Question

assuming that the mass of the largest stone that can be moved by flowing of water depends .....​

Answer 1

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Answer 2

${\bold {\huge {\mathcal{\green {\boxed {\red {DERIVATION}}}}}}}$

The relation between velocity, density and acceleration due to gravity is given as

$V^{6} \: dl \: and \: {g}^{ - 3}$

The given question has mass of the largest stone movable by river flow depending upon three factors, their relation can be ruled out using dimensional analysis.

Given are three dependent factors as Velocity V with density d and acceleration due to gravity as g.

The mass M of the largest stone is directly proportional to the V, d and g

M proportional to V d g, M=

$k \: {V}^{x} d^{y} g ^{z}$

$\mathrm{k}\left[\mathrm{L} \mathrm{T}^{-1}\right]^{\mathrm{x}}\left[\mathrm{M} \mathrm{L}^{-3}\right]^{\mathrm{y}}\left[\mathrm{L} \mathrm{T}^{-2}\right]^{\mathrm{z}}=\mathrm{k}\left[\mathrm{M}^{\mathrm{y}}\right]\left[\mathrm{L}^{\mathrm{x}-3 \mathrm{y}+\mathrm{z}}\right]\left[\mathrm{T}^{-\mathrm{x}-2 \mathrm{z}}\right]$

Now, we have, on comparison

y = 1

x – 3y + z = 0

x – 3×1 +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

Thereby, x + z = 3 => 6 + z = 3 => z = -3.

Therefore, M=

$k \: V ^{6} {d}^{1} g^{ - 3}$

hence, Mass M of the largest stone is proportional to

${V}^{6}$

with

${d}^{1}$

and

${g}^{ - 3}$