Physics
Physics
Problem
A uniform thin hoop of mass and radius can be rotated about two different fixed axes, each axis being perpendicular to the plane of the hoop: • Axis C passes through the center of the hoop. • Axis S is parallel to Axis C but intersects the plane of the hoop at a point on the hoop’s rim, a distance from the center. Assume the hoop is always rotated with its plane remaining horizontal. (a) Without performing a numerical calculation, state which axis—C or S—gives the larger rotational inertia for the hoop. Provide a clear physical explanation based on the distribution of mass relative to each axis. (b) Describe qualitatively, in words or with a sketch, how the rotational inertia of the hoop changes as the axis is moved continuously from Axis C to Axis S. Identify the feature of the system that controls the rate of change. (c) Suppose the hoop is set spinning at the same angular speed about each axis separately. Compare the rotational kinetic energies and for the two cases. Explain your reasoning in terms of the results from parts (a) and (b).Intuitive explanation
The key here is understanding how the moment of inertia changes when you shift the rotation axis—this is where the parallel axis theorem becomes essential. For a hoop, all the mass sits at a fixed distance from any axis in its plane, so you can calculate directly for each axis without integration. Once you know both moments of inertia, you can compare rotational dynamics (like angular acceleration under the same torque) or energy requirements to see how the axis location fundamentally affects the hoop's resistance to rotation.
Intuitive explanation
The key here is understanding how the moment of inertia changes when you shift the rotation axis—this is where the parallel axis theorem becomes essential. For a hoop, all the mass sits at a fixed distance from any axis in its plane, so you can calculate directly for each axis without integration. Once you know both moments of inertia, you can compare rotational dynamics (like angular acceleration under the same torque) or energy requirements to see how the axis location fundamentally affects the hoop's resistance to rotation.