Question

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 value of a, b, c.

Answer 1

$T\propto p^a{d}^b{E}^c$

$p\to M{L}^{-1}{T}^{-2}$

$d\to M{L}^{-3}$

$E\to M{L}^{-2}{T}^{-2}$

$t\propto$$(M{L}^{-1}{T}^{-2}{)}^a$$(M{L}^{-3}{)}^b$$(M{L}^{-2}{T}^{-2}{)}^c$

$t\propto M^{a+b+c}$$L^{a-3b+2c}$$T^{-2a-2c}$

since, $a+b+c=0$

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

$a=3b+2c$

$\therefore (3b+2c)+b+c=0$

$4b+3c=0⟹b=\frac{-3c}{4}$

$-2a-2c=1$

$\therefore a=\frac{-(1+2c)}{2}$

So, $-\frac{1+22c}{2}-\frac{3c}{4}+c=0⟹c=\frac{2}{3}⟹a=\frac{-7}{6}⟹b=\frac{-1}{2}$