Introduction – Centre of Mass
Each object around us has certain shape and size. When we study motion of such objects, we will have to study motion of all the different particles of the object. Practically speaking, this is not possible and we will have to consider alternative ways to study motion of the object. Scientists have defined a concept called `Centre of Mass’ to take care of this situation. To consider motion of an object having finite size, they consider motion of the centre of mass of the object. The basic objective of this chapter is to understand this concept.
OBJECTIVE
We will attempt to define the concept of a point where the mass of the whole body can be assumed to be concentrated. This helps us in analysis of problems that involve application of forces that result in subsequent motion. We will study systems which have a non uniform distribution of mass. And finally we will understand the motion of centre of mass as the body moves.
PRE-REQUISITE
The following pre-requisites apply
Þ Vector Representation.
Þ Integration in one variable.
Þ Force & Acceleration.
CORE CONCEPTS
Classically each and every object has some finite size, but we have assumed them as particles i.e. having mass but no size so far.
In translatory motion each point on an object undergoes the same displacement as any other point as time goes on, so that the motion of one particle represents the motion of whole object. Thus, the translatory motion of an object of finite size can be studied by analysing the motion of any constituent particle.
But, in cases where the motion is not translatory, rather than considering any point on the object, a point known as the centre of mass is defined, and the motion of this point is studied. The motion of the centre of mass leads to the analysis of the object as a whole
Conceptually, the point where the whole mass of body or system can be assumed to be concentrated for simplified study of its motion is called the centre of mass.
For a discrete system of particles the positions (see figure given below) of the centre of mass is
