Problem 58 - 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 hydrogen filled balloon stops rising after it has attained a certain height in the sky. Reason The atmospheric pressure decreases with height and becomes zero when maximum height is attained.

Step-by-Step Solution

Answer

(b) Both Assertion and Reason are true, but Reason is not the correct explanation of the Assertion.

Step 1: Understand the Assertion

The Assertion states that a hydrogen-filled balloon stops rising after it reaches a certain height in the sky. We need to understand that as a balloon rises, the atmospheric pressure outside decreases, while the pressure inside the balloon remains constant. At some point, the buoyant force equals the gravitational force, causing the balloon to stop rising.

Step 2: Evaluate the Reason

The Reason states that atmospheric pressure decreases with height and becomes zero at maximum height. It is true that atmospheric pressure decreases with altitude; however, it never becomes zero, even at very high altitudes.

Step 3: Assess the Truth of the Assertion and Reason

The Assertion is true: a hydrogen-filled balloon stops rising at a certain height due to equalization of forces. The Reason is partially true in that atmospheric pressure decreases with height, but it falsely claims that pressure becomes zero when maximum height is reached.

Step 4: Determine the Correct Option

Since both statements are partially true but the Reason does not correctly explain the Assertion (because the pressure never actually reaches zero), the correct option is (b): Both Assertion and Reason are true, but the Reason is not the correct explanation of the Assertion.

Key Concepts

Buoyant Force

The buoyant force is an essential concept in understanding why objects float or sink in a fluid. It originates from the pressure difference in the fluid between the top and the bottom of the immersed object. This force acts in the upward direction.

• The higher pressure at the bottom of an object submerged in a fluid exerts an upward force.
• This is why objects like balloons rise when the buoyant force is greater than their weight.
• The buoyant force is determined by Archimedes' principle, which states that it is equal to the weight of the fluid displaced by the object.

As a balloon rises, the atmospheric pressure decreases, reducing the buoyant force. Meanwhile, the gravitational force remains constant, and when these two forces equalize, the balloon stops rising.

This balance of forces is crucial in determining the altitude a balloon or any floating object can reach.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere above us. It is crucial to various natural processes and everyday activities. As altitude increases, atmospheric pressure decreases.

• Atmospheric pressure is higher at sea level due to the dense layers of air above.
• As we ascend through the atmosphere, the number of air molecules decreases, leading to a decrease in pressure.
• Pressure never truly reaches zero; even at extreme altitudes, a small amount of pressure exists.

Understanding atmospheric pressure is key to explaining why a balloon rises to a certain height. The decreasing pressure outside reduces the buoyant force until it balances with the balloon's weight, causing it to stop ascending.

This concept is vital for fields such as meteorology, engineering, and even aviation, where pressure variations significantly affect weather patterns and flight dynamics.

Gravitational Force

Gravitational force is the force attraction that acts between masses, typically noticeable between celestial bodies, such as the Earth and an object on it. It remains consistent regardless of altitude changes.

• Gravitational force acts downward, pulling objects toward the center of the Earth.
• This force is what gives objects weight, calculated as the product of mass and gravitational acceleration ( \( F = mg \) ).
• It continues to act on a hydrogen balloon as it rises, countering the buoyant force.

For a floating object such as a balloon, the gravitational force must be in balance with the upward buoyant force for stable altitude.

Conceptually, understanding gravitational force adds insight into why certain heights stop the balloon's ascent. Though the buoyant force diminishes with decreasing pressure during the rise, the constant gravitational pull holds significant influence in stopping the upward climb of the balloon once forces are balanced.