Question

The mass '$M$' of a stone that can be moved by water current depends upon velocity '$V$' of the stream, density of water '$d$' and acceleration due to gravity '$g$' .The relation between the mass and velocity is:

• A. $M\propto V^6$
• B. $M\propto V^2$
• C. $M\propto d^2$
• D. $M\propto V$

Answer 1

The correct option is A $M\propto V^6$

$M\propto v^{\alpha }{d}^{\beta }{g}^{\gamma }$

$M^1{L}^0{T}^0=\left(L{T}^{-1}{)}^{\alpha }\right(M{L}^{-3}{)}^{\beta }(L{T}^{-2}{)}^{\gamma }$$M^1{L}^0{T}^0=M^{\beta }{L}^{\alpha -3\beta +\gamma }{T}^{-\alpha -2\gamma }$

$\Rightarrow \beta =1$
$\alpha -3\beta +\gamma =0$

$\Rightarrow \alpha +\gamma =3$

$-\alpha -2\gamma =0$

$\Rightarrow \alpha =2\gamma$

$\Rightarrow \alpha =6$

Hence, $M\propto V^6$