Dynamics of a Particle

We have seen that in an inertial reference frame, Euler’s first law (5.43) for the motion of the center of mass “particle” of a rigid body ℬ, a fictitious material point of mass m(ℬ) that moves with the body, has the same form as Newton’s second law (5.39) for the motion of a particle P of mass m(P). Hence, the motion of any such “material point” or “particle” is governed by the Newton-Euler law of motion, here written in its various forms as $$ F = \dot p = ma = m\dot v = m\ddot x, $$ (6.1) in which m is the mass of the “particle,” p=m v, and x, v, and a are its respective current position, velocity, and acceleration in an inertial reference frame.