A rigid body of arbitrary shape is made up of point masses (for )
PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion
Problem
A rigid body of arbitrary shape is made up of N point masses mi (for i=1 to N), each located at a fixed distance ri from a vertical axle that passes through a stationary point O. The axle is fixed in space so that point O remains at rest.
The body is set into rotation about the axle with constant angular speed ω. Throughout the motion the center of mass of the body remains at point O.
(a) Starting from the definition of kinetic energy for each point mass, derive a general expression for the total kinetic energy of the body in terms of the set {mi}, the set {ri}, and the angular speed ω.
(b) Identify the single quantity in your expression from part (a) that summarizes the distribution of mass relative to the axle, and write a concise verbal definition of that quantity.
(c) Explain why your result in part (a) shows that the kinetic energy of any rigid body rotating about a fixed point depends on how its mass is distributed, not merely on its total mass.
(d) For the special case of a continuous mass distribution with surface mass density σ(r) on a thin flat disk of radius R (center fixed), outline the integral you would set up to evaluate the quantity identified in part (b). Do not perform the integration.