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Question: The largest mass (m) that can be moved by a flowing river depends on velocity (v), density (\(\rho\)...

The largest mass (m) that can be moved by a flowing river depends on velocity (v), density (ρ\rho) of river water and acceleration due to gravity (g). The correct relation is

A

m∝ρ2v4g2m \propto \frac{\rho^{2}v^{4}}{g^{2}}

B

m∝ρv6g2m \propto \frac{\rho v^{6}}{g^{2}}

C

m∝ρv4g3m \propto \frac{\rho v^{4}}{g^{3}}

D

m∝ρv6g3m \propto \frac{\rho v^{6}}{g^{3}}

Answer

m∝ρv6g3m \propto \frac{\rho v^{6}}{g^{3}}

Explanation

Solution

Given, m = mass = [M], v = velocity = [LT−1]\lbrack LT^{- 1}\rbrack, ρ = density = [ML−3]\lbrack ML^{- 3}\rbrack, g = acceleration due to gravity = [LT–2]

By substituting, the dimension of each quantity we can check the accuracy of the formula

m=Kρv6g3m = K\frac{\rho v^{6}}{g^{3}}

⇒[M]=[ML−3][LT−1]6[LT−2]3\Rightarrow \lbrack M\rbrack = \frac{\lbrack ML^{- 3}\rbrack\lbrack LT^{- 1}\rbrack^{6}}{\lbrack LT^{- 2}\rbrack^{3}}

= [M]

L.H.S. = R.H.S. i.e., the above formula is Correct**.**