If $v$ stands for velocity of sound, $E$ is elasticity and $d$ the density, then find $x$ in the equation $v = (d / E)^{x}$

1

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Solution

The correct option is C $- 0.5$
We have to solve this problem using dimension.
The dimension of $v$ is $[v] = [L T^{- 1}]$
Density $d =$ mass/volume so $[d] = [M] / [L^{3}] = [M L^{- 3}]$
Elasticity $E =$ stress/strain, where strain is dimensionless
So dimesionally, $E =$ stress $=$ force/area or $[E] = [M L T^{- 2}] / [L^{2}] = [M L^{- 1} T^{- 2}]$
Putting the dimension of each quantity in given equation,
$[L T^{- 1}] = {([M L^{- 3}] / [M L^{- 1} T^{- 2}])}^{x} = ([L^{- 2} T^{- 2}])^{x}$
By the principle of homogeneity of dimension, equating the power,
$1 = - 2 x$ or $x = - 1 / 2$

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If v stands for velocity of sound, E is elasticity and d the density, then find x in the equation v=(d/E)x

If $v$ stands for velocity of sound, $E$ is elasticity and $d$ the density, then find $x$ in the equation $v = (d / E)^{x}$