Question

A gas bubble from an explosion under water oscillation with a period proportional to $P^a,d^b,E^c$ where $P$ is the static pressure, $d$ is the density of water and $E$ is the energy of explosion. Then $a,b$ and $c$ respectively are

• A. $-\frac{5}{6},\frac{1}{2},\frac{1}{3}$
• B. $\frac{1}{2},\frac{-5}{6},\frac{1}{3}$
• C. $\frac{1}{3},\frac{1}{2},\frac{-5}{6}$
• D. $1,1,1$

Answer 1

Answer: (A)

$-\frac{5}{6},\frac{1}{2},\frac{1}{3}$

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

Since $P$ has dimension

${\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[M{L}^{-3}\right]}^b{\left[M{L}^2{T}^{-2}\right]}^c$

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

$a+b+c=0$

$=-a+\left(3b\right)(-1)+2c=0$

$-[a+c]2=1$

$a+c=-1/2$

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