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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