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A gas bubble oscillates with a time period T proportional to $P^{a} d^{b} E^{c}$ where P is pressure, d is the density and E is the energy. The values of $a, b$ and $c$ are

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$a = \frac{- 5}{2}, b = \frac{1}{3}, c = \frac{1}{2}$

$a = - \frac{5}{6}, b = \frac{1}{2}, c = \frac{1}{3}$

$a = \frac{3}{2}, b = - \frac{1}{3}, c = \frac{1}{2}$

$a = \frac{3}{2}, b = - 1, c = \frac{1}{2}$

answer is C.

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Detailed Solution

$[T] = {[P]}^{a} {[d]}^{b} {[E]}^{c}$

$M^{0} L^{0} T^{1} = {(M L^{- 1} T^{- 2})}^{a} {(M L^{- 3})}^{b} {(M L^{2} T^{- 2})}^{c}$

comparing dimensions on both sides we get

$\Rightarrow a + b + c = 0$

$- a - 3 b + 2 c = 0$

$- 2 a - 3 b - 2 c = 1$

solving we get

$a = \frac{- 5}{6}, b = \frac{1}{2}, c = \frac{1}{3}$

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