How Does Angular Momentum Vary with Orbital Distance in Physics?

[SignaturePF]

Homework Statement

https://aapt.org/physicsteam/2010/upload/2009_F-ma.pdf

Homework Equations

L = mrv
L = Iω

The Attempt at a Solution

For a circular orbit:
Fc = Fg
mv^2/r = Gmm/r^2
v = √(GM/R)
Thus:
l = mR√(GM/R)
l = m√(GMR)

This means that LA > LC, eliminating choices B, C, and E.

Now, to compare B, C
I'm interested in finding a more rigorous approach, but here goes.
The point of intersection between the Circlular path that C orbits on and the elliptical path that B orbits.
We know that the velocity at the perihelion is greater than the aphelion, that is, the velocity of the intersection is the maximum velocity that B ever achieves. I then made an intelligent guess and postulated that thus B > C,
leading to LA > LB > LC

Could you suggest more rigor/principles to do this question?

 

[Simon Bridge]

If choices B,C & E are eliminated - what is left are:

(A) LA > LB > LC
(E) The relationship between the magnitudes is different at various instants in time.

Look at E.
Consider: conservation of angular momentum.

 

[tms]

l = m√(GMR)
...
Could you suggest more rigor/principles to do this question?

You've already done that: for B, $r$ is never less than C's and never more than A's.

 

[SignaturePF]

Ya I see that but isn't it root(GM/a), where a is the semi major axis for object B. Doesn't that mean that the radius in the numerator won't cancel with the semi major axis on the denominator?
That's where I was worried.

 

[Nickie]

I would first clarify the question and make sure I understand what is being asked. The question is comparing the angular momentum of three different orbits, A, B, and C, around a central object. The equations provided are the equations for linear momentum and angular momentum. In order to compare the angular momentum of these orbits, we need to use the equation L = mrv, where m is the mass of the orbiting object, r is the distance from the central object, and v is the velocity of the orbiting object.

To compare the angular momentum of these orbits, we need to consider the variables that affect angular momentum. These include the mass of the orbiting object, the distance from the central object, and the velocity of the orbiting object. Looking at the equations provided, we can see that the only difference between the orbits is the distance from the central object, as all objects have the same mass and are orbiting at the same velocity.

Based on this, we can conclude that the angular momentum will be directly proportional to the distance from the central object. This means that the orbit with the greatest distance, A, will have the greatest angular momentum, followed by B, and then C. This is because the farther away an object is from the central object, the greater the distance it travels in a given time, resulting in a greater angular momentum.

To further support this conclusion, we can also use the equation L = Iω, where I is the moment of inertia and ω is the angular velocity. In this case, the moment of inertia for all three orbits will be the same, as they all have the same mass and are orbiting at the same velocity. This means that the only variable that affects angular momentum is the angular velocity, which is directly proportional to the distance from the central object.

In conclusion, based on the principles of angular momentum, we can determine that the angular momentum of orbit A will be greater than that of orbit B, which will be greater than that of orbit C. This is due to the fact that the distance from the central object is the only variable that affects angular momentum in this scenario.