Q. A gas bubble from an explosion under water oscillates with a time period $T$, depends upon static pressure $p$, density of water $\rho$ and the total energy of explosion $E$. Find the expression for the time period $T$.(where, $k$ is a dimensionless constant)
A) $T = kp^{- 5/6}\rho^{1/2}E^{1/3}$
B) $T = kp^{- 4/7}\rho^{1/2}E^{1/3}$
C) $T = kp^{- 5/6}\rho^{1/2}E^{1/2}$
D) $T = kp^{- 4/7}\rho^{1/3}E^{1/2}$
Explanation
Time period $T \propto p^{a}\rho^{b}E^{c}$ Or, $T = kp^{a}\rho^{b}E^{c}$ $k,$ is a dimensionless constant. According to homogeneity of dimensions, LHS=RHS $\therefore\ \ \ \lbrack T\rbrack = \left\lbrack ML^{- 1}T^{- 2} \right\rbrack^{a}\left\lbrack ML^{- 3} \right\rbrack^{b}\left\lbrack ML^{2}T^{- 2} \right\rbrack^{c}$ $\lbrack T\rbrack = \left\lbrack M^{a + b + c} \right\rbrack\left\lbrack L^{- a - 3b + 2c} \right\rbrack\lbrack T^{- 2a - 2c}\rbrack$ Comparing the powers, we obtain $a + b + c = 0$ $- a - 3b + 2c = 0$ $- 2a - 2c = 1$ On solving, we get $a = - \frac{5}{6},b = \frac{1}{2},c = \frac{1}{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.