Question

A gas bubble from an explosion under water oscillates with a period `T` proportional to `P^(a) d^(b) E^(c )`, where `P` is the pressure, `d` is density of water and `E` is the total energy of the explosion. Find the value of a,b and c`.

Answer 1

`T prop P^(a) d^(b) E^(c )`
`[T] prop [ML^(-1) T^(-2)]^(a) [ML^(-3)]^(b) [ML^(2) T^(-2)]^(c )`
`M^(0) L^(0)T^(1) prop M^(a + b + c)L^(-a-3b+2c) T^(-2a - 2c)`
Comparing powers of `M, L` and `T`
`a + b + c = 0`
`- a - 3b + 2c = 0`
`- 2a - 2c = 1`
From (iii), `a + c = - (1)/(2)` in (i), we get `b = (1)/(2)`, putting the value of `b` in (ii)
`-a-3 ((1)/(2)) + 2c = 0 implies - a + 2c = (3)/(2)`
From (iii), `a + c = - (1)/(2)`
`- a + 2c = (3)/(2)`
Adding `(v)` and `(vi)`, `3c = 1 implies c = (1)/(3)`
`a + c = - (1)/(2)`
` a + (1)/(3) = - (1)/(2) implies = - (5)/(6)`
`a = - (5)/(6), b = (1)/(2), c = (1)/(3)`