A gas bubble, from an explosion under water oscillates with a period $T$ proportional to $p^{a} d^{b} E^{c}$ where $p$ is static pressure, $d$ is the density of water and $E$ is the total energy of the explosion. Find the values of $a, b a n d c$ .

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

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

$a = - \frac{5}{7}, b = \frac{1}{2}, c = \frac{1}{3}$

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

answer is B.

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Detailed Solution

$[T] = [p^{a} d^{b} E^{c}] = {[{ML}^{- 1} T^{- 2}]}^{a} {[{ML}^{- 3}]}^{b} {[{ML}^{2} T^{- 2}]}^{c}$

Equating the powers of both sides, we have

$a + b + c = 0 \dots (i) - a - 3 b + 2 c = 0 \dots (ii) - 2 a - 2 c = 1 ... (iii)$

Solving these three equations, we have

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

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