The time period $T$ of oscillation of a gas bubble from an explosion under water is proportional to

Question

The time period $T$ of oscillation of a gas bubble from an explosion under water is proportional to $P^{a} d^{b} E^{c}$ where $P$ is the static pressure, $d$ is the density of water and $E$ is the total energy of explosion. Find the values of $a, b$ and $c$.\

Professor PC Thomas Classes · Answer

(c) : Given $T=k P^a d^b E^c$, where $k$ is a proportionality constant. Substituting the dimensions of $T, P, d$ and $E$, we have \ $ \left[T^1ight]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}ight]^a\left[\mathrm{ML}^{-3}ight]^b\left[\mathrm{ML}^2 \mathrm{~T}^{-2}ight]^c \ $ Equating powers of $\mathrm{M}, \mathrm{L}$ and T on both sides, we have $$ \begin{array}{ll} a+b+c=0 \ -2 a-2 c=1 \end{array} \quad-a-3 b+2 c=0 $$ Solving these equations, we get $a=-\frac{5}{6}, b=\frac{1}{2}$ and $c=\frac{1}{3}$.