Physics

# The air bubble formed by explosion under water performs oscillations with a time period $T$ which depends on pressure $P$, density $d$ and energy of explosion $E$. Establish a formula for the time period $T$ by dimensional analysis

##### SOLUTION
$T\propto { P }^{ a }{ d }^{ b }{ E }^{ c }$
$P=\left[ M \right] \left[ L \right] ^{-1}\left[ T \right] ^{-2}$
$d=\left[ M \right] \left[ L \right]^{-3}$
$E=\left[ M \right] \left[ L \right]^2 \left[ T \right] ^{-2}$
$T=\left[ M \right] ^{(a+b+c)}\left[ L \right]^{(-a-3b+2c)} \left[ T \right] ^{(-2a-2c)}$
$a+b+c=0$
$a+c=-b$
$-a-3b+2c=0$
$-2a-2c=1$
$2b=1$
$b=\dfrac{1}{2}$
$a+c=-\dfrac{1}{2 }\quad and\quad2c-a=3b=\dfrac{3}{2}$
$3c=1$
$c=\dfrac{1}{3}\quad and\quad a=-\dfrac{5}{6}$
$T= { P }^{ -\dfrac{5}{6} }{ d }^{ \dfrac{1}{2} }{ E }^{ \dfrac{1}{3} }$

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Subjective Medium Published on 18th 08, 2020
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