^1 T^(-2)]=[M^x L^(y+z) T^(-y)]`
On equating the dimensions on both sides, we have x=1
y+z=1
and y=2
So that z=-1
Thus the expression for F is `F=a mv^2 r^(-1)`
or` F=(amv^2)/r`
Dimensionless constant a, in this case is 1.
`F=(mv^2)/r`
The force F is said to be the centripetal force .