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.(adsbygoogle = window.adsbygoogle || []).push({});

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?

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# How to handle negative momentum

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