Question

Assume that the mass M of the largest stone that can be moved by a flowing river depends upon the velocity v, the density of water d and the acceleration due to gravity g. Deduce the relationship between these quantities by dimensional analysis method.

Answer 1

Let V is the volume, d is the density and g

Let $M\propto V^a{d}^b{g}^c$

Now,

$M=k{V}^a{d}^b{g}^c$

$\left[M\right]=\left[V{]}^2\right[d{]}^b[g{]}^c$

$M^1{L}^0{T}^0=\left(L^1{T}^{-1}{)}^a\right(M^1{L}^{-3}{)}^b(L^1{T}^{-2}{)}^c$

$M^1{L}^0{T}^0=M^b{L}^{a-3b+c}{T}^{-a-2c}$

Comparing the powers

$b=1$

$a-3b+c=0$

$a-3\left(1\right)+c=0$

$a+c=\mathrm{3......}\left(1\right)$

$-a-2c=\mathrm{0....}\left(2\right)$

Solving (1) and (2)

$a+c=3$

$-a-2c=0$

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$-c=3$

Substitute $c=-3$ in equation (1)

We get $a=6$

Thus

$M=k{V}^{6}dg$

Hence, it can be said that M varies with 6th power of velocity.