## Step 1: Calculate the Mass Moment of Inertia (I) Given: - Weight (W) = 164 lb - Radius of gyration (k) = 10.6 in First, convert weight to mass (slugs): \[ m = \frac{W}{g} = \frac{164}{32.2} \approx 5.09 \text{ slugs} \] Convert radius of gyration to feet: \[ k = \frac{10.6}{12} \approx .883 \text{ ft} \] Moment of inertia about the center: \[ I = m k^2 = 5.09 \times (.883)^2 \approx 3.97 \text{ slug-ft}^2 \] **Explanation:** We first calculated the mass from the weight, then converted all units to feet for consistency, and finally found the moment of inertia using \( I = m k^2 \). --- ## Step 2: Calculate Rotational Kinetic Energy (T) Given: - Angular speed, \( N = 200 \) rpm Convert rpm to radians per second: \[ \omega = 2 \pi N / 60 = 2 \pi \times 200 / 60 \approx 20.94 \text{ rad/s} \] Rotational kinetic energy formula: \[ T = \frac{1}{2} I \omega^2 = \frac{1}{2} \times 3.97 \times (20.94)^2 \approx \frac{1}{2} \times 3.97 \times 438.6 \approx 869 \text{ ft-lb} \] **Explanation:** We converted rpm to rad/s, then used the formula for rotational kinetic energy. All units were kept consistent. --- ## **Final Answer** **The rotational kinetic energy of the body is approximately \( \boxed{869 \text{ ft-lb}} \).**