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)`
`therefore 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 -2 c =1 " "...(3)`
From (3) and (1),
`a+c = - (1)/(2) and b= (1)/(2) = - (1)/(2) , (-a+ 2c = 3 xx (1)/(2) )/( 3c=1)`
`rArr c = (1)/(3) and a = - (1)/(3) - (1)/(2) = -(5)/(6)`
`therefore` The values of a, b, e are respectively, `- (5)/(6) , (1)/(2) , (1)/(3)`.