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)`
`:. T^(1)alpha (ML^(-1)T^(-2))^(a) (ML^(-3))^(b)(ML^(2)T^(-2))^(c )`
Comparing powers 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)=-(1)/(2), (-a+2c=3xx(1)/(2))/(3c=1)`
`implies c = (1)/(3) ` and `a = -(1)/(3)-(1)/(2)= -(5)/(2)`
`:. ` The values of a,b,c are respectively , `-(5)/(6), (1)/(2), (1)/(3)`