Question

using the methos of dimensions derive an expression for the centripetal force f acting on a particle of mass m mivingv with velocity v in a circle of radius r? ​

Answer 1

Answer:

According to the Question:

$\footnotesize\implies F \propto [m]^x [v]^y [r]^z$

$\footnotesize\implies F =k [m]^x [v]^y [r]^z \:...(i)$

• K = Dimensionless Constant

Now, substituting Dimensions of each physical quantity in equation (I):

$\footnotesize\implies [M^1L^1T^{-2}] =[M^1L^0T^0]^x [M^0L^1T^{-1}]^y [M^0L^1T^0]^z$

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

On comparing the powers of LHS and RHS:

$\implies \bf x = 1$

$\implies - y = - 2$

$\implies \bf y = 2$

$\implies y + z = 1$

$\implies z = 1 - 2$

$\implies \bf z = - 1$

$\footnotesize\implies F = [m]^x [v]^y [r]^z$

$\footnotesize\implies F = [m]^1 [v]^2 [r]^{ - 1}$

$\implies\footnotesize \underline{ \boxed{ \bf \red{ F = \frac{m {v}^{2} }{r} }}}$