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^(1)L^(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=amv^(2)r^(-1)`
or `F=(amv^(2))/r`
Dimensionless constant a, in this case is 1.
`thereforeF=(mv^(2))/r`
The force F is said to be the centripetal force.