PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Energy and Momentum of Rotating Systems · Energy and Momentum of Rotating Systems
Problem
A rigid body is modeled as N point masses, each of fixed mass mi, located at positions that do not change relative to one another. The body translates with center-of-mass speed VCM and simultaneously rotates with angular speed ω about an axis through the center of mass.
Answer the following in a clear, logically ordered manner, showing all steps in your reasoning.
(a) Write a single mathematical statement that defines the total kinetic energy of the system in terms of the individual masses and their instantaneous speeds.
(b) Prove that the vector sum of all position vectors measured from the center of mass is zero for this system.
(c) Starting from your expression in part (a) and using the result from part (b), demonstrate that the total kinetic energy can be expressed as the sum of two distinct contributions: one associated only with the motion of the center of mass and the other associated only with rotation about the center of mass.
(d) Clearly state which physical parameter of the body determines each of the two contributions identified in part (c).