General Physics Question 28
Question 28
- A gas bubble, from an explosion under water, oscillates with a period $T$ 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 the explosion. Find the values of $a, b$ and $c$.
(1981, 3M)
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Answer:
Correct Answer: 36. $a=\frac{-5}{6}, b=\frac{1}{2}, c=\frac{1}{3}$
Solution:
- $[T]=\left[p^{a} d^{b} E^{c}\right]=\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]^{a}\left[\mathrm{ML}^{-3}\right]^{b}\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]^{c}$
Equating the powers on both sides, we have
$$ \begin{aligned} a+b+c & =0 \ -a-3 b+2 c & =0 \ -2 a-2 c & =1 \end{aligned} $$
Solving these three equations, we have
$$ \begin{aligned} a & =-\frac{5}{6}, b=\frac{1}{2} \ \text { and } \quad c & =\frac{1}{3} \end{aligned} $$
