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. $v^4$
• C. $v^5$
• D. $v^6$

Answer 1

The correct option is D $v^6$

Let $M$ be proportional to $V^a{P}^b{g}^c$

$\Rightarrow$ $\left[{ML}^0{T}^0\right]={\left[M^0{LT}^{-1}\right]}^a{\left[{ML}^{-3}{T}^0\right]}^b{\left[M^0{LT}^{-2}\right]}^c$

Comparing coefficients

$b=1$

$0=a-3b+c\Rightarrow 3=-2c+c=-c\Rightarrow c=-3$

$0=-a+-2c$

$a=-2c=6$

$\therefore$ $M\alpha V^6$