Question

Assume that the mass $\left(m\right)$ of the largest stone that can be moved by the flowing river depends only upon the velocity $v$ and the density $\rho$ of the water, along with the acceleration due to gravity $g$. Show that $m$ varies with sixth power of the velocity of the flow.

Answer 1

It is given that mass is proportional to velocity, density and acceleration due to gravity

$m\propto v\rho g$

$\left[m\right]=k{\left[v\right]}^a{\left[\rho \right]}^b{\left[g\right]}^c............\left(1\right)$

Dimensional formula for mass is $\left[M{L}^0{T}^0\right]$

Dimensional formula for velocity is $\left[L{T}^{-1}\right]$

Dimensional formula for density is $\left[M{L}^{-3}\right]$

Dimensional formula for acceleration due to gravity is $\left[L{T}^{-2}\right]$

So, $\left[M{L}^0{T}^0\right]=k{\left[L{T}^{-1}\right]}^a{\left[M{L}^{-3}\right]}^b{\left[L{T}^{-2}\right]}^c$

$M{L}^0{T}^0=M^b{L}^{a-3b+c}{T}^{-a-2c}$

Now, on comparing the powers

$b=1$

$a-3b+c=0ora+c=3$

$-a-2c=0ora+2c=0$

$so,a=6$

$c=-3$

Put values of a,b and c in equation (1)

$m=k{v}^6\rho g^{-3}$

$m\propto v^6$

So, mass varies with sixth power of velocity.