A gas bubble from an explosion under water, oscillates with a time period T proportional to p ^a d ^b E ^c , where p,d, and E are pressure, density and total energy respectively. Find the value of a, b and c.

given,

T ∝ P^a d^b E^c

 = K P^a d^b E^c  ( K being some constant)

putting the dimensions of pressure,density,energy and time

[ T ] = K[M L^-1 T^-2]^a [M L^-3 ]^b [M L² T^-2]^c

 = K [ M ]^a+b+c [ L ]^-a-3b+2c[ T ]^-2a-2c

compairing powers on LHS and RHS

a+b+c = 0  ...........................(1)

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

-2a-2c = 1  ............................(3)

from (3)

 a+c = -1/2

or,     c = -1/2 - a

putting in (1) gives b= 1/2

from (2)

-a -3(1/2) + 2(-1/2 - a) = 0

or,          a= -5/6

from (3)

-2(-5/6) - 2c = 1

or, 5/3 -2c = 1

or, 2c = 5/3 - 1

or, c = 1/3