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Chapter 6: Problem 809

If \(\mathrm{r}\) denotes the distance between the sun and the earth, then the angular momentum of the earth around the sun is proportional to (A) \(1^{3 / 2}\) (B) \(\mathrm{r}\) (C) \(r^{1 / 2}\) (D) \(r^{2}\)

Short Answer

Expert verified
The angular momentum of the Earth around the Sun is proportional to (C) \(r^{1/2}\).

Step by step solution

01

Recall the formula for angular momentum

The angular momentum (L) of an object in circular motion is given by the formula: L = mvr where m is the mass of the object, v is its tangential velocity, and r is the distance between the object and the center point around which it is moving (in this case, the Sun).
02

Relating tangential velocity to distance

We know from Kepler's laws of planetary motion that the earth's orbital period (T) is proportional to the distance (r) raised to the power (3/2): T ∝ r^(3/2) Additionally, we know that T = 2πr/v. Combining these two relationships, we get v ∝ r^(-1/2).
03

Substituting the velocity relationship in the angular momentum formula

We substitute the relationship for tangential velocity in terms of r from Step 2 into the angular momentum formula: L = m(r^(-1/2))r L ∝ r^(1/2)
04

Match the proportionality with the given options

Comparing our result with the given options, we see that the angular momentum of the Earth around the Sun is proportional to: (C) \(r^{1/2}\)

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