Chapter 19: Center of Mass and Moment of Inertia

Center of Mass and Moment of Inertia , "Center of Mass; Moment of Inertia," establishes crucial concepts in classical mechanics necessary for analyzing the motion of extended bodies and complex systems. The discussion begins with the center of mass (CM), defined mathematically as the mass-weighted average position of a body's constituent parts. This centralized point is physically significant because the net external force acting on the system determines the acceleration of the CM, simplifying Newton's Second Law for large-scale objects. Techniques for locating the CM are detailed, including the use of integration for continuous objects, and a practical shortcut known as Pappus' Theorem is introduced, which relates the volume or surface area generated by rotating a plane curve or area to the distance moved by its CM. The chapter then transitions to rotational dynamics by defining the moment of inertia (I), which serves as the rotational analogue of mass, quantifying an object’s resistance to changes in angular velocity. The moment of inertia is calculated by summing the mass elements multiplied by the square of their respective distances from the axis of rotation. To simplify calculations involving parallel axes, the Parallel-Axis Theorem is derived, stating that the moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the CM plus the total mass times the square of the distance between the axes. Finally, the lecture introduces rotational kinetic energy, which is related to the moment of inertia and the square of the angular velocity. This concept is explored through the example of a figure skater changing their moment of inertia, demonstrating the relationship between angular momentum conservation and the work required to change the rotational kinetic energy. The chapter concludes by examining apparent forces within non-inertial reference frames, offering a detailed look at the radially outward centrifugal force and the sideways acting Coriolis force experienced by objects moving relative to a rotating system.