Question

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

Answer 1

Given time period T

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

$T=k{P}^a{d}^b{E}^c$ .....eq(1)

where K is a constant of proportionality and dimensionless quantity.

Inserting the dimensions of Time, pressure, density and Energy in equation (1) we get

$\left[T\right]={\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[M{L}^{-3}\right]}^b{\left[M{L}^2{T}^{-2}\right]}^c$

Equating powers of M ,L,T on both sides we get

$0=a+b+c$ ...(2)

$0=-a-3b+2c$ ..(3)

$1=-2a-2c$ ..(4)

By solving (2),(3),(4)

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