Orientation of cylinder around massive body

1. May 16, 2014

8BitTRex

http://i.imgur.com/ZH9huJt.png

Lets suppose this perfectly rigid cylinder has a uniform mass and has a length on the order of the distance between the Earth and Mars or some similar situation. This cylinder is orbiting around the Sun in a 2-body universe. What would this cylinder look like on the opposite side of the Sun?

If the cylinder obeyed consv of ang momentum, the cylinder would point non-radially immediately after the initual condition was set up. Since the gravitational force acts as 1/r^2, there would be a net moment on the rod, http://i.imgur.com/J7W3OUQ.png, because the closer end would perceive a lorger force than the the far end.

I guess my question is... Would the cylinder point radially on the opposite end of orbit like in the initial picture, or would the moment cause it to assume some other orientiation?

I don't really have a specific question I'm asking but would like some insight to how the cylinder would act as a function of time.

EDIT: At t=0, the COM of the cylinder is assumed to be in a circular orbit. The cylinder is not spinning with respect to its initial position. The total mass of the cylinder is <<M. Sun is point mass. Width of cylinder is negligeable, assume it is a line with a mass.

Last edited: May 16, 2014
2. May 16, 2014

voko

I do not think the problem is solvable as stated. The initial condition is not specified, so anything is possible.

3. May 16, 2014

8BitTRex

How about now? What if we know its COM was in a circular orbit at t=0. The heart of my question is whether the cylinder will point radially or not after t=0.

4. May 16, 2014

voko

You cannot say "circular orbit at t = 0". At "t = 0" you can only specify the velocity of the centre of mass, the angular velocity of the body, and the location of the CoM. Whether that will result in circular motion of the CoM is something that you need to figure out as a first step. Any ideas?

5. May 16, 2014

8BitTRex

What mechanism would cause it to be non-circular?

The only way the orbit would be non-circular is if the radius of the COM changed. What would cause the COM orbital radius to change?

All forces in the problem are radial between individual points on the cylinder and the mass M. To maintain the circular orbit, the NET gravitational force and NET centrifugal forces on the cylinder would have to remain equal at all times. I don't think they need to be equal everywhere along the cylinder.

I can't justify why the COM would change its orbital radius. What am I missing/How should I proceed?

6. May 16, 2014

voko

I do not think this a correct approach. It is invalid to say "it is circular because I cannot think of anything else". What theorem about the CoM motion might be applicable here?

7. May 16, 2014

8BitTRex

Help me out, I don't really know what theorem you are talking about. We could analyze the energy of the cylinder and compute the trajectory since energy will be conserved in the Hamiltonian of the system.

8. May 17, 2014

voko

There is a theorem that applies to a system of self-interacting particles which are also acted upon by external forces. It says something about the motion of the CoM. Your cylinder is a particular case of such a system, as any rigid body is.

9. May 17, 2014

8BitTRex

Can you be a little more specific what this theorem is named and how it specifically applies. I'm at a loss. Is it one of these

https://en.wikipedia.org/wiki/Category:Physics_theorems

I'd love to be able to understand this problem, I just need a little bit more on the specifics of how you would approach this problem. Thanks.

10. May 17, 2014

voko

Say the particles are $m_i$, where $i \in [1, n]$, and the force on particle $i$ from particle $j$ is $\vec f_{ij}$, and the external force on particle $i$ is $\vec F_i$. Then Newton's second law for particle $i$ is $$m_i \ddot {\vec r_i} = \sum\limits_{j = 1}^n \vec f_{ij} + \vec F_i$$ Can you derive the second law for the CoM from that?

11. May 17, 2014

8BitTRex

You lost me. Why do we need to consider the self interaction? The structure of the cylinder is rigid and the internal forces will not effect the CoM because they cancel each other out axially. The only effect on the CoM radius must be from an external force.

12. May 17, 2014

voko

What you say makes sense, but then you should be able to do proceed with the derivation requested in #10. How would you show that $\vec f_{ij}$ cancel each other?

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