Question

Assuming that the mass $m$ of the largest stone that can be moved by a flowing river depends upon the velocity $v$ of the water, its density $\rho$ and the acceleration due to gravity $g$. Then $m$ is directly proportional to

• A. $v^3$
• B. ${\rho }^6$
• C. $g^{-4}$
• D. $v^6$

Answer 1

The correct option is D $v^6$

As per the question we can write
$m\propto v^a{\rho }^b{g}^c$
$\Rightarrow \left[M\right]=\left[L{T}^{-1}{]}^a\right[M{L}^{-3}{]}^b[L{T}^{-2}{]}^c$
$\Rightarrow \left[M^1{L}^0{T}^0\right]=\left[M^b{L}^{(a-3b+c)}{T}^{(-a-2c)}\right]$
On comparing L.H.S and R.H.S we get
$b=1$....(i)
$a-3b+c=0\Rightarrow a+c=3$.....(ii)
and, $-a-2c=0$.....(iii)
On solving (ii) & (iii) we get
$c=-3,a=6$
So, the relation would be $m\propto v^6\rho g^{-3}$