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

To solve this question, we must have the concept of the principle of homogeneity of dimensions. The given question has the mass of the largest stone movable by river flow. Depending upon three factors, their relation can be ruled out using dimensional analysis.

Complete step by step answer:
Given are three dependent factors as Velocity $v$ with density $\rho$ and acceleration due to gravity as $g$ .
Let, the mass M of the largest stone is directly proportional to the $v$ , $\rho$ and $g$ .
i.e.,
$m = k{v^x}{\rho ^y}{g^z}$
Where k is a dimensionless constant.
Now, here we will write all the dimensions of each of the physical quantities.
$m = k{\left[ {L{T^{ - 1}}} \right]^x}{\left[ {M{L^{ - 3}}} \right]^y}{\left[ {L{T^{ - 2}}} \right]^z}$
On multiplication, we get
${M^1}{L^0}{T^0} = k\left[ {{M^y}} \right]\left[ {{L^{x - 3y + z}}} \right]\left[ {{T^{ - x - 2z}}} \right]$
Now, comparing the powers,
$y = 1 \\\ x - 3y + z = 0 \\\$
Now, putting the value of $y$ in the above equation,
$x - 3 \times (1) + z = 0$
$x + z = 3$ -----(1)
And,
$- x - 2z = 0$ -----(2)
Multiply equation 1 with 2 and compare with equation 2,
$2x + 2z = 6 \\\ \- x - 2z = 0 \\\ x = 6 \\\$
Therefore,
$x + z = 3 \\\ \Rightarrow 6 + z = 3 \\\ \Rightarrow z = - 3 \\\$
Therefore, $m = k{v^6}{d^1}{g^{ - 3}}$
Hence, the mass $m$ of the largest stone is proportional to ${6^{th}}$ power of velocity.

Note:

In any type of problems of dimensional analysis where it is given that this quantity depends on so and so…. solve by this method always. It would be easier.
Dimensional homogeneity is the concept where the dimensions of variables on both sides of an equation are the same. An equation could be dimensionally homogeneous but invalid if the equation is also not fully-balanced.