Final answer:
Comparing the moment of inertia for a solid cylinder, hoop, and solid sphere, all with equal mass (6 kg) and radius (0.5 m), the hoop has the largest moment of inertia (1.5 kg·m²) because its mass is concentrated furthest from the axis of rotation.
Step-by-step explanation:
When comparing the moment of inertia for a solid cylinder, hoop, and solid sphere of equal mass (6 kg) and radius (0.5 m), we have to consider the distribution of mass in relation to the axis of rotation. The moment of inertia depends on how the mass of the object is distributed with respect to the axis about which it is rotating. In the provided examples, all objects have an equal mass and radius, but they will have different moments of inertia given by their specific formulas.
For a solid cylinder, the moment of inertia about its symmetry axis can be calculated using the formula ½MR², which for M = 6 kg and R = 0.5 m gives us an inertia of 0.75 kg·m². For a hoop, the moment of inertia about its symmetry axis is MR², which results in 1.5 kg·m². In the case of a solid sphere, the moment of inertia about its diameter is ⅗MR², leading to a value of approximately 0.6 kg·m².
Therefore, the hoop will have the largest moment of inertia because all of its mass is distributed at the outer edge, at maximum distance from the axis of rotation. This results in greater resistance to rotational acceleration compared to the other shapes where the mass is more evenly spread or closer to the axis.