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# A gas bubble from under water explosion oscillates with time period $T$ propotional to $P^xd^yE^z$ where $P$-pressure, $d$-density, $E$-energy, values of $x,y,z$ are

$(a)\;\frac{1}{2},\frac{1}{3},\frac{1}{2}\quad (b)\;\frac{5}{6},\frac{1}{2},\frac{1}{3}\quad (c)\;\frac{-5}{6},\frac{1}{2},\frac{-1}{3} \quad (d)\;\frac{-5}{6},\frac{1}{2},\frac{1}{3}$

$[T]=P^xd^yE^z$
$\quad=[ML^{-1}T^{-2}]^x[ML^{-3}]^y [ML^2T^{-2}]^z$
$x+y+z=0$
$-x-3y+2y=0$
$-2x-2z=1$
$x=\large\frac{-5}{6}$$\qquad y=\large\frac{1}{2}$$\qquad z=\large\frac{1}{3}$
Hence D is the correct.
edited Jul 11, 2014