Problem 51 - Master Resource Book in JEE Main Physics

Question

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 \(E\). The expression for the time period \(T\). (where, \(k\) is a dimensionless constant) is (a) \(T=k p^{-5 / 6} \rho^{1 / 2} E^{1 \phi}\) (b) \(T=k p^{-4 / 7} \rho^{t / 2} E^{1 / s}\) (c) \(T=k p^{-5 / 6} \rho^{1 / 2} E^{\prime / 2}\) (d) \(T=k p^{-4 \hbar} \rho^{1 / 2} E^{1 / 2}\)

Step-by-Step Solution

Answer

The correct expression is (a) \( T = k p^{-5/6} \rho^{1/2} E^{1/3} \).

Step 1: Dimensional Analysis

To determine the correct expression for the time period \(T\), we will perform dimensional analysis. We know the dimensions of time \([T] = T\), pressure \(p\) as \([M L^{-1} T^{-2}]\), density \(\rho\) as \([M L^{-3}]\), and energy \(E\) as \([M L^2 T^{-2}]\).

Step 2: Setup Dimensional Equation

Assume the time period \(T\) depends on pressure \(p\), density \(\rho\), and energy \(E\) as:\[ T = k p^a \rho^b E^c \]where \(a\), \(b\), and \(c\) are constants to be determined. The dimensions are:\[ [T] = [M L^{-1} T^{-2}]^a [M L^{-3}]^b [M L^2 T^{-2}]^c \]

Step 3: Equate Dimensions

Equating the dimensions on both sides, we have:For \(M\): \(0 = a + b + c\) For \(L\): \(0 = -a - 3b + 2c\) For \(T\): \(1 = -2a - 2c\)

Step 4: Solve Dimensional Equations

Solve the equations derived from dimensional analysis:1. From \(1 = -2a - 2c\), we find \(a + c = -1/2\).2. From \(a + b + c = 0\), we substitute \(c = -1/2 - a\) into \(a + b + c = 0\) to get \(b = 1 - a\).3. Substitute \(c = -1/2 - a\) into \(-a - 3b + 2c = 0\), leading to \(-a - 3(1-a) + 2(-1/2 - a) = 0\). Simplify and solve to find values for \(a\), \(b\), and \(c\).The solutions are \(a = -5/6\), \(b = 1/2\), \(c = 1/3\).

Step 5: Determine Correct Option

Using the values \(a = -5/6\), \(b = 1/2\), \(c = 1/3\), substitute back into the expression:\[ T = k p^{-5/6} \rho^{1/2} E^{1/3} \]Hence, the correct expression for the time period \(T\) is option (a):\[ T = k p^{-5/6} \rho^{1/2} E^{1/3} \]

Key Concepts

Oscillation Time Period

Oscillation time period is the duration for one complete cycle of oscillation. For a gas bubble in an underwater explosion, this can give us insight into the dynamics of the explosion. This is an important aspect because it helps us understand how the product of an explosive reaction behaves underwater, where forces like pressure and density play significant roles.
The period of oscillation, denoted as \( T \), is usually influenced by three major factors: pressure, density, and the energy of the explosion. In essence, we're assessing how long it takes for a bubble, caused by combustion or explosion, to oscillate in size due to these influences. This correlation can be used to estimate unknown parameters by measuring observable quantities, such as the time period of the bubble's oscillation.

Pressure Dependence

Pressure in a liquid medium like water affects the time period of oscillation of an underwater gas bubble. Pressure is typically defined as force per unit area, and it exerts a compressive force on the bubble from all sides.
When analyzing the relation, we find that as pressure increases, the time period of oscillation diminishes. This inverse relationship is reflected in our expression for the time period, \( T = k p^{-5/6} \), which indicates that the time period decreases when pressure increases. This happens because higher pressure compresses the bubble more, causing quicker oscillations.

Density Effects

Density is another critical factor affecting the oscillation of a gas bubble. In this context, density refers to the mass per unit volume of water that surrounds the bubble. Higher density means more mass around the bubble, which affects its ability to expand and contract.
The relationship between density and time period is found through dimensional analysis, which shows a direct yet moderated dependence: \( T = k \rho^{1/2} \). This implies that the time period increases with density, but not as strongly as it does with pressure. This is because a denser medium adds more 'inertia' or resistance to the bubble's expansion and contraction cycles.

Energy of Explosion

The energy of the explosion is the force driving the formation and initial expansion of the gas bubble. It's the total energy released when the explosion occurs under water. This energy is crucial because it dictates the potential size and vigor of the bubble’s motion.
In the expression for time period, the explosion energy \( E \) has a direct relationship: \( T = k E^{1/3} \). This suggests that as more energy is imparted to the explosion, the bubble has a more significant force of expansion, increasing the time period of oscillation. It's a proportionality that indicates energy plays a vital role in defining how oscillations unfold, though the effect is less pronounced compared to pressure and density.