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I've seen several posts with questions regarding transfer of energy and momentum through a gear box. Following is a problem I'm trying to work out but I clearly don't understand the physics involved:

A flywheel with inertia of I

As a starting point, I assumed that I could determine the ending velocity of the flywheel by using the conservation of momentum law. The flywheel sees the load inertia as I

I can show where energy plus the work done to decelerate the flywheel and accelerate the load is conserved (treating this as an in-eleastic connection), but momentum only is conserved by taking the momentum as the flywheel ending velocity times the sum of the flywheel inertia and the reflected load inertia. I feel certain this is not correct: The flywheel does have an ending momentum of its ending velocity times its inertia and the load does have momentum equal to the ending velocity of the load times its inertia. And momentum must be conserved. Any ideas?

(The worked out math is shown in the attachment)

A flywheel with inertia of I

_{fw}initial velocity of ω_{fwi}= 10,000 rpm is connected (assume through a lossless clutch) to a gear reducer with a ratio of n = 0.5. The output of the reducer is connected to a load inertia, I_{ld}which has an initial velocity of ω_{ld}= 0. The flywheel will decelerate (giving up momentum) and the load will accelerate (picking up momentum) until the velocity of the load is equal to the velocity of the flywheel times n. At that point, the system will be in equilibrium with both flywheel and load slowing due only to friction.As a starting point, I assumed that I could determine the ending velocity of the flywheel by using the conservation of momentum law. The flywheel sees the load inertia as I

_{ld}* n^{2}. So, the ending flywheel velocity should be the initial total momentum, L_{fwi}(which is the same as the ending total momentum) divided by the flywheel inertia, I_{fw}, plus the reflected inertia of the load, I_{ld}* n^{2}.I can show where energy plus the work done to decelerate the flywheel and accelerate the load is conserved (treating this as an in-eleastic connection), but momentum only is conserved by taking the momentum as the flywheel ending velocity times the sum of the flywheel inertia and the reflected load inertia. I feel certain this is not correct: The flywheel does have an ending momentum of its ending velocity times its inertia and the load does have momentum equal to the ending velocity of the load times its inertia. And momentum must be conserved. Any ideas?

(The worked out math is shown in the attachment)