Physics

# A gas bubble, from an explosion under water oscillates with a period T proportional to $P^ad^bE^c$, 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.

##### SOLUTION
Given time period T
$T\propto P^ad^bE^c$

$T=kP^ad^bE^c$                                          .....eq(1)
where K is a constant of proportionality and dimensionless quantity.

Inserting the dimensions of Time, pressure, density and Energy in equation (1) we get
$\left [ T \right ]=\left [ ML^{-1}T^{-2} \right ]^a\left [ ML^{-3} \right ]^b\left [ ML^{2}T^{-2} \right ]^c$

Equating powers of M ,L,T on both sides we get
$0=a+b+c$                          ...(2)
$0=-a-3b+2c$                   ..(3)
$1=-2a-2c$                           ..(4)

By solving (2),(3),(4)
$a=\dfrac{-5}{6}$ $b=\dfrac{1}{2}$ and $c=\dfrac{1}{3}$

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Subjective Medium Published on 18th 08, 2020
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