First slide

Dimensions of physical quantities

Question

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

Moderate

A

$v^{3}$
B

$v^{4}$
C

$v^{5}$
D

$v^{6}$

Solution

$\begin{array}{l} m \propto v^{a} ρ^{p} g^{c} \\ M L^{0} T^{0} \propto {(L T^{- 1})}^{a} {(M L^{- 3})}^{b} {(L T^{- 2})}^{c} \\ M L^{0} T^{0} \propto [M^{b} L^{a - 3 b + c} T^{- a - 2 c}] \\ b = 1 - - - - (1) \\ a - 3 b + c = 0 - - - (2) \\ - a - 2 c = 0 - - - (3) \\ o n s o l v i n g w e g e t \end{array}$
$b = 1, c = - 3, a = 6 \Rightarrow m \propto v^{6}$

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