Mr.p: Good morning. Billy, please read the problem and Bo, please translate.
"Flipping Physics" Intro
Billy: A satellite is in an elliptical orbit.
Part A: is its linear momentum conserved? And part B: is its angular momentum conserved?
Mr.p: Okay. We have a satellite in an elliptical orbit
and, part A, conservation of linear momentum with a question mark
and, part B, conservation of angular momentum with a question mark.
Mr.p: Alright, this seems pretty straightforward. Bobby, please begin solving the problem.
Bobby: Okay, umm...
We know linear momentum is conserved when the net external force acting on the system equals zero
because then the change in linear momentum over change in time equals zero
which means the linear momentum of the system is not changing.
So, we need to to determine if the net external force acting on the system equals zero.
And the system in this case would be just the satellite.
Billy: Draw a free body diagram of the forces acting on the satellite?
Bobby: Yeah, sure. So, the forces acting on the satellite in elliptical orbit are...
well there's only one force acting on the satellite
and it is the force of gravitational attraction pulling it toward the object it is orbiting.
Bo: The object the satellite is orbiting is called the primary. Satellites orbit around the primary.
Bobby: Okay. Thanks.
The only force acting on the satellite is the gravitational force pulling it towards the primary.
Oh, that means, because there is only one force, the net external force on thesatellite cannot equal.
Therefore, the linear momentum of the satellite does change and its linear momentum is not conserved.
Mr.p: Great! And Billy, what about angular momentum?
Is the angular momentum of the satellite conserved?
Billy: Well, this is very similar to what we just did for linear momentum.
Angular momentum is conserved when the net external torque acting on a system equals zero
because then change in angular momentum over change in time equals zero
and therefore the angular momentum of the system is not changing. The system is just the satellite
and we already drew the free body diagram of all the forces acting on the satellite.
So, the question is, is the net torque acting on the satellite equal to zero.
Bobby: We cannot answer that question yet. Bo: Why not?
Bobby: Torque and angular momentum require an axis of rotation and we have not defined an axis of rotation,
so we cannot determine torque or angular momentum. Bo: Right.
Billy: Okay. The satellite is revolving around the primary.
So, the axis of rotation should be the center of mass of the primary.
Bo: That assumes the primary is much much larger than the satellite.
Mr.p: That is fine. You may assume the primary is much much larger than the satellite
and therefore the satellite revolves around the center of mass of the primary.
Billy: Alright. The force of gravity always points directly toward the center of mass of the primary,
which is the axis of rotation. Torque equals the r vector times force times
the sine of the angle between the direction of the r vector and the direction of the force.
The r vector always points from the primary toward the satellite.
The force of gravity acting on the satellite from the primary
always points from the satellite toward the primary.
The angle between those two directions is always 180 degrees. The sine of one 180 degrees equals zero,
so, with an axis of rotation at the primary, the net torque acting on the satellite always equals zero
so the angular momentum of the satellite about the center of mass of the primary is conserved.
Bo: Did you really have to identify the axis of rotation so many times?
Billy: Absolutely. Without a clearly defined axis rotation, you cannot define torque or angular momentum.
Bo: Bobby already said that. Billy: And I repeated it because I am awesome.
Mr.p: Y'all are correct that the linear momentum of the satellite is not conserved
and the and the angular momentum of the satellite,
about the axis of rotation of the center of mass of the primary, is conserved.
Billy: Did you hear that? Mr.p said that the axis of rotation as well.
Bo: Yeah, I heard that.
Mr.p: I will point out that if we assume the satellite's orbit is circular, linear momentum is still not conserved
and angular momentum, about the center of mass of the primary, is still conserved.
Billy: See. He even said it again. Bo: I did not see. But I did hear.
Billy: Okay.
Mr.p: I simply identified the orbit as elliptical because most satellites have nearly elliptical orbits
and very few satellites have nearly circular orbits.
Thank you very much for learning with me today, I enjoyed learning with you.