Problem 60 - Master Resource Book in JEE Main Physics

Question

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes \((a)\), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion A bubble comes from the bottom of a lake to the top. Reason Its radius increases.

Step-by-Step Solution

Answer

Both Assertion and Reason are true and Reason is the correct explanation. So, the answer is (a).

Step 1: Identify the Assertion

The Assertion states: 'A bubble comes from the bottom of a lake to the top.' This implies that we are observing the behavior of a bubble as it rises through a column of water, from the bottom of a lake to the surface.

Step 2: Identify the Reason

The Reason given is: 'Its radius increases.' This suggests that the radius of the bubble becomes larger as it ascends from the bottom to the top of the lake.

Step 3: Analyze the Physics of the Assertion

As a bubble rises from the bottom of the lake to the surface, the pressure surrounding the bubble decreases. According to Boyle's Law, which states that pressure is inversely proportional to volume for a given amount of gas at constant temperature, the volume of the bubble will increase as pressure decreases.

Step 4: Connect the Reason to the Assertion

Given that volume increases with decreasing pressure (as per Boyle's Law), the radius of the bubble must also increase to accommodate the larger volume, as the bubble maintains its shape as a sphere. Hence, the Reason that its radius increases supports the phenomenon described in the Assertion that the bubble ascends to the top of the lake.

Step 5: Determine the Correct Relationship Between Assertion and Reason

Both the Assertion and the Reason are true. Additionally, the Reason effectively explains why the bubble's radius increases as it moves from the bottom to the top of the lake. Thus, the Reason is the correct explanation of the Assertion.

Key Concepts

Boyle's Law

Boyle's Law is a fundamental principle in physics that describes how the pressure and volume of a gas relate to each other. According to this law, at a constant temperature, the pressure of a gas is inversely proportional to its volume. This can be mathematically represented by the equation: \[ P_1V_1 = P_2V_2 \]Where \( P_1 \) and \( V_1 \) are the initial pressure and volume, while \( P_2 \) and \( V_2 \) are the final pressure and volume. Simply put, if a gas's volume doubles, its pressure is halved, assuming no temperature change. Boyle's Law is a critical concept in understanding various scientific phenomena, such as how balloons expand when taken to higher altitudes or why your ears pop during a plane ascent.

Physics Concepts

Physics concepts often involve understanding how physical properties like pressure, volume, and temperature interact. These properties are crucial for explaining various natural phenomena and everyday occurrences. In the case of a rising bubble in a lake, understanding how pressure decreases as the bubble rises helps to predict changes in its volume and consequently its radius. Key concepts that relate to this scenario include:

• **Pressure:** the force exerted by a substance per unit area on another substance. It decreases with altitude in fluid environments like water.
• **Volume:** the amount of space occupied by an object. In the context of gases, volume changes inversely with pressure when temperature is constant, as described by Boyle's Law.
• **Buoyancy:** the upward force experienced by objects submerged in fluid, which helps a bubble rise to the surface.

Grasping these concepts writes the narrative for many phenomena beyond just the bubble in the lake, including weather patterns and engineering applications.

Pressure and Volume Relationship

The relationship between pressure and volume is a cornerstone in both chemistry and physics. It describes how changes in pressure affect the volume of gases. When a bubble rises from the bottom of a lake to the top, this relationship is at play. As a bubble ascends:

• The water pressure around it decreases because water pressure is higher at the bottom due to the weight of the water above.
• According to Boyle's Law, as pressure decreases, the volume of the bubble increases.
• The increase in volume means the bubble's molecules have more space, resulting in an increase in its radius as it reaches the surface.

Understanding this pressure-volume relationship helps explain why and how the bubble grows as it rises, and is key to numerous applications, including diving physiology and the design of pressure vessels.