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A gas bubble from an explosion under wat...

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`.

Text Solution

Verified by Experts

`T propto P^(a)p^(b)E^(c )`
`implies [T]=[P]^(a)[p]^(b)[E]^(c )`
` implies [M^(0)L^(0)T]=[ML^(-1)T^(-2)]^(a)[ML^(-3)]^(b)[ML^(2)T^(-2)]^(c )`
`[M^(0)L^(0)T]= [M^(a+b+c )L^(-a-3b+2c)T^(-2a-2c)]`
On comparing individual dimensions we can write the following equations :
`a+b+c=0" ""............."(i)`
`-a-3b+2c= 0" "".........."(ii)`
`-2a -2c= 1" ""..........."(iii)`
On solving above equations we get the following :
`a= -5"/"6, b= 1"/"2, " and "c= 1"/"3`.
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