Motion of the center of mass

The instantaneous velocity of a particle is v = d r / dt. Hence, if we take the time derivative of Eq. 5.10 we find

Velocity of the center of mass

vCM = ∑ mi v i (5.13)

This may be written in a form that highlights the importance of the CM:

Total linear momentum of a system of particles

P= M vCM = m1 v 1 + m2 v 2 +…+ mN v N (5.14)

The total momentum P = ∑ p _i of a system of particles is equivalent to that of a single (imaginary) particle of mass M = ∑ m_i moving at the velocity of the center of mass v _cm.

This result provides us with an enormous simplification: We may deal with the translational motion of extended objects or systems of particles, as if they were point particles with all the mass concentrated at the CM.

If we take the derivative of Eq. 5.14 we find M a _CM = ∑ m_i a _i = ∑ F _i = F _EXT, where F _i is the net force on the i th particle. We conclude that Newton's second law for a system of particles is

F _EXT = M a _CM (5.15)

The CM accelerates as if it were a point particle of mass M = ∑ m_i and the net external force were applied at this point. If we had started with the second law in the form F = d p / dt, we would have found

F _EXT = P (5.16)

The rate of change of the total momentum of a system is equal to the net external force.

Equations 5.15 and 5.16 allow us to apply the second law to a system of particles in a very simple way—provided we are interested only in the translational motion of the CM. For a complete description of the motion of the system, we would have to apply the second law to each individual particle, which can be a formidable task.

We may deduce Newton's first law as it applies to a system of particles either from the conservation of linear momentum or from the second law. Using either Eq. 5.15 or 5.17, we may state

If F _EXT = 0, then v _CM = constant

If the net external force on a system of particles is zero, the velocity of the center of mass remains constant.

Figure 5.5 shows a spinning wrench moving over a frictionless surface. Al­though the motion of any given particle is quite complex, the velocity of the CM stays constant.

FIGURE 5.5

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