A rigid body of mass M and radius R rolls without slipping across a laboratory floor with its center of mass moving at speed v. The body’s moment of inertia about its central axis can be expressed as I=kMR2, where the dimensionless constant k depends only on the object’s shape.
Three bodies, each of mass M and radius R, roll side-by-side at the same speed v:
• a solid sphere with k=0.40
• a solid cylinder with k=0.50
• a thin hoop with k=1.00
(a) In terms of k, derive an expression for the ratio of rotational kinetic energy to total kinetic energy for any object that rolls without slipping.
(b) Using your result from part (a), rank the three bodies from greatest to least in total kinetic energy. Clearly indicate any ties.
(c) Explain physically why the ranking obtained in part (b) is consistent with differences in how mass is distributed within each object.