Consider two identical gears touching each other. The system is friction less. One has a handle that you use to apply a torque on the entire system. If you turn the handle, there will be a non-zero torque on the system, but since the gears are identical and would have equal angular speeds but rotate in opposite senses their angular momenta would add to zero at any time. So how can the angular momentum of the system remain zero (i.e. never change) when there is an external torque being applied to the system?

Edit: The two gears are fixed on axles. And what if, to simplify, you are just applying a tangential force on the edge of one of the gears (no handles involved) to supply the torque?

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    $\begingroup$ If the gears are fixed on axles, the torque can be transferred to the other end of the axles. You don't have a closed system, so the conservation law does not apply. It is just like when you are standing and start walking forward. Your momentum increases. Does that violate momentum conservation? No, you have transferred it to the earth, which has changed velocity and angular velocity ever so slightly. $\endgroup$ Dec 8 '15 at 6:18

Here's the key question: what is supporting the two gears' axles? What happens when you turn that handle depends on your answer.

If the two axles are connected by some floating frame, then when you turn the handle on gear 1, the whole system will start rotating in just such a way to conserve angular (and linear) momentum.

If the two axles are mounted on some externally-supported frame that doesn't move, then that frame supplies the "missing" angular momentum to keep that second gear's axle motionless. And again, angular momentum will be conserved.

Edit: One more issue: how do you apply torque to that handle? Usually the handle on a wheel pivots, so that you are applying a linear force that only results in torque relative to the axle of the wheel. But again: what is that axle held by? If it's a floating frame, then pushing on the handle will mostly just translate the whole system (although there will be some rotation since your force through the handle probably won't be through the CG of the system). If it's a fixed frame, then you aren't applying a torque; the torque is the result of your force and the opposing force at the axle.

TL;DR: when you're talking about dynamics, you have to be absolutely clear about the environment and constraints. Otherwise you'll end up with conclusions that have no basis in reality.

  • $\begingroup$ So do you mean that the angular momentum isn't changing because there is an extra torque on the second gear's axle which counter acts the torque from my hand meaning the net torque is zero and the total angular momentum won't change? $\endgroup$ Dec 8 '15 at 22:47
  • $\begingroup$ Correct: there is a torque that you aren't accounting for that keeps the gears from moving. $\endgroup$ Dec 9 '15 at 0:23

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