Classical Mechanics, angular momentum

[not-einstein]

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

 

[quantumdude]

[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.

If they give you the potential, you should be able to find the force. And if you have an expression for the force, then you should be able to write down Newton's second law. And if you can do that, then you should be able to solve the resulting differential equation for the 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.

So here you are being asked to study the case of hyperbolic trajectories that you find in [1].

[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.

This should be easy, if you correctly implement the assumption that the Sun is "very heavy". Do you know how that translates into mathematics?

[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?

What is "finite motion"? Does that refer to a closed orbit?

[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.

Again, you can find the force and solve Newton's second law. You should be looking for a critical case for which the particle will not orbit, but rather fall towards the center (as it would if the tangential component of its velocity were to drop to zero).