Question 52: Gravitation

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:

Accepted Answer

The assertion (A) states that the Ã¢ÂÂradius vector from the Sun to a planet sweeps out equal areas in equal intervals of timeÃ¢ÂÂ which is just another way of stating Kepler's second law. This implies that the areal velocity is constant.
The reason (R) states that "For a central force field the angular momentum is a constant." In a central force system (like that of a planet orbiting the Sun under gravity), the force always acts along the line joining the two bodies. This means there is no torque acting on the planet, and thus, its angular momentum is conserved.
The connection between angular momentum and areal velocity is given by:
$$	ext{Areal velocity} = \frac{1}{2}r^2\dot{	heta} = \frac{L}{2m}$$
where $$L$$ is the angular momentum and $$m$$ is the mass of the planet. Since the angular momentum $$L$$ is constant, the areal velocity must also remain constant.
Therefore, both the assertion and the reason are correct, and the reason correctly explains why the areal velocity (as stated in the assertion) is constant.
So, the most appropriate answer is:
Option C: Both (A) and (R) are correct and (R) is the correct explanation of (A).

Answer

(A) is not correct but (R) is correct

Answer

Both (A) and (R) are correct but (R) is
not
the correct explanation of (A)

Answer

Both (A) and (R) are correct and (R) is the correct explanation of (A)

Answer

(A) is correct but (R) is not correct

Gravitation

Choose the correct answer: