A gas bubble from an explosion under water, oscillates with a period proportional to `P^a rho^b E^c` where p is the state pressure, `rho` is the density of water and is the total energy of the explosion. Find the values of a, b, and c.

Explanation

Let , `T alpha P^a rho^b E^c`
Dimensional formula of
`p to ML^(-1) T^(-2) , rho to ML^(-3) , E to ML^(2) T^(-2)`
`:. T^(1) alpha (ML^(-1) T^(-2))^a (ML^(-3))^b (ML^2 T^2)^c`
Comparing powers of M, L and T on both sides.
`a + b + c = 0 " " ....(1)`
`-a - 3b + 2c = 0 " " .........(2)`
`-2a - 2c = 1 " " ......(3)`
From (3) and (1) : `a + c = - 1/2 and b = 1/2`
`a + c = -1/2 , (-a + 2c = 3 xx 1/2)/(3c = 1)`
`implies c = 1/3 and a = -1/3 - 1/2 = -5/6`
`:.` The values of `a, b, c ` are respectively , `-5/6 , 1/2, 1/3.`