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## Homework Statement

Any help would be much appreciated, even just a nod in the right direction cos I don't know where to start!

[1] A particle of mass m moves in 3D space around a fixed attractive centre with the potential U(r)=-u/r. List all possible types of trajectories for this particle. Which values of energy-angular momentum corresponds to those trajectories.

[2] Write expression for the angle of the asymptotes of the hyperbolic motion and study its dependence on values of energy and angular momentum.

[3] Two planets of the same mass rotate around a very heavy Sun with the mass much larger than that of the planets. Assuming the orbits are circular and that the ratio of their radii is 4, find the ratio between their rotation periods.

[4] A particle of mass m moves in 3D space around a fixed attractive centre with the potential U(r)=-u/(1+(r/a)4). Find the effective potential for the radial movement of this particle. Under what conditions (values of energy E and angular momentum M) finite motion is possible?

[5] A particle of mass m moves in 3D space around a fixed attractive centre with the potential U(r)=-u/r4. Find the effective potential for the radial movement of this particle. Under what conditions (values of energy E and angular momentum M) the particle will fall to the attractive centre.

I have been trying to use the orbital equation but I can't find values for different orbits.

## Homework Equations

orbital equation, angular momentum equation

## The Attempt at a Solution

I know it has something to do with M r theta or M-L=m(rxr) i think but need values for each orbit: elliptical, hyperbolic, and parbolic