Hope this helps.

[assaftolko]

A body moves in circular motion around the Earth with orbit radius of 3R_E
At a certain time the body detaches to 2 identical parts, each one with a mass of m: A and B. A moves in an angle of 34 deg and B moves straight to the center of the earth.

What is the angle between the velocity vector and the radius-vector of part A when it gets to a distance of 4R_E from the center of the earth?

I think I'm suppose to use angular momentum conservation with respect to the center of the Earth for part A and calculate it's angular momentum just after the detachment and when it's at 4R_E. The problem is that I don't really know how can I justify angular momentum conservation... I think that gravity produces torque and so this is a problem...

 

[gneill]

Gravity produced by a point source will not produce a torque on an orbiting point mass. Assuming that the Earth is taken to be a spherically symmetric distribution of mass, it will produce the same field as a point mass located at its center. Angular momentum is always conserved.

 

[assaftolko]

Gravity produced by a point source will not produce a torque on an orbiting point mass. Assuming that the Earth is taken to be a spherically symmetric distribution of mass, it will produce the same field as a point mass located at its center. Angular momentum is always conserved.

I'm sorry I got confused for a second... r and the gravity force lay on the same line for every moment so of course gravity doesn't produce torque... thanks

 

[gneill]

1. After the detachment - can you still say A is "orbiting" something?
2. Can you reffer me to a source that shows why this is true for the angular momentum?

1. Assuming that the mass of the Earth is much greater than that of A, then Earth's field will dominate the motion of A with respect to the Earth. Whether or not the orbit is closed is another matter (check the velocity of A versus escape velocity at R = 3Re).

2. Angular momentum is ALWAYS conserved. For orbiting objects its a constant of the motion. For a proof, any text on astrodynamics should have a derivation of the equation of motion for the two-body system. One of my favorites is "Fundamentals of Astrodynamics" by Bate, Mueller, and White (very inexpensive yet amazingly good).

 

[Nickie]

I would approach this problem by first clarifying the scenario and assumptions. It seems that we are considering a body moving in circular motion around the Earth with a given orbit radius. At a certain point, the body splits into two identical parts, A and B, with mass m each. Part A continues in circular motion at an angle of 34 degrees from the original orbit, while part B moves straight towards the center of the Earth.

To answer the question of the angle between the velocity vector and radius-vector of part A when it reaches a distance of 4RE from the center of the Earth, we can use conservation of angular momentum. This principle states that the total angular momentum of a system remains constant unless acted upon by an external torque. In this case, we can assume that the system is isolated and there are no external torques acting on the two parts A and B.

To justify angular momentum conservation, we can consider the forces acting on the system. The only force acting on the two parts is the gravitational force from the Earth. Since this force acts along the line connecting the two parts, it does not produce any torque on the system. Therefore, we can assume that the total angular momentum of the system remains constant.

To calculate the angular momentum of part A, we can use the formula L=mr^2ω, where m is the mass of the object, r is the distance from the center of rotation, and ω is the angular velocity. Initially, part A has an angular momentum of L=mr^2ω, where r is the original orbit radius and ω is the angular velocity corresponding to this orbit.

When part A reaches a distance of 4RE from the center of the Earth, its angular momentum will be L=mr^2ω', where r is now 4RE and ω' is the new angular velocity. Since we know that the total angular momentum of the system is conserved, we can equate these two values to get:

L=mr^2ω=mr^2ω'

Solving for ω', we get ω'=ω/4, which means that the new angular velocity is one-fourth of the original one. This also means that the new angular velocity is perpendicular to the radius-vector, as the angle between the two vectors is 90 degrees. Therefore, the angle between the velocity vector and the radius-vector of part A when it reaches 4RE from the center