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Question

The angular momentum of a planet of mass M moving around the sun in an elliptical orbit is L→. The magnitude of the areal velocity of the planet is


A

LM

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B

2LM

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C

L2M

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D

4LM

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Solution

The correct option is C

L2M


Step 1: Given data

Mass of planet=M

The angular momentum of a planet=L

Step 2: Formula used:

We know that the formula of angular momentum is

L=Mvr Where M is the mass of the planet, V is the velocity and r is the radius of the orbit.

v=rω where, ω is angular velocity

Step 3: Determine the magnitude of the areal velocity of the planet

Consider the following figure:

From the formula,

L=Mvr----1

The area covered in the figure is -

Suppose dθ is the angle covered for making area dA

From

angle=arcradiusdθ=dlrdl=rdθ

The area will be calculated as-

dA=12rrdθdA=12r2dθ

On differentiating with respect to t-

dAdt=12r2dθdt=12r2ω.................(2)asω=dθdt

The areal velocity of the planet is dAdt.

From equation (1)-

L=MvrL=MrωrL=Mr2ωr2ω=LM

Now putting the values from equation (1)-

dAdt=L2M

So, the magnitude of the areal velocity of the planet will be L2M.

Hence, option C is the correct answer.


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