GRAVITATION

Film Loop: Orbiting bodies in force fields: Part II: Negative power laws.

Length: 3:35 min., Black and White, No sound

Using computer generated animation these demonstrations show the behavior of two bodies attracted to each other by a force varying as the negative power of R: the force gets smaller as the bodies move farther apart. The behavior is shown for three forces; F proportional to: 1/R, 1/(R squared) and 1/(R cubed).

NOTES: The conventions used in the film are all shown in the figure: the size of the circle shows the mass (2:1 ratio), the direction of the force is along the arrow and the magnitude of the force is proportional to the length of the arrow, the center of mass of the two bodies is marked by a cross and a line traces the path of each body. the initial velocities ae chosen so that the center of mass does not move.

In general, if the period of the X and Y or the radial (r) and the angular (q) components of motion are equal or have an integer ratio, then the orbits will be reentrant - they will close. This is the case shown in the second sequence where F proportional to 1/(R squared): the gravitational force field. In the other cases of negative power laws shown, the orbits do not close; the figure shows the orbit for F proportional to 1/R. Pure circular orbits are always posible for a body acted on by a force dependent on only the displacement R: a central force. However, as the last sequence shows if the force decreases as 1/(R cubed),or faster, the orbits are unstable; a slight change in velocity from that needed to hold a body in a circular orbit will cause the body to either spiral into the force center or spiral away from it.