Classical Mechanics, angular momentum

What is that condition?In summary, the conversation discusses various problems involving a particle moving in 3D space around a fixed attractive center with different potential functions. These include finding the types of trajectories and their corresponding values of energy and angular momentum, determining the angle of asymptotes for hyperbolic motion, finding the ratio of rotation periods for two planets orbiting a very heavy Sun, and finding the effective potential and conditions for finite motion or falling towards the attractive center. The conversation also mentions using the orbital equation, angular momentum equation, and Newton's second law to solve these problems.
  • #1
not-einstein
1
0

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
 
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  • #2
not-einstein said:
[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).
 

1. What is classical mechanics?

Classical mechanics is a branch of physics that studies the motion of objects and the forces acting on them. It includes concepts such as Newton's laws of motion, energy, and momentum.

2. What is angular momentum?

Angular momentum is a measure of an object's rotational motion, calculated by multiplying its moment of inertia by its angular velocity. In simpler terms, it is the tendency of an object to continue rotating at a constant rate.

3. How is angular momentum related to classical mechanics?

Angular momentum is one of the fundamental principles of classical mechanics. It helps us understand the rotational motion of objects and how forces act on them to change their rotational speed or direction.

4. What are some real-life examples of angular momentum?

Some examples of angular momentum in everyday life include spinning tops, the rotation of the Earth on its axis, and the motion of a gymnast during a somersault. It is also important in the operation of machines, such as turbines and flywheels.

5. How can angular momentum be conserved?

In an isolated system (where there are no external forces acting on the system), angular momentum is conserved. This means that the total angular momentum of the system remains constant, even if individual objects within the system are experiencing changes in their angular momentum due to internal forces.

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