# Is Angular Momentum really conserved

1. Apr 15, 2003

### ObsessiveMathsFreak

I'm having a lot of trouble with this.

Is it really a conserved quantity, like linear momentum, or is this just a rule applied to certain systems.

2. Apr 15, 2003

### arivero

It is even more important than linear momentum, as the main problem of dynamics was, for a long time, movement under central forces.

Angular momentum is preserved undir this kind of force. It can be a bit confusing because now it is customary to present this as a consequence or rotational symmetry, but angular momentum is preserved even in some cases where the force field is not rotationally symmetric.

3. Apr 15, 2003

### Hurkyl

Staff Emeritus

There's a newton's law for angular momentum just like for linear momentum:

F = dp/dt

&tau = dL/dt

(&tau is torque and L is angular momentum)

So in a system with no external torque (such as any closed system), angular momentum must be conserved via Newton's law.

Hurkyl

4. Apr 15, 2003

### ObsessiveMathsFreak

You have a straight rod, floating in space.

Now two projectiles are fired at the rod, from opposite directions and with velocity perpendicular to the line of the rod.

They both strike the rod at its ends and are fixed there by some mechanism.

Won't the impact of the particles constiute a moment about the rod causing it to rotate and essentially creating angular momentum where none existed before?

5. Apr 15, 2003

### FZ+

In that case, you have applied an external torque, haven't you?

6. Apr 15, 2003

### damgo

Also, the total angular momentum of the two projectiles before they hit will be equal to the angular momentum of the spinning projectiles+rod system that results.

7. Apr 16, 2003

### arivero

by the way, kepler second law is related to the discussion

8. Apr 17, 2003

### ObsessiveMathsFreak

So does a particle moving in a straight line have angular momentum?

If so what is the formula?

9. Apr 17, 2003

### Janus

Staff Emeritus
The same as as the regular formula you just have to define the Point from which you are measuring the angular momentum.

For instance for your example, the system of the rod and two projectile, this would be the center of the rod.

Now imagine two rays extending from the center of the rod to each projectile. As the projectiles approach the rod, these rays will shorten and rotate around the center of the rod. And any given instant each ray will have a length(radius)and angular velocity. the cloeser the projectile to the rod, the shorter the radius and the greater the angular velocity. Thse two properties, plus the mass of the projectile determine the angular momentum of the projectiles with respect to the center of the rod. The decrease of the radius and increase of the angular velocity exactly compensate for each other leading to a constant angular momentum.

10. Apr 17, 2003

### pmb

ObsessiveMathsFreak basically asked whether angular momentum was a conserved in physics.

The answer is yes. The angular momentum of a closed system is always conserved. However it may change form. If might change from mechanical angular momentum to angular momentum of radiation. Consider a charge moving in a circle, say at the end of a string attached to a pole or something like that. The angular momentum is all mechanical at first. But as the charge moves it radiates and thus looses energy and will eventually slow down and stop. However the *total* angular momentum remains the same - The angular momentum was carried off in the form of electromagnetic energy.

And you can't say if something has angular momentum or not unless you say with respect to what point. If a particle is moving along the x-axis of a Cartesian coordinate system then the angular momentum with respect to the origin is zero - and the total angular momentum with respect to the origin will remain zero if the system is closed. However, for the same particle, the angular momentum is not zero with any point other than the origin.

Pete