A gas bubble from an explosion under water, oscillates with a period proportional to P^arho^b E^cwhere 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.

A gas bubble from an explosion under water, oscillates with a period p

1.	A gas bubble from an explosion under water, oscillates with a period proportional to P^arho^b E^cwhere 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.
Answer» Solution :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)`.