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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