6. What is torque and angular momentum?

            In predicting the time-phased behavior of a system of many basic particles a force F can be applied which does not pass through the system mass center. The center of mass will still respond as predicted by F = ma, where m is the total system mass and a is the acceleration of the center of gravity. However the system of particles can also rotate. We thus need to know how to compute the rotational behavior of a system of particles.

           Consider a force applied to one particle which force does not pass through the center of mass of the system of particles. This particle will accelerate and due only to the acceleration of this one particle the system of particles will have a linear acceleration equal to the value of the force divided by the total system mass. Further, this force produces a torque about the center of mass equal to the value of the force times the perpendicular distance from the force line of action to the center of the mass. The angular momentum of a system of particles about the center of mass is defined as the linear momentum times the perpendicular distance from the force to the center of mass. In the case of the single force on a single basic particle (of mass m) being part of a larger system the angular momentum is mvr, where v is the particle velocity and r is the distance from the force to the center of mass of the system. The time rate of change of the angular momentum is d(mvr)/dt = mrdv/dt , in this special case where r is perpendicular to the force. Now mdv/dt is the force applied and (mdv/dt) times r is the torque applied, i.e., the force times the perpendicular distance. The equation giving the response to this torque of the complete system of particles thus is T = (mdv/dt)r =d(mvr)/dt. The resulting angular acceleration of the complete system of particles, of course, would generally be very small for a system consisting of a large number of particles.