Chapter 4 – Kinetics of Systems of Particles

Section 4.1 – Introduction We have primarily only discussed the kinetics of a single particle. We wish to extend this analysis to describe the motion of a general system of particles. This will unify the remaining topics of dynamics and permit us to treat the motion of rigid bodies, and fluids. A Rigid Body is a solid system of particles wherein the distances between the particles remains unchanged.

Section 4.2 – Generalized Newton’s 2nd Law We must generalize the Equation of Motion to describe a mass system which we will model by n mass particles bounded by a closed surface in space. The System under investigation is the mass within the boundary. This mass must be clearly defined and isolated.

There are two types of forces acting on each of the n particles in isolation: External Forces: due to sources external to the boundary such as contact forces with external bodies, external gravitational, electric, magnetic, … Internal Forces: due to sources internal to the system boundary. These are due to the interaction of the masses within the system with one another.

Consider a general system of particles: Where G is the Centre of Mass of the isolated system of particles. Let us further consider a representative particle i of mass mi within the system.

The forces acting on the isolated particle are: The ith particle is located by the position vector ri from the origin of an inertial frame. The location of the centre of mass is given by r

We apply the Equation of Motion to the isolated ith particle: And for the entire system of particles:

Notes: represents the acceleration of the instantaneous position of the centre of mass of the system of particles. For a non-rigid body this acceleration does not necessarily represent the actual acceleration of any of the individual mass particles that make up the system. Although F =ma requires the acceleration to be in the same direction as F, it does not require that F actually passes through G! In general, F does not pass through G  rotation of the system of particles about G in addition to the translation of G.

Section 4.4 Impulse & Momentum Linear Momentum The Linear Momentum of the ith particle is: The Linear Momentum of the Entire System is: