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A gas bubble, from an exlosion under wat...

A gas bubble, from an exlosion 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 `'E'` is the total energy of the explosion. Find the values of `a, b and, c`.

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Let, `T prop P^(a)rho^(b)E^(c)`
Dimensional formula of
`P rarr ML^(-1)T^(-2) , rho-ML^(-3) , E rarr ML^(2)T^(-2)` `:. T^(1) prop (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)`
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