Question

Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R) : For a central force field the angular momentum is a constant. In the light of the above statements, choose the most appropriate answer from the options given below :

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

When dealing with Assertion-Reason type questions, first verify the correctness of each statement individually. Then, try to establish a causal link between the Reason and the Assertion. Ask yourself: "Does the Reason logically explain why the Assertion is true?" In this case, the conservation of angular momentum (due to the central nature of gravity) is the direct cause of the constant areal velocity.

Answer 1

The Correct Option is A

To solve the given assertion and reason problem, we first need to analyze the statements individually and then collectively:

1. Understanding the Assertion (A): "The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant."
   • This statement is based on Kepler's Second Law of Planetary Motion, which states that a line joining a planet to the sun sweeps out equal areas in equal times. This implies that the areal velocity, the area swept per unit time, is constant for a planet in orbit. Thus, Assertion (A) is correct.
2. Understanding the Reason (R): "For a central force field the angular momentum is a constant."
   • A central force field is a force field in which the force acts along the line joining the bodies and depends only on the distance between them, such as the gravitational force between a planet and the Sun.
   • For a central force, the angular momentum is conserved because there is no external torque acting on the system, which supports the reason (R) being correct.
3. Connecting Assertion and Reason:
   • The constancy of areal velocity as mentioned in Assertion (A) is due to the conservation of angular momentum, as stated in Reason (R). Hence, (R) is indeed the correct explanation of (A).

Therefore, the correct answer is: Both (A) and (R) are correct and (R) is the correct explanation of (A).

Answer 2

Step 1: Analyze Assertion (A).
Assertion (A) states Kepler's second law of planetary motion: the radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time, implying constant areal velocity. This is a fundamental law of planetary motion and is correct. 

Step 2: Analyze Reason (R).
Reason (R) states that for a central force field, the angular momentum is a constant. Gravitational force, which governs planetary motion around the Sun, is a central force. Under a central force, the torque on the planet with respect to the Sun is zero, leading to the conservation of the planet's angular momentum. 
Thus, Reason (R) is also correct. 

Step 3: Determine if Reason (R) is the correct explanation of Assertion (A).
The areal velocity \( \frac{dA}{dt} \) of a planet is mathematically related to its angular momentum \( L \) by \( \frac{dA}{dt} = \frac{L}{2m} \), where \( m \) is the mass of the planet. Since the gravitational force is central, the angular momentum \( L \) is conserved. As the mass \( m \) is also constant, the areal velocity \( \frac{dA}{dt} \) remains constant. 
Therefore, the conservation of angular momentum (Reason (R)) directly explains the constant areal velocity (Assertion (A)).