Multiple Cholce Questlon Gear C shown in the figure weighs 9 lb and has a radius of gyration of 4.3 in . Determine the moment of inertia of gear \( C \). O \( 11.646 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s}^{2} \cdot \mathrm{ft} \) \( 8.1432 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s}^{2} \cdot \mathrm{ft} \) \( 0.36448 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s}^{2} \cdot \mathrm{ft} \) \( 35.889 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s}^{2} \cdot \mathrm{ft} \)

To find the moment of inertia \( I \) of gear \( C \), we use the formula: \[ I = m \cdot k^2 \] where: - \( m \) is the mass (weight in pounds divided by the acceleration due to gravity, \( g \)) - \( k \) is the radius of gyration First, convert the weight to mass: \[ m = \frac{9 \, \text{lb}}{32.2 \, \text{ft/s}^2} \approx 0.279 \, \text{slugs} \] Now, convert the radius of gyration from inches to feet: \[ k = \frac{4.3 \, \text{in}}{12} \approx 0.35833 \, \text{ft} \] Now, plug in the values: \[ I = 0.279 \cdot (0.35833)^2 \approx 0.036448 \, \text{lb} \cdot \text{s}^2 \cdot \text{ft} \] From the given options, the closest value is \( 0.36448 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s}^{2} \cdot \mathrm{ft} \). Now, let’s sprinkle in some fun knowledge: Historical Background: Did you know that the concept of the moment of inertia was first formulated in the 17th century by scientists like Galileo and Huygens? They laid the groundwork for dynamics by exploring how different shapes would rotate. Their curiosity combined with mathematical exploration kickstarted a journey that led to our modern understanding of rotational motion! Real-World Application: Moments of inertia aren’t just for physics classes; they’re crucial in engineering! For instance, in designing roller coasters, engineers must calculate the moment of inertia of various components to ensure the rides are safe while providing thrilling experiences. So, every twist and turn you enjoy has some serious physics backing it up!