# Is there a mathematical relationship here or am I looking for relations when there are none?

When I was taking classical mechanics, we dealt a lot with pendulums, and orbiting bodies problems. This lead me to think about the two situations depicted above. Left: Shows two balls of equal mass suspended from a ceiling with string, Right: Shows to large orbiting bodies in outer space.

For some reason, my intuition is begging that there these two cases are somehow mathematically related. However, I need help finding this relationship. It seems that the as the string twists and the balls get closer and closer to each other, its awfully similar to the case when two bodies in space orbit each other until they collide too! (Or am I just crazy and thinking too much?)

What relevant equations / geometric relationships are needed here to establish a relationship between these two cases? (Kepler's Laws, what else?)

Going further: Given that the strings magically "go through each other" at every rotation, can't this be related to the case in which the two orbiting masses orbit a third body, say a star, located at the center of the axis of rotation? (Picture not provided here, but I hope you understand my further question)

Again, What relevant equations are needed here to establish a relationship between these two cases? Going even further: This is getting pretty hand wavy, but I was thinking that perhaps this could be a new way of looking at the 3-body problem or multi-body problem using the string and ball model? Would Knot theory come into play?

• One potential problem, in your first diagram, they'll describe circular (well spiral if no magic strings!) orbits, there seems to be no scope for elliptical orbits like you'd want for inverse square law. Commented Jan 3, 2013 at 17:31
• @twistor59 they'll be ellipses if their initial motion is non circular... just like orbits. Angular momentum and energy are independent parameters. Commented Feb 2, 2013 at 20:42

In the second problem you have just gravitational force between the two bodies : $$F = G\frac{m_1m_2}{r^2}$$ in the pendulum problem you have the "centrifugal force" and the force of the gravity of the earth acting in perpendicular directions balanced by string tension. The rate of the spheres coming together will likely be different, I don't have time to prove it at the moment however...