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`.

Options

A. `a=1, b=1, c=2`
B. `a=1, b=2, c=1`
C. `a=(5)/(6), b=(1)/(2), c=(1)/(3)`
D. `a=-(5)/(6), b=(1)/(2), c=(1)/(3)`

Answer

D

Explanation

Given, `Tpropp^(a)d^(b)E^(c)`
We have `["M"^(0)"L"^(0)"T"]=k["ML"^(-1)"T"^(-2)]^(a)["ML"^(-3)]^(b)["ML"^(2)"T"^(-2)]^(c)`
where, `k` is a constant.
On comparing dimensions of similar terms, we have
`["M"^(0)"L"^(0)"T"]=k["M"^(a+b+c)"L"^(-a-3b+2c)"T"^(-2a-2c)]`
On comparing powers of `M` we have
`" "0=a+b+c" "...(i)`
On comparing powers of `L`, we have
`" "0=-a-3b+2c" "...(ii)`
On comparing powers of `T`, we have
`" "1=-2a-2c" "...(iii)`
On solving Eqs. (i), (ii) and (iii), we have
`" "a=-(5)/(6),b=(1)/(2),c=(1)/(3)`