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

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

Answer 1

The correct option is C $a=-\frac{5}{6},b=\frac{1}{2},c=\frac{1}{3}$

$\begin{array}{c}Given,timeperiodisproportinaltoT\alpha p^a{d}^b{E}^c\\ equatingbothsidesofaboveequationensionally,weget\\ \left[M^0{L}^0{T}^{-1}\right]={\left[M^1{L}^{-1}{T}^{-1}\right]}^a{\left[M^1{L}^{-3}{T}^0\right]}^b{\left[M^1{L}^2{T}^2\right]}^c\\ \left[M^0{L}^0{T}^{-1}\right]=\left[M^{a+b+c}{L}^{-a-3b+2c}{T}^{-2a-2c}\right]\\ Now,comparingtheensionsofM,LandTonbothsodes,weget\\ a+b+c=0---------\left(1\right)\\ -a-3b+2c=0-------\left(2\right)\\ -2a-2c=0--------\left(3\right)\\ solving\left(1\right),\left(2\right)and\left(3\right),weget\\ a=-\frac{5}{6},b=\frac{1}{2}andc=\frac{1}{3}\end{array}$

Hence,

option $\left(C\right)$ is correct answer.