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 rarr ML^(-1) T^(-2), rho rarr ML^(-3), E rarr ML^(2) T^(-2)`
`therefore T^(1) alpha (ML^(-1) T^(2))^(a) (ML^(-3))^(b) (ML^(2)T^(2))^(c )` Comparing power 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)`
`rArr c= (1)/(3) and a = -(1)/(3)-(1)/(2)= -(5)/(6)`
`therefore` The values of a, b, c are respectively, `-(5)/(6), (1)/(2), (1)/(3)`