To drive a typical car at 40 mph on a level road for one hour requires about $3.2 \times 10 ^ { 7 } \mathrm {~J}$ of energy. Suppose we tried to store this much energy in a spinning, solid, uniform, cylindrical flywheel. A large flywheel cannot be spun too fast or it will fracture. If we used a flywheel of diameter 1.2 m and mass 400 kg, what angular speed would be required to store $3.2 \times 10 ^ { 7 } \mathrm {~J} ?$
A) 940 rad/s
B) 3600 rad/s
C) 5500 rad/s
D) 530 rad/s
E) 1800 rad/s

Question

To drive a typical car at  40 mph on a level road for one hour requires about $3.2 \times 10 ^ { 7 } \mathrm {~J}$ of energy. Suppose we tried to store this much energy in a spinning, solid, uniform, cylindrical flywheel. A large flywheel cannot be spun too fast or it will fracture. If we used a flywheel of diameter  1.2 m and mass  400 kg, what angular speed would be required to store $3.2 \times 10 ^ { 7 } \mathrm {~J} ?$
A)  940 rad/s
B)  3600 rad/s
C)  5500 rad/s
D)  530 rad/s
E)  1800 rad/s

Quizplus · Accepted Answer

The energy stored in a spinning flywheel is given by the formula $E = \frac{1}{2} I \omega^2$, where $E$ is the energy, $I$ is the moment of inertia of the flywheel, and $\omega$ is the angular speed. For a solid, uniform cylindrical flywheel, the moment of inertia $I = \frac{1}{2} m r^2$, where $m$ is the mass of the flywheel and $r$ is its radius. Given that the diameter of the flywheel is 1.2 m, its radius $r = 0.6$ m. Substituting the given values, we get $I = \frac{1}{2} \times 400 \times (0.6)^2 = 72 \, \text{kg} \cdot \text{m}^2$. To find the angular speed $\omega$, we rearrange the energy formula to $\omega = \sqrt{\frac{2E}{I}}$, substituting $E = 3.2 \times 10^7 \, \text{J}$ and $I = 72 \, \text{kg} \cdot \text{m}^2$, we get $\omega = \sqrt{\frac{2 \times 3.2 \times 10^7}{72}} \approx 940 \, \text{rad/s}$.

To Drive a Typical Car at 40 Mph on a Level