We have a variable-inertia flywheel coupled by a shaft to a fixed inertia flywheel. The VIF (variable-inertia flywheel) has a moment of inertia range of 10 to 5 (m2*kg) and an initial velocity of 100 (rad/s). The fixed inertia flywheel (fw) has a MI (moment of inertia) of 10. The initial momentum is Lvif=100*10=1000 (Nms) plus Lfw=100*10=1000 equals 2000 Nms. Since the ending momentum equals the beginning momentum, ωvife = (100*(10+10)/(5+10) = 133 rad/s. Now, we add a simple reversing gear pair between the fixed and the variable flywheels. The initial velocities are: ωvifi = 100; ωfwi = -100; the initial momentum for the VIF is 1000 Nms and for the FW it’s -1000 Nms, so the total momentum is zero. When the inertia of the VIF is changed from 10 to 5, it’s velocity will be 100*(10+10)/5+10 = 133 rad/s. This makes sense because the VIF still sees the inertia of the fixed flywheel even though the fixed flywheel is now reversed in direction (there is no negative inertia). But the total ending momentum is now Lvife = 133*5 = 665 Nms and for Lfwe = -133*10 = -1330 Nms equals -665 Nms – obviously incorrect (we started with zero momentum). It’s easy to understand that if two spinning flywheels (at equal but opposite directions and therefore having a net zero momentum) are coupled via a clutch the result will be zero momentum – even if they are coupled via an infinitely-variable transmission. What is the general rule when two sources of angular momentum are coupled via reversing gears?
The reversing gears embody a connection to the outside world. That connection can and will carry a torque. You no longer have a closed system. Angular momentum is no longer conserved. To go into painful detail... As your variable-inertia-flywheel has its moment of inertia reduced, it will be speeding up against the resistance of the reversing gear mechanism. It will be exerting a positive torque on that mechanism. As your fixed-inertia-flywheel speeds up due to the linkage, it will be getting an increasingly negative angular momentum. It must be under a negative torque. So it will also be exerting a positive torque on the reversing gear mechanism. So the reversing gear mechanism is under a net torque from the flywheels and, consequently, must be applying a net torque to the flywheels. If it is anchored to the outside world then you have a net torque from the outside world which must appear in the books when you try to balance angular momentum. If it is unanchored then it will start spinning and must be included in the books when you try to balance angular momentum.
Thank you - I can hardly believe I failed to consider gear housing torque. Seems I'm going backwards as I get older.