A gas bubble from an explosion under water oscillates with a time period T, depends upon static pressure p, density of water r and the total energy of explosion E. The expression for the time period T. (where, k is a dimensionless constant) is

Time period,

$T\propto p^a{\rho }^b{E}^c$

or $T=k{p}^a{\rho }^b{E}^c$

k, is a dimensionless constant.

According to homogeneity of dimensions,

LHS = RHS

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

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

Comparing the powers, we obtain

a + b + c = 0

-a -3b + 2c = 0

-2a -2c =1

On solving, we get

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