Rods with Masses orbiting Earth

[postfan]

Homework Statement

Two small, equal masses are attached by a lightweight rod. This object orbits a planet; the length of the rod is
smaller than the radius of the orbit, but not negligible. The rod rotates about its axis in such a way that it remains
vertical with respect to the planet.
• Is there a force in the rod? If so, is it tension or compression?
• Is the equilibrium stable, unstable, or neutral with respect to a small perturbation in the angle of the
rod? (Assume this perturbation maintains the rate of rotation, so that in the co-rotating frame the rod
is still stationary but at an angle to the vertical.)

(A) There is no force in the rod; the equilibrium is neutral.
(B) The rod is in tension; the equilibrium is stable.
(C) The rod is in compression; the equilibrium is stable.
(D) The rod is in tension; the equilibrium is unstable.
(E) The rod is in compression; the equilibrium is unstable.

A picture may be found here http://www.aapt.org/physicsteam/2014/upload/exam1-2013-1-6-unlocked.pdf
page 6, problem 12

Homework Equations

The Attempt at a Solution

I know that there is a force of tension because the Earth is pulling the black mass (according to diagram) down and so the rod must being pushing the white mass up to keep it in equilibrium as it rotates. I'm just having trouble about whether the equilibrium is stable or not. Any thoughts?

 

[Nathanael]

I know that there is a force of tension because the Earth is pulling the black mass (according to diagram) down and so the rod must being pushing the white mass up to keep it in equilibrium as it rotates.

The force of gravity (Earth) is pulling on both masses...

I'm just having trouble about whether the equilibrium is stable or not. Any thoughts?

What would happen if you slightly displaced the rod so that it wasn't perfectly vertical with respect to the planet?

 

[postfan]

OK, even though the Earth is pulling on both masses, isn't the rod still pulling up on the white mass to keep it taut as it revolves?

 

[Nathanael]

isn't the rod still pulling up on the white mass to keep it taut as it revolves?

I'm not sure what you mean by this.

 

[postfan]

Like the system isn't collapsing as it revolves.

 

[Nathanael]

But why would it collapse?

 

[postfan]

Because the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.

 

[Nathanael]

Because the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.

There we go :)
But if there was no tension then the system would not collapse, it would stretch apart! That is why I thought you were misunderstanding.

About the stability of the object. I'm not sure if you already figured it out, but what would happen if you slightly displaced the object so it wasn't vertical? Would it start rotating more and more away from being vertical, (unstable) or would it go back to being vertical (stable)?

 

[postfan]

I think it would go back to vertical, but I can't explain why.

 

[Nathanael]

Is there any torque on the system?

 

[postfan]

Yes because there is a force (tension) and it acts over the length of a rod, giving it both a force and a distance.

 

[Nathanael]

Yes because there is a force (tension) and it acts over the length of a rod, giving it both a force and a distance.

Tension acts in the direction through the center of mass, therefore it produces no torque. Just like when the object is rotating vertically, gravity acts through the center of mass, so it provides no torque. But what about when we displace the object slightly; does gravity still act through the center of mass?

 

[postfan]

Yes, gravity does acts through the center of mass.

 

[Nathanael]

The center of mass is located in the center of rod. But the force of gravity acts on the two masses on the edges of the rod (because it said the rod's weight was negligible). Perhaps draw a free body diagram of the object slightly tilted from it's vertical position.

 

[postfan]

Ok ok ,doing the FBD I see now that gravity does not act through the center of mass.

 

[Nathanael]

Ok ok ,doing the FBD I see now that gravity does not act through the center of mass.

So is there a net torque? If so, does it cause the rod to return to a vertical position or to rotate away from being vertical?

 

[postfan]

Yes because the force is acting some distance away perpendicular to the center of mass.

 

[postfan]

And it returns it to a vertical position.

 

[Nathanael]

Yes because the force is acting some distance away perpendicular to the center of mass.

Your explanation is a bit unclear again. The torque from gravity on the white mass acts to rotate the object away from the vertical, but the torque on the black mass acts to rotate the object back to the vertical. Does one of these torques win? (Is one of them stronger?)

 

[postfan]

I think that the torques are the same because even though the black mass is experiencing more force ,the white mass has more distance, hence balancing the torques.

 

[Nathanael]

When measuring torque you want to use the distance to the axis of rotation (in this case the center of mass). You also want to only use the component of the for force that is perpendicular to the rod. Equivalently, you can use the full force and the "Lever arm"

 

[postfan]

I honestly don't understand what you're trying to get me to do.

 

[Nathanael]

I think that the torques are the same because even though the black mass is experiencing more force ,the white mass has more distance, hence balancing the torques.

The lever arm is the same for both the white and black masses. Therefore the larger force produces a larger torque, so there is a net torque.

 

[postfan]

Ohh OK, I get it so since the torque is not equal the black mass experiences a torque clockwise making it vertical again and making it stable.

 

[BvU]

I'm going to make this one more difficult. So far you two looked at gravity (from the planet, it didn't mention Earth !). From the wording of the exercise (co-rotating frame of reference, rod length non-negligible wrt planet radius --either a big stick or a small planet :) ) I get the impression you also want to look at the apparent force that comes with such a frame of reference. There too, the black one gets more than the white one, and the torque is in the other direction !

 

[BvU]

The lever arm is the same for both the white and black masses.

Yes.

Therefore the larger force produces a larger torque

Only if the stick length can be igored. (The exercise says it can't)

 

[haruspex]

Because the Earth is pulling the black mass with more force than the white mass. You need tension to balance it.

Yes, there's tension, but the explanation is not quite as simple as that.
Closer to the Earth means greater gravitational force, but it also means a greater centripetal force is needed to keep it in orbit.
To analyse it properly, do a free body diagram for each mass. But a quick way is to compare the black mass, say, with the mass centre. The mass centre must be orbiting at the right rate for its altitude. We know that a body at a lower orbit needs to move faster, so the black mass must be traveling below its proper orbital speed. That means it would tend to fall towards Earth if not restrained by the rod.

From the wording of the exercise (co-rotating frame of reference,

Arguably, that wording is merely to clarify what is being said. But if comfortable with working in non-inertial frames (I never have been) it's probably the simplest approach.

since the torque is not equal the black mass experiences a torque clockwise making it vertical again and making it stable.

Again, it's not enough to think about gravitational pull. Need to consider centripetal/centrifugal force too. I believe that is the point BvU made here:

you also want to look at the apparent force that comes with such a frame of reference.