Section 18–1 The center of mass

(Motion of center of mass / Center of mass vector / Stationary center of mass)

In this section, Feynman discusses the motion of center of mass, a definition of center of mass vector, and an example whereby the center of mass is stationary. This section only gives a short introduction to the concept of center of mass. The next chapter elaborates on this concept further with two sections: 19–1 Properties of the center of mass and 19–2 Locating the center of mass. We may understand this section from the perspectives of idealization, approximation, and exception.

1. Motion of center of mass:

“That is called the theorem of the center of the mass, and the proof of it is as follows… (Feynman et al., 1963, section 18–1 The center of mass).”

According to Feynman, the first theorem concerning the motion of complicated objects can be observed if we throw an object made of some blocks held together by strings, into the air. It demonstrates that there is a mean position (or effective center) which moves parabolically, but it is not definitely a point of the material itself. The theorem of the center of mass’s motion can be developed from a simpler model involving two particles first. The simplified theorem may be stated as “[t]he motion of the center of mass of a system of two particles is the same as the motion of a single particle of mass equal to the total mass of the system acted on by the resultant of all the external forces which act on the individual particles (French, 1971, p. 353).” Then, we can let students deduce a theorem involving a system of many particles.

It is not straightforward to develop an equation to describe the motion of complicated objects such as “flowing water” or whirling galaxies.” The equation describing the center of mass of an extended body involves idealizations and simplifications. Firstly, we idealize a rigid body as “a collection of particles whose relative distances are constrained to remain absolutely fixed (Thornton & Marion, 2004, p. 411).” The rigid body is an idealization that does not exist in nature because the atoms within the body are always in motion. In addition, we simplify the motion using the equation F = MR that describes only the translation of the body (or the motion of center of mass). This idealized equation does not provide information on the body’s orientation in space as well as the location of every atom within the body.

2. Center of mass vector:

“… M is the sum of all the masses, i.e., the total mass. Then if we define a certain vector R to be R = S_im_ir_i/M, Eq. (18.3) will be simply F = d²(MR)/dt² = M(d²R/dt²), since M is a constant (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman defines the center of mass vector using R = S_im_ir_i/M and expresses the external force F on the total mass M of a body as F = d²(MR)/dt² = M(d²R/dt²). The point at R is called the center of mass of the body and it is a kind of average r in which the different r_i’s have weights that are proportional to the masses. In a sense, one may assume the weights of the masses are under a parallel gravitational field. Importantly, the center of gravity for an extended object that is relatively large does not necessarily coincide with the center of mass of the same object. By assuming the “same” gravity, we may say that the center of gravity of a body is approximately located at the same point as the body’s center of mass.

Another approximation is based on the assumption in which the parts of a body are moving at speeds very much slower than the speed of light. Using the nonrelativistic approximation for all quantities, we may deduce that the mass of the moving body is constant such that we can write F_i = d²(m_ir_i)/dt². We could include internal forces within the body using Newton’s second law by writing S_iF_i^ext + S_iF_i^int = M d²R/dt² and stating S_iF_i^int = 0 due to Newton’s third law. However, Newton’s third law is not always valid for moving charged particles because electromagnetic forces are velocity dependent. To be precise, magnetic forces exerted on a moving charge q in a magnetic field B obey the weak form of the third law (F_ab = -F_ba), but not the strong form (action and reaction must act along the line joining the particles).

3. Stationary center of mass:

“… then this little blob of gas goes one way as the rocket ship goes the other, but the center of mass is still exactly where it was before (Feynman et al., 1963, section 18–1 The center of mass).”

Feynman discusses a special case (or exception) of a center of mass’s motion. He asks whether it is absolutely impossible to have rocket propulsion because one cannot move the center of mass. Although the center of mass is stationary, he clarifies that we can propel an interesting part of the rocket by throwing away the uninteresting part (exhaust gases). On the other hand, there could be a discussion about the relativistic limitation to the concept of an absolutely rigid body. For instance, if we strike a blow at one end of the rocket, it is impossible for the effect to be felt instantaneously at the opposite end. This is related to the transmission of a signal with an infinite velocity that is forbidden by the theory of relativity.

In the real world, the lift-off of a rocket has to overcome the Earth’s gravitational forces while it accelerates to the outer space. Similarly, a system of (colliding) particles cannot be completely isolated from gravity. In the words of French (1971), “[t]he conservation of momentum in a collision process holds good only to the extent that the effect of any external forces can be ignored. If external forces are indeed present, the duration Dt of the collision must be so short that the product F Dt is negligible. A different way of stating this same condition is that the forces of interaction between the colliding particles must be much greater than any external forces which may be acting (p. 353).”

Questions for discussion:

1. What are the idealizations or simplifications in conceptualizing the center of mass’s motion?

2. What are the approximations needed in writing the equation of the center of mass’s motion?

3. What are the exceptions that are related to the center of mass’s motion?

The moral of the lesson: we may define the center of mass vector using R = S_im_ir_i/M and express the external force F on the total mass M of an idealized object as F = M(d²R/dt²).

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. French, A. (1971). Newtonian Mechanics. New York: W. W. Norton.

3. Thornton, S. T. & Marion, J. B. (2004). Classical Dynamics of Particles and Systems (5th Edition). Belmont, CA: Thomson Learning-Brooks/Cole.