Question

# From this week's lab you learned that the change in gravitational potential energy of a falling...

From this week's lab you learned that the change in gravitational potential energy of a falling body can be captured and stored as a rotational kinetic energy of a spinning disk. This gave you the idea of a regenerative driver for an elevator in a tall building. When an elevator goes down the corresponding change in the gravitational potential energy of the elevator is transformed into the rotational kinetic energy of a solid disk-shape flywheel which can be extracted and used to lift up the elevator when needed.

(a) You consider a flywheel in a solid disk geometry of radius R = 0.85 m and thickness t = 26 cm made out of carbon fiber reinforced polymer (CFRP) of mass density ρ = 1650 kg/m3. The rotation axis of the flywheel is the center line of the disk geometry along the thickness direction. If you want the maximum rotational kinetic energy of the flywheel to be Emax = 200 MJ (mega-Joule), what should be the maximum angular speed of the flywheel ωmax?

Consider an elevator in a 35-story tall building driven by this flywheel. The floor-to-floor (vertical) distance is 3.7 m. The mass of the elevator cart is 2900 kg (when the elevator is empty). For simplicity, you ignore the mass of the cables connecting the elevator cart and the flywheel, and assume a perfect energy conservation between the rotational kinetic energy and the gravitational potential energy. The gravitational acceleration is 9.8 m/s².

(b) The elevator is used to transport down 3 identical heavy cargos from the floor number 26 to the floor number 4. Each cargo's mass is 841 kg. The cargos are big, and only 1 cargo can be put in the elevator. When the elevator reaches the floor number 4 carrying the last cargo, the angular speed of the flywheel is 555 rad/s. If the elevator is used only to transport these cargos, what must be the angular speed of the flywheel when the first cargo is loaded at the floor number 26?

solution: In this problem, we have to use the principle of conservation of mechanical energy. follow the steps, I have shown below.

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