A gas bubble oscillates with a time period T proportional to `p^(a) d^(b) E^(c)` where P is pressure , d is the density and E is the energy. The values of a,b and C are :

Options

A. `-5//6,1//2,1//3`
B. `5//6,1//3,1//2`
C. `5//6,1//2,1//3`
D. None of these

Answer

A

Explanation

Given that, time, period,
`T prop p^(a)d^(b)E^(c)" "...(i)`
The dimensions of theses quantities are given as
`p=[ML^(-1)T^(-2)], d[Ml^(-3)]`
`E=[ML^(2)T^(-2)]`
In Eq. (i) on writing the dimensions on both sides,
`[M^(0)L^(0)T] prop [ML^(-1)T^(-1)]^(a)[ML^(-3)]^(b) [ML^(2)T^(-2)]^(c)`
`rARr [M^(0)L^(0)T] prop [M^(a+b+c)L^(-a-3b+2c)T^(-2a-2c)]`
On comparing the powers of M,L, T on both sides, we get
`rArr a+b+c=0 " "...(ii)`
`-a-3b+2c=0 " "...(iii)`
`-2a-2c=1 " "...(iv)`
Solving above equations we get value of a, b, and c are `-(5)/(6),(1)/(2) and (1)/(3)`, respectively.