Question

Is conservation of angular momentum a direct consequence of the conservation of linear momentum?

Answer 1

I wouldn't say that it is a direct consequence of conservation of "linear" momentum. It is part of a broader statement of the conservation of momentum, period. Whether it is linear, angular or a combination of both.

Answer 2

No! Energy is always conserved and momentum is a form of storing energy. I think (from very poor memory) it is "m" "r" "theta" where m = mass, r = radius at which the mass occurrs and theta is the radians per second..get a bunch of buddies on a kids roundabout. All stand on the outside of the roundabout and push it...not too fast mind! then, make your way to the middle as a group....it will get SIGNIFICANTLY FASTER! That is the conservation of angular momentum in action.(also figure skaters when they do the pirrouette) Noone must jump off, but gradually work your way (as a group) to the outside again to slow it down and get off.

Excellent fun....but you may need some paper-bags!

Answer 3

Both are consequences of newton's laws, bot conservation of angular momentum is not a direct consequence of conservation of linear momentum. To see why just consider the roundabout mentioned above. When you move on the roundabout the law of conservation of angular momentum is present, but linear momentum is not conserved. Why? Because you must use a force to pull yourself toward the center of the roundabout. Since the force is radial, there is no torque in question (so angular momentum is conserved), but linear momentum and kinetic energy are in fact not preserved because of the work done in the system.

Answer 4

Yeah, but since you're moving in a circle, you can't think of forces like you do in inertial reference frames... like since you're directing a force inwards, since you're rotating, part of that force is going to have an effect that is perpendicular to the direction of the force (i.e., the force is deflected along a diagonal), which accounts for the change in speed. Can't we use that notion to derive conservation of angular momentum from conservation of linear momentum?

Answer 5

The problem is this: Conservation of linear momentum is valid if there are no external forces on the system. In that case the angular momentum will always be 0. The theorem of conservation of angular momentum is valid for any value of angular momentum, hence the theorem of conservation of angular momentum is valid for systems where there in fact are external forces acting, however the theorem limits itself to deal only with forces that produce no torque. To conclude the premises for the two laws are different, hence we must bring in other properties than just the conservation of linear momentum in order to prove the law of conservation of angular momentum.

Answer 6

I don't understand how no external forces on the system implies zero angular momentum. Consider a particle moving at a constant velocity \(\vec v\). About any point not on its path, it will have angular momentum. Or consider a wheel spinning in space. No external forces are acting on it and it still has angular momentum.

Answer 7

You're right I goofed about the angular momentum being zero. However, I still believe my conclusion is right. Conservation of angular momentum only requires that the external torque on the system is zero, but not that the external forces on the system is zero. Hence the law of conservation of angular momentum is valid in cases where the law of conservation of linear momentum is not...

Just consider a planet moving about the sun. Let the origin of the coordinate system be at the center of the sun (not rotating). Then it is obvious that linear momentum is not conserved since the direction (and if we assume that the planet moves in an ellipse also magnitude) of the speed changes. However, we do have conservation of angular momentum in this case. Therefore conservation of angular momentum cannot follow directly from the conservation of linear momentum...

Answer 8

You need to consider also that the sun orbits the earth as well; it is not stationary. Linear momentum really is conserved because the center of mass either doesn't move or travels at constant velocity*. Look at the following animation (the same idea holds for elliptical orbits).

http://en.wikipedia.org/wiki/File:Orbit2.gif

*To show this, consider the conservation of momentum \(\frac{d}{dt}\sum \vec p_i = 0\) (i.e., the sum of all momenta \(\vec p_i\) in a closed system does not change). Since in classical mechanics, we have \(\vec p_i = m_i \vec v_i = m_i \frac{d}{dt} \vec x_i\), so the conservation of momentum says \(\frac{d}{dt}\sum \vec p_i = \frac{d}{dt}\sum m_i \frac{d}{dt} \vec x_i = \frac{d^2}{dt^2}\sum m_i\vec x_i = 0\), which thus implies that \(\sum m_i\vec x_i \) moves at constant velocity, which is, except for a scalar multiple \(\frac{1}{\sum m_i}\), the center of mass.

Answer 9

Basically, an orbit is a continuous collision.

Answer 10

I agree with that. The difference is in what we define as the "system". If I exclude the sun from the system, then in that system linear momentum is not conserved. Then the sun acts as an external force on my system and angular momentum of the planet is conserved since the force is radial.