Question

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

• A. $1$
• B. $0.5$
• C. $-0.5$
• D. $-1$

Answer 1

Answer: (C)

We have to solve this problem using dimension.

The dimension of $v$ is $\left[v\right]=\left[L{T}^{-1}\right]$

Density $d=$ mass/volume so $\left[d\right]=\left[M\right]/\left[L^3\right]=\left[M{L}^{-3}\right]$

Elasticity $E=$ stress/strain, where strain is dimensionless

So dimesionally, $E=$ stress $=$ force/area or $\left[E\right]=\left[ML{T}^{-2}\right]/\left[L^2\right]=\left[M{L}^{-1}{T}^{-2}\right]$

Putting the dimension of each quantity in given equation,

$\left[L{T}^{-1}\right]={\left(\right[M{L}^{-3}\left]/\right[M{L}^{-1}{T}^{-2}\left]\right)}^x=\left(\right[L^{-2}{T}^{-2}]{)}^x$

By the principle of homogeneity of dimension, equating the power,

$1=-2x$ or $x=-1/2$