# When are energy, mechanical energy, momentum, and angular momentum conserved?

I am in AP Physics and my only real hangup is knowing when the said quantities are conserved. Please define what is the SYSTEM in your answer.

I kind of have the basic idea. For example, if there are no external forces, momentum is conserved, and the same is true of torque and angular momentum.

It all just seems so complicated so it would be great if you could clear this up and show me that it's really quite a simple thing.

ONE EXAMPLE of a problem that's tricking me is this:

A 64.0-kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 505 kg · m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim counterclockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. Consider the woman–turntable system as motion begins.

Apparently, angular momentum is conserved but not mechanical energy or regular momentum.

I need some serious help with this.

• Hmmm ... "if there are no external forces" is a stronger condition than is absolutely needed for momentum conservation, but the less stringent version can require careful consideration and can be tricky to apply. Are you interested in answer that are easy to evaluate, answers that give the maximum flexibility or something in between? In your example there are (possibly strong) forces on the bearings of the turntable, but because of the small radius and the (unstated) assumption that the bearings are good you neglect the torque that might develop because of these forces. Nov 21, 2015 at 19:06

For these kinds of system we often define a pair of quantities, one which is characteristic of objects or systems and one which is characteristic of interactions. Examples of these pairs are work (interaction) and energy (system) or impulse (interaction) and momentum (system). There is no commonly applied name for the interaction quantity that pairs with angular momentum, but it would be the integral of torque applied over time in strict analogy with impulse and linear momentum.

Then the general rule is that the system-quantity is conserved if the system is subject to zero net interaction-quantity from external sources.

So

• Energy is conserved for system that experience zero net external work.
• Linear momentum is conserved for systems that experience zero net external impulse.
• Angular momentum is conserved for systems that experience zero net $\int\! \tau_{ext} \,\mathrm{d}t$.

and so on.

The "zero net external [interaction-quantity]" formulation can require considerable math to check, and can develop in a way that means the conservation rule only applies between certain states of the system. In each case you replace it with a stronger requirement that either external force or external torque be zero at all times allowing you to conserve the system-quantity between any two states of the system.

This whole discussion is pitched at a level where only the Newtonian formulation of physics is available. Once you have access to Lagrangian physics, Noether's theorem provides a precise and mathematically rigorous way to reformulate these considerations.

• Very concise, indeed.
– Gert
Nov 21, 2015 at 19:27