Coupled Flywheels

Conservation of angular momentum does not always help in understanding the behavior of rotating devices. The diagram shows two flywheels, 1 and 2, of moments of inertia I₁ and I₂, mounted on parallel horizontal shafts along with pulleys of diameters D₁ and D₂. The belt is slack at first, and the two flywheels are

running at angular velocities ω₁₀ and ω₂₀. Suddenly the belt is tightened. One can write out the torque equations and the angular momentum equation to get the relation I₁ ω₁ + I₂ ω₂ = k – (N – 1) I₁ ω₁. Here, k is a constant of integration and N = D₂ / D₁, the ratio of pulley diameters. When N = 1, angular momentum is conserved. If N ≠ 1 and ω1 changes, the angular momentum is not conserved! Why not?

Answer

The overall angular momentum of the system must be conserved, so including just the change in angular momenta of the flywheels leads to an incomplete calculation. The tension is different in the two sides of the belt, so the belt exerts a downward force on pulley 2 and an upward force on pulley 1. These forces are counteracted by reactions at the bearings, in addition to the reactions to the weight of the components. These additional reactions produce a torque that accounts for the change in angular momentum.

If the pulleys are the same size, this additional torque does not exist unless the belts are crossed.