# Why is angular momentum conserved?

1. Nov 5, 2014

### Pejeu

Why do things which spin tend to keep spinning in the absence of external forces such as friction with the environment?

In order for objects to keep spinning doesn't their periphery (relative to their centre of rotation - which would be their centre of mass, right? - ) have to be constantly acted upon by forces external to it in order to constantly change its direction of travel all the while preserving its (instantaneous) speed?

Where does this force come from? How come the spin doesn't immediately start to slow down after torque is no longer applied to impart spin?

At one moment in time a clump of atoms on the periphery of a flywheel spinning freely in interplanetary space is going one way and some time later (after that clump of atoms has completed half of a rotation around the centre of rotation) the same clump of atoms is now travelling the opposite way, with the same speed as before.

Where did the energy for decelerating and accelerating that clump of atoms come from?

Please note I am not asking whether angular momentum is conserved or not. I am asking why it is conserved.

I would very much appreciate it if replies to this thread made that distinction.

Thank you very much for your interest in this inquiry and your time. Especially if you decide to help elucidate this for me.

2. Nov 5, 2014

### A.T.

Conservation angular momentum doesn't require changing the direction. It applies to linear motion too.

If their speed is the same, then so is their kinetic energy, therefore no energy input was required.

Physics describes nature as it is. It doesn't tell why it is that way. At best you can justify it based on some more general assumptions:
http://en.wikipedia.org/wiki/Noether's_theorem#Basic_illustrations_and_background

3. Nov 5, 2014

### Pejeu

But the clump is now going the opposite way, regardless that with the same speed as it was going before.

So the clump of atoms has to have been first decelerated and then accelerated, at least in the direction described by the tangent it was on half of a revolution before.

4. Nov 5, 2014

### Andrew Mason

Linear momentum is conserved because the time rate of change of momentum is force. So if no external forces act on a system, the total momentum cannot change with time.

Angular momentum is conserved because the time rate of change of angular momentum is torque. If no external torques act on a system, then angular momentum cannot change.

In answer to your question about the change in direction of atoms in a rotating body, there is no change in energy. There is no work done on the rotating atoms. Work is the dot product of Force and displacement: $W = \int \vec{F}\cdot\vec{d}$. Since the force is at right angles to the direction of motion and, therefore, at right angles to the direction of displacement over which the force acts, there is no energy required to keep the system of atoms rotating.

AM

Last edited: Nov 5, 2014
5. Nov 5, 2014

### A.T.

For kinetic energy the direction is irrelevant.
In uniform circular motion there is no tangential acceleration, just centripetal acceleration, which doesn't require energy input.

6. Nov 5, 2014

### Pejeu

Well their kinetic energy is indeed the same.

But it required energy to change their direction of movement. Especially preserving their speed.

Are you saying no energy needs to be expended in order to change the direction of movement of a particle with mass, especially if the particle retains its speed in the new direction of travel?

Yes. But that's something else entirely.

We have a clump of atoms at 12 o'clock which are being accelerated towards 6 o'clock by the centripetal force. But when they've arrived to 3 o'clock they're being accelerated by the centripetal force in the opposite direction as the one they were travelling at 12 o'clock.

Or, I should say, they've been decelerated to a stand still and are starting to be accelerated in the opposite way on the same direction they were travelling in when they were at 12 o'clock.

So objects or parts of objects with mass can be accelerated without energy input?

And aren't objects being accelerated against their previous direction of travel, 90 degrees before?

Aren't they being decelerated in the direction of travel they had 90 degree of revolution before? Isn't energy being expended at some fundamental level in order to decelerate that clump of atoms in the direction of travel they had before?

When the clump of atoms is at 3 o'clock they've been completely decelerated in the direction of travel they used to have when at 12 o'clock.

And then they're re-accelerated in the opposite direction by the time they reach 6 o'clock.

Isn't this reciprocating motion?

Where is the energy coming from to cyclically or continuously accelerate and decelerate all those clumps of atoms?

7. Nov 5, 2014

### Andrew Mason

No. This is not correct. The energy required to change their direction of motion is equal to the work done on the atoms in changing their motion. That work is zero since the force is at right angles to the displacement through which the force acts. $W = \int\vec{F}\cdot d\vec{s} = \int Fds\cos(\frac{\pi}{2}) = \int Fds(0) = 0$

Not in general. But if the force acts at right angles to the direction of motion at all times, which is the case in circular motion, there is no work done on the particle in changing its direction. For elliptical orbits, energy is required because the force is not at right angles to the motion (except at critical points). That energy comes from changes in potential energy of the particle. Total energy remains constant. As the kinetic energy of the particle in elliptical orbit increases the potential energy decreases by exactly the same amount.

AM

8. Nov 6, 2014

### A.T.

No, it doesn’t require energy to change the direction while preserving speed.
What energy? The kinetic energy doesn't increase, so why would there be energy coming from somewhere?

9. Nov 6, 2014

### CWatters

No. If it moves in a circle the acceleration is towards the centre (eg at right angles to the direction of travel) so there is no component "against" the direction of travel.

10. Nov 6, 2014

### rcgldr

Getting back to the original question ...

For the same reason why objects with a specific velocity retain that veloctiy in the absence of external forces. In the absence of external forces, then momentum, both linear and angular are conserved.

In the case of a spinning object the forces that keep the object from flying apart are internal centripetal forces, but even if the object does separate into multiple pieces, and absent any external forces, the angular momentum will continue to be conserved.

11. Nov 6, 2014

### CWatters

Sure but why does that matter? At all times the acceleration is towards the centre BUT the distance from the centre is constant.

Work = force * displacement

No energy is expended because there is no displacement in the direction of the force.

That's like asking if I bounce a ball off a wall where does the energy come from to decelerate and accelerate the ball? No extra energy is required to decelerate and accelerate the ball in the new direction. The KE of the ball is simply stored temporarily in the rubber spring of the ball.

12. Nov 6, 2014

### maajdl

The periphery is acted upon by the internal forces of the solid object.
If the object was not a solid, the angular momentum would also be conserved, but the "periphery" would not be rotating.
If the object was made of sand, the periphery would not rotate, but each part would go on a straight line.
Angular momentum would still be conserved, but the object would dislocate.

The fundamental reason why the angular momentum is conserved is the rotational symmetry of the system.
The orientation of the object in space has no influence.