Question

A gas bubble from an explosion under water oscillates with a period '$T$' which depends on $p^a,d^b,E^c$ where $p$ is static pressure, $d$ is density of water, $E$ is the total energy of the explosion. The values of $a,b$ and $c$ respectively are:

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

Answer 1

The correct option is B $\frac{-5}{6},\frac{1}{2},\frac{1}{3}$

$T\propto P^{\alpha }{d}^b{E}^c$
$M^0{L}^0{T}^1=\left(M^1{L}^{-1}{T}^{-2}{)}^a\right(M^1{L}^{-3}{)}^b(M^1{L}^2{T}^{-2}{)}^c$

Comparing dimensions on both sides
$a+b+c=0$
$-a-3b+2c=0$
$-2a-2c=1$
Solving 3 Equation mutually, We get

$\Rightarrow a=-\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$