Kepler Second Law and Angular Momentum

Understand Kepler’s second law and its relation to conservation of angular momentum in a central force field. This helps solve assertion reason...

❓ Question

Given below are two statements: one is labelled as Assertion (A) and the other 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 the areal velocity of the planet is constant.
Reason (R): For a central force field, the angular momentum is a constant.

🖼️ Question Image

This problem is based on:

👉 Kepler’s second law
👉 Areal velocity
👉 Conservation of angular momentum.

Main idea:

For motion under central force:

\boxed{ \vec\tau=0 }

Therefore:

\boxed{ \vec L=\text{constant} }

and hence areal velocity remains constant.

Kepler’s second law states:

\boxed{ \text{Radius vector sweeps equal areas in equal times} }

This means:

\boxed{ \text{Areal velocity is constant} }

So Assertion (A) is correct.

For a central force:

\vec F \parallel \vec r

Thus torque:

\vec\tau=\vec r\times\vec F=0

Hence angular momentum remains constant:

\boxed{ \vec L=\text{constant} }

So Reason (R) is also correct.

Areal velocity is:

\frac{dA}{dt} = \frac12 rv\sin\theta

Angular momentum:

L=mrv\sin\theta

Therefore:

\boxed{ \frac{dA}{dt}=\frac{L}{2m} }

Since:

L=\text{constant}

Thus:

\frac{dA}{dt}=\text{constant}

Hence Reason (R) correctly explains Assertion (A).

❌ Kepler’s second law directly ratta maar lena without physics connection

❌ Torque in central force ko non-zero maan lena

❌ Areal velocity formula bhool jaana

Remember:

\boxed{ \frac{dA}{dt}=\frac{L}{2m} }

\boxed{ \text{Both (A) and (R) are correct and (R) is the correct explanation of (A)} }

(Option 2)