Understanding Center of Mass

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Understanding Centre of Mass

Before we proceed to understand the concept of Centre of Mass, let us define a few terms we will use subsequently.

System of bodies (or masses): It is a collection of two or more distinct bodies interacting with each other through some form of force. For example, the solar system is a system of bodies (comprising the sun, planets, moons of the planets etc.) interacting with each other through the gravitational force. The configuration of the bodies comprising the system with respect to each other may or may not be fixed. For example, in the case of the solar system the configuration of the planets, moons etc. is constantly changing. However, in the case of two steel balls connected rigidly at the ends of a thin, massless rod (which constitutes a two body system) the configuration of the balls is fixed.

Rigid body: A rigid body as the term suggests is a body whose shape does not change. Sphere, rod, plate, rock are examples of rigid bodies.

The text book definition of centre of mass is as follows:

Centre of mass (CM) of a system of bodies or a rigid body is that imaginary point whose dynamic behaviour (i.e. motion) will remain unchanged if the entire mass of the bodies comprising the system or the rigid body were to be concentrated at that point and the resultant of the external forces acting on different bodies comprising the system or the rigid body were to act at that point.

This definition of centre of mass is somewhat abstract and needs some elaboration to really grasp the concept. Before we do so, let's recall from Newton's Laws of Motion that the motion of a body can be defined by i) its instantaneous momentum and ii) the rate of change of momentum if the body is in accelerated (or decelerated) motion. In this case the rate of change of momentum equals the net external force acting on the body. We can approach the concept of Centre of Mass either via momentum or via force. We take up both approaches as follows:

Centre of Mass via momentum approach: We can now explain the definition of CM of a system of bodies using the momentum to define motion as follows: At any given instant each body in the system has some momentum. The resultant momentum of the system is the vector sum of the momenta of different bodies in the system. Now imagine that the net mass of the ...