Q:

A spherical planet in a vacuum is orbited by a satellite such that it keeps a fixed distance from the planet surface and has a forward motion with constant acceleration. A flexible chain of negligible weight is suspended from the satellite. The top link is attached to the satellite the other links hang down under gravity and tension. The chain does not reach the surface. An acute angle is formed between the chain and a perpendicular line to the surface. What is the mathematical relationship between the angle and the acceleration? I suspect acceleration is a function of the tangent of the angle.

- Bill (age 61)

UK

- Bill (age 61)

UK

A:

This is a very interesting question, requiring some thought.

Let's try to answer it first ignoring any gravity from the satellite itself. Let's use the free-fall reference frame of the satellite. In that frame, all that matters is the way that the gravitational field changes as one leaves the satellite- the "tidal" effects. These are pretty simple. The downward gravity grows as one goes straight down. The gravity weakens as one goes upward, so in this frame there's a growing upward gravitational field above the satellite. As one goes horizontally to any side, there's a gravitational field pointing in toward the satellite. So far as I can see, there's just two stable positions for the chain, either straight up or straight down.

Now of course if you count the gravity from the satellite, things are a lot different. It pulls everything in toward the satellite. near the satellite, it should be more important than those tidal effects from the earth. So the chain would just fold in on the satellite.

Mike W.

Let's try to answer it first ignoring any gravity from the satellite itself. Let's use the free-fall reference frame of the satellite. In that frame, all that matters is the way that the gravitational field changes as one leaves the satellite- the "tidal" effects. These are pretty simple. The downward gravity grows as one goes straight down. The gravity weakens as one goes upward, so in this frame there's a growing upward gravitational field above the satellite. As one goes horizontally to any side, there's a gravitational field pointing in toward the satellite. So far as I can see, there's just two stable positions for the chain, either straight up or straight down.

Now of course if you count the gravity from the satellite, things are a lot different. It pulls everything in toward the satellite. near the satellite, it should be more important than those tidal effects from the earth. So the chain would just fold in on the satellite.

Mike W.

*(published on 10/22/2012)*