Problem

Energy storage devices that use spinning flywheels to store energy are starting to become...

Energy storage devices that use spinning flywheels to store energy are starting to become available. To store as much energy as possible, it is important that the flywheel spin as fast as possible. Unfortunately, if it spins too fast, internal stresses in the flywheel cause it to come apart catastrophically. Therefore, it is important to keep the speed at the edge of the flywheel below about 1000 m/s. In addition, it is critical that the flywheel be almost perfectly balanced to avoid the tremendous vibrations that would otherwise result. With this in mind, let the flywheel D, whose diameter is 0.3 m, rotate at ω = 60,000 rpm. In addition, assume that the cart B is constrained to move rectilinearly along the guide tracks. Given that the flywheel is not perfectly balanced, that the unbalanced weight A has mass mA, and that the total mass of the flywheel D, cart B, and electronics package E is mB, determine the following as a function θ, the masses, the diameter, and the angular speed of the flywheel:

(a) the amplitude of the motion of the cart,

(b) the maximum speed achieved by the cart.

Neglect the mass of the wheels, assume that initially everything is at rest, and assume that the unbalanced mass is at the edge of the flywheel. Finally, evaluate your answers to Parts (a) and (b) for mA = 1 g (about the mass of a paper clip) and mB = 70 kg (the mass of the flywheel might be about 40 kg).