Given below are two statements: 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.

 

❓ 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.

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🖼️ Question Image

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✍️ Short Solution

To check whether the Assertion and Reason are correct, and whether R explains A, let’s analyze both step by step.

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🔹 Assertion (A) Explanation

The statement says that the radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time.

This is Kepler’s Second Law of Planetary Motion, also known as the Law of Equal Areas.
Mathematically, the areal velocity (dA/dt) is given by:

$$\frac{dA}{dt}=\frac{1}{2}{r}^2\frac{d\theta }{d\mathrm{t}}$$

If dA/dt = constant, then the area swept per unit time remains the same — meaning areal velocity is constant. ✅
So, Assertion (A) is true.

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🔹 Reason (R) Explanation

A central force is one that always acts along the line joining the particle and a fixed point (like the Sun).
In such a case, the torque (τ) acting on the particle is zero because:

$$\vec{\tau }=\vec{r}\times \vec{F}=0$$

Hence, the angular momentum (L = r × p) remains constant in magnitude and direction.
Therefore, Reason (R) is also true.

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🔹 Connecting (A) and (R)

Since angular momentum (L) is constant:

$$L=m{r}^2\frac{d\theta }{dt}=\text{constant}$$

Dividing both sides by $2m$, we get:

$$\frac{1}{2}{r}^2\frac{d\theta }{dt}=\text{constant}$$

But this is exactly the areal velocity (area swept per unit time)!
Hence, the constancy of angular momentum directly explains the constancy of areal velocity.

✅ Therefore, R is the correct explanation of A.

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🧮 Image Solution

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✅ Conclusion & Video Solution

✅ Final Answer:

• Assertion (A): True

• Reason (R): True

• R is the correct explanation of A.

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📘 Concept Recap:

• Kepler’s 2nd Law arises due to the law of conservation of angular momentum.

• A central force (like gravity) exerts no torque, so angular momentum is conserved.

• This leads to a constant areal velocity, meaning the planet covers equal areas in equal times.

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