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Extended Object Momentum

An extended object means a composite system of many particles, whether loosely bound or rigid. Loosely bound system is one in which any two particles can move relative to each other in time. A rigid system is one in which, by definition, there is no relative motion between the constituent particles. (A rigid body is an idealization; according to special theory of relativity, a rigid body cannot exist.)

Suppose we have an (extended) object consisting of n particles. In an inertial frame S, at any instant of time, let the velocity of ith particle be i; hence its momentum is p_i= m_i v_i. The total momentum of the object is defined as the vector sum of momenta of its constituent particles:

How do we actually determine P_sys for even a simple extended object in motion, say a rolling sphere, where it is practically impossible to determine the momenta of all the particles (which are different, either in magnitude or direction, for each particle at any instant of time)? How do we write Newton’s equation of motion for such a composite system? A physically useful meaning of P_sys emerges from the concept of center-of-mass of an extended object.