Angular momentum conservation of two point masses

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Lets suppose we have 2 point masses one fixed and imoveable the other rotateing in a perfect circle around the first imoveable one with constant speed.

Regardless of vich force is keeping the rotating mass on its circular orbit.
It may be a string or gravitational force or whatever.

The velocity of the rotating mass will allways be tangential to the circle the mass follows but the velocity vector intensity is the same as stated above (constant speed).

Now lets pull the rotating mass closer to the fixed mass, shorten the radius.
The conservation of angular momentum states that the the tangential spped of the rotating object should increase

If we aply a force orthogonal to the velocity vector to shorten the radius then this force could never ever change the intensity of the velocity.
Yet the conservation of angular momentum states different.

What is going on

Answers and Replies

  • #2
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If you are pulling the object in using a central force then that force is perpendicular to the object's original circular path. But it is not perpendicular to the object's spiral path as it is reeled in. The object speeds up because the force has a forward-pointing component along its path.

If, instead, you were pulling the object in using a force that was perpendicular to the object's spiral path then that force could not be centrally directed. In this case, angular momentum would not be conserved because the force would be an unbalanced external torque on the system. The circling object would retain its original speed, but would lose angular momentum.
  • #3
The spiral motion happends because the tangential velocity vector gets a component perpendicular to it because of the pulling force and the new vector is composed of the tangential component vitch remains constant and the ortogonal component vitch increases as long as rhe force is acting.

So there is no non-orthogonal component of the force on the tangential part of the velocity vector.

When force comes to hold the orthogonal component of the velocity dies out and one is left with the same velocity vector as at the begining.
Tangential to the now again circular path and same in magnitude as before.

So what is going on
  • #4
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the tangential velocity vector gets a component perpendicular to it because of the pulling force
No. When it's moving in a spiral the instantaneous centre about which the object is rotating is NOT the center of the spiral. The instantaneous centre of rotation moves. See diagram. In this case the point is rotating around a point to the right of the centre of the spiral. This means that a force acting towards the centre of the spiral has a forward component.