I have an inertia torque source (e.g. a flywheel) which is the input to a simple gear reducer, say with a ratio of 3:1, and the output is coupled to a purely inertial load (e.g. no friction or external torque). Let's say the flywheel has an initial ωfw of 120, so then the output of the gear reducer and velocity of the load is 40. By inertia laod, I mean to accelerate it and the simplest way is to decrease the moment of inertia of the flywheel. The initial kinetic energy is 1/2*Ifwo*ωfwo2 + 1/2*Ild*(ωfwo/3)2. If I reduce the moment of inertia of the flywheel by 1/2, then the ending kinetic energy is: 1/2*1/2*Ifwo*(ωfwo*Ifwo/1/2*Ifwo)2. The impulse momentum is the integral of torque wrt time. I can show that the work done to change the moment of inertia due to mpulse momentum on the gear housing is (-Le2/2*Ie)-(-Li2/2*Ii). The momentum balance is correct when the impulse momentum is added and the energy balance is correct for for a ratio of 1 (I.E. no gear reducer) but is incorrect for any ratio not equal to 1. I think this must be because there is energy associated with the impulse momentum on the gear housing. Can anyone confirm and if so explain how it's determined? If not, any other ideas as to the energy imbalance. This couldn't get much simpler but I can't get the energy balance straight.