Question

A gas bubble formed from an explosion under water oscillates with a period $T$ proportional to $P^a{d}^b{E}^c$, where $P$ is pressure, $d$ is the density of water and $E$ is the total energy of explosion. The value of $a,b,c$ are

• A. $a=1b=1,c=2$
• B. $a=1,b=2,c=1$
• C. $a=\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$
• D. $a=-\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$

Answer 1

Answer: (D)

Every equation relating Physical quantities should be in dimensional balance.
Given $T\propto p^a{d}^b{E}^c$
Since, only similar quantities can be equated, therefore dimensions of the terms on both sides of the equation must be same.
Hence, we have Dimensions of $=\left[M^o{L}^o{T}^1\right]$
Dimensions of pressure $P=\left[M{L}^{-1}{T}^{-2}\right]$
Dimensions of density $d=\left[M{L}^{-3}\right]$
Dimensions of energy $E=\left[M{L}^2{T}^{-2}\right]$
$\therefore$ We have
$\left[M^o{L}^o{T}^1\right]=k{\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[M{L}^{-3}\right]}^b{\left[M{L}^2{T}^{-2}\right]}^c$
where, $k$ is a constant.
Comparing dimensions of similar terms, we have
$\left[M^o{L}^o{T}^1\right]=k\left[M^{a+b+c}{L}^{-a-3b+2c}{T}^{-2a-2c}\right]$
Comparing powers of $M$, we have $0=a+b+c\dots$(i)
Comparing powers of $L$, we have $0=-a-3b+2c\dots$(ii)
Comparing powers of $T$, we have $1=-2a-2c\dots$(iii)
Solving Eqs. (i), (ii) and (iii), we have
$a=-\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$