Final answer:
The moment of inertia for a pulley system is the product of its weight and the squared centroidal radius of gyration. Given the unconventional notation in the question, the correct answer is a) 58 lb×k₀, assuming k₀ represents the squared radius of gyration.
Step-by-step explanation:
The moment of inertia of a double pulley system is calculated as the product of the weight of the pulley system and the centroidal radius of gyration squared (I = mass × k₀²). In US customary units, mass can be related to weight by using the acceleration due to gravity (g = 32.2 ft/s²). However, the question presents the weight in pounds and does not convert it to mass, which indicates a potential misuse of units or an unconventional notation. To stay consistent with the information provided in the question and the answer choices given which seem to imply a directly proportional relationship between weight and the radius of gyration, the moment of inertia would be 58 lb × (k₀)^2. But none of the answer options appropriately represent the square of the radius of gyration (k₀). This indicates that there might be a typo in the question or answer choices since the question's unit usage is not consistent with the standard definition of the moment of inertia in physics. Assuming the question intends for k₀ to represent the squared radius of gyration (an unconventional notation), the correct answer choice would be a) 58 lb×k₀. However, it's important to clarify the notation with an instructor or consult additional resources to ensure the question is interpreted correctly based on the context it is presented.