The operation of the Princeton Tokomak Fusion Test Reactor requires large bursts of energy. The power needed exceeds the amount that can be supplied by the utility company. Prior to pulsing the reactor, energy is stored in a giant flywheel of mass \(7.27 \times 10^{5} \mathrm{kg}\) and rotational inertia $4.55 \times 10^{6} \mathrm{kg} \cdot \mathrm{m}^{2} .$ The flywheel rotates at a maximum angular speed of 386 rpm. When the stored energy is needed to operate the reactor, the flywheel is connected to an electrical generator, which converts some of the rotational kinetic energy into electric energy. (a) If the flywheel is a uniform disk, what is its radius? (b) If the flywheel is a hollow cylinder with its mass concentrated at the rim, what is its radius? (c) If the flywheel slows to 252 rpm in 5.00 s, what is the average power supplied by the flywheel during that time?

Step 1: Calculate the radius of the uniform disk flywheel Using the given values, we can find the radius of the uniform disk flywheel: $$ r_d = \sqrt{\frac{2 \times 4.55 \times 10^{6}}{7.27 \times 10^{5}}} $$ $$ r_d ≈ 2.53\, \mathrm{m} $$ Step 2: Calculate the radius of the hollow cylinder flywheel Similarly, we can calculate the radius of the hollow cylinder flywheel: $$ r_c = \sqrt{\frac{4.55 \times 10^{6}}{7.27 \times 10^{5}}} $$ $$ r_c ≈ 2.00\, \mathrm{m} $$ Step 3: Convert angular velocities to rad/s Converting the angular velocities to rad/s: $$ \omega_i = 386 \times \frac{2 \pi}{60} ≈ 40.37\, \mathrm{rad/s} $$ $$ \omega_f = 252 \times \frac{2 \pi}{60} ≈ 26.39\, \mathrm{rad/s} $$ Step 4: Calculate the average power supplied by the flywheel Using the given rotational inertia and angular velocities, we can find the total energy lost by the flywheel: $$ \Delta E = \frac{1}{2} \times 4.55 \times 10^6 \times (26.39^2 - 40.37^2) ≈ - 2.67 \times 10^9\, \mathrm{J} $$ Now, we can find the average power supplied during the time interval: $$ P = \frac{-2.67 \times 10^9}{5.00} ≈ -5.34 \times 10^8\, \mathrm{W} $$ In conclusion, the radius of the flywheel as a uniform disk is approximately 2.53 m, and as a hollow cylinder is approximately 2.00 m. The average power supplied by the flywheel during the given time interval is approximately -5.34 x 10^8 W.