Question

centripetal force acting on a body moving along a circular path depends on (1) mass of body(m), velocity of body (v),radious circular path(r). using the method dimensions drive the expression
for centripetal force

Answer 1

The centripetal force acting is
$m {v}^{2} \div r$

Answer 2

Answer:

The centripetal force, F acting on a particle moving uniformly in a circle depend upon the mass (m), velocity (v) and radius (r) of the circle.

Explanation:

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.

Derivation of the formula for 'F' using the method of dimensions is based on the principle of comparing the dimesions of the

Fundamental unit and the derived formula.

We assign a coffeicient of proportionality 'k' and compare the powers of the similar base (physical quantity)

After getting the values of the powers of the derived formula, we obtain the formula in terms of the mentioned physical quantitites.

Let $F=k(m)^{x}(v)^{y}(r)^{z}$

Here, $k$ is a dimensionless constant of proportionality. Writing the dimensions of RHS and LHS in Eq. (i), we have

$\left[M L T^{2}\right]=[M]^{x}\left[L T^{-1}\right]^{y}[L]^{z}=\left[M^{x} L^{y+z} T^{-y}\right]$

Equation the powers 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

$\begin{aligned}&F=k m v^{2} r^{-1}=k \frac{m v^{2}}{r} \\&F=\frac{m v^{2}}{r}(\text { where } \mathrm{k}=1)\end{aligned}$