# A rod in free fall with varying length

Suppose you have a weightless rod with at the endpoints two masses. This is a special rod whose length varies in time as a function L(t). If I consider this rod in free fall my intuition is that this rod will rotate for suitable L(t). However, I want to write down the equations of motions for this rod. I am not so sure how to do this in the sense of what kind of coordinates are useful and which physics formulas I should consider. Any help is welcome.

1. Which are the initial conditions (configuration and velocities)?
2. What makes L vary? Internal or external forces? How, exactly?

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lightarrow

1) I assume that the rod is at rest. So the velocity is zero but I will keep the orientation of the rod free. However, I assume the motion is fully planar.

2) The length of the rod is determined purely by the internal forces so I the length of the rod, L(t), is a function only dependent on the independent variable t for time

davenn
Gold Member
Suppose you have a weightless rod with at the endpoints two masses. This is a special rod whose length varies in time as a function L(t).

its pretty much impossible to give real answers to something that defies physics
so how about restating your question using real world materials that obey real world physical laws

Regards
Dave

jfizzix
Gold Member
Suppose you have a weightless rod with at the endpoints two masses. This is a special rod whose length varies in time as a function L(t). If I consider this rod in free fall my intuition is that this rod will rotate for suitable L(t). However, I want to write down the equations of motions for this rod. I am not so sure how to do this in the sense of what kind of coordinates are useful and which physics formulas I should consider. Any help is welcome.

For this unusual rod, it would at least be straightforward to describe its center of mass.

The trajectory of the center of mass of a free falling rod in a vacuum (using local gravity ($F_{g}=m g$)) would be the same as the trajectory of a point mass at the same location as the center of mass of the rod. Whatever angular momentum it had about its center of mass would be a conserved quantity, though the rate of rotation would change with the changing length of the rod (so that angular momentum is kept constant)

Using Universal gravity ($F_{g} = -G\frac{m M}{r^{2}}$), the rod would experience tidal forces, and would gradually become aligned with the gravitational field of the planet it's falling towards. In this case, however, the angular momentum of the rod is not a conserved quantity, since tidal forces apply a torque onto the rod. By changing the length of the rod over time in the right way, one could imagine building up angular momentum until the rod is spinning very fast, though this would take an incredibly long time, as tidal forces are very weak unless you're in very extreme conditions.

Long story short, the difficulty of this free falling variable length rod depends on whether or not gravitational tidal forces are significant (and of course, if there are any other external forces).

In order to get some insight into how to write the equations for such a rod, I would look at how the physics of driven systems is described in Lagrangian or Hamiltonian mechanics, since this variable length could be considered a driving term.

jfizzix
Gold Member
Actually, now that I think of it, it's not really clear if you could build up angular momentum by using the tidal force, since the amount of torque would depend on the length of the rod..

davenn
Gold Member
For this unusual rod, it would at least be straightforward to describe its center of mass.

if it is weightless as he said ... then it is massless ie. it cannot have a centre of mass

Staff Emeritus