The centripetal force F acting on a particle 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.

Options

A. `f=(mv)/(r)`
B. `F=(mv)^(2)/(r)`
C. `F=(mv)^(2)/(r)^(2)`
D. None of these

Answer

B

Explanation

Let `F="k" (m)^x(v)^y(r)^(z)`
Here, `k` is a dimensionless constant of proportinality. Writing the dimensions of RHSand LHS in Eq. (i), we have `[MLT^(-2)]=[M]^(x)[LT^(-1)]^(y)[L]^(z)`
`=[M^(x)L^(y+z)T^(-y)]`
Equaliting the power of M, L and T of both sides, we have
`x=1,y=2`and` y+z=1`
or `z=1-y=-1`
putting the values in Eq. (i), we get
`F="k"mv^(2)r^(-1)="k"(mv^(2))/(r)`
`F=(mv^(2))/(r)`