Question

The air bubble formed by explosion under water performs oscillations with a time period $T$ which depends on pressure $P$, density $d$ and energy of explosion $E$. Establish a formula for the time period $T$ by dimensional analysis

Answer 1

$T\propto P^a{d}^b{E}^c$

$P=\left[M\right]{\left[L\right]}^{-1}{\left[T\right]}^{-2}$

$d=\left[M\right]{\left[L\right]}^{-3}$

$E=\left[M\right]{\left[L\right]}^2{\left[T\right]}^{-2}$

$T={\left[M\right]}^{(a+b+c)}{\left[L\right]}^{(-a-3b+2c)}{\left[T\right]}^{(-2a-2c)}$

$a+b+c=0$

$a+c=-b$

$-a-3b+2c=0$

$-2a-2c=1$

$2b=1$

$b=\frac{1}{2}$

$a+c=-\frac{1}{2}and2c-a=3b=\frac{3}{2}$

$3c=1$

$c=\frac{1}{3}anda=-\frac{5}{6}$

$T=P^{-\frac{5}{6}}{d}^{\frac{1}{2}}{E}^{\frac{1}{3}}$