Section 19–1 Properties of the center of mass

(Symmetrical properties / Scaling properties / Accelerated reference frame)

In this section, Feynman discusses symmetrical properties and scaling properties of the center of mass, as well as a theorem of motion of the center of mass in an accelerated reference frame.

1. Symmetrical properties:

“… if it is just any symmetrical object, then the center of gravity lies somewhere on the axis of symmetry because in those circumstances there are as many positive as negative x’s (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

Center of mass (CM) is a point “inside” an object where the net external force may produce an acceleration of an imaginary particle at this point as if the whole mass of the object were to be concentrated there. Using circular symmetry, Feynman clarifies that CM does not have to be in the material of a body, for example, the CM of a hoop is in the center of the hoop that is not in the hoop itself. In the case of a rectangle that is symmetrical in two planes, we can easily determine its CM that lies on their line of intersection. Similarly, if a body is symmetrical about an axis, its CM also lies on the same axis. More importantly, we can use possible symmetrical properties of CM to simplify physics problems.

Idealization (or simplification): we idealize a rigid body as a system of discrete particles (or a continuous distribution of matter) and gravitational forces are uniform in a small region of space. Feynman also proves a theorem that simplifies the location of the center of mass by assuming a body is composed of two or more parts whose centers of mass are known. However, one may elaborate that the location of the center of mass of a rigid body is uniquely defined, but the center of mass vector is dependent on the selected coordinate system. Furthermore, the center of mass is defined without reference to the gravity of an object. These properties of the center of mass could have been summarized at the beginning or at the end of the section.

2. Scaling properties:

“…Newton’s law has the peculiar property that if it is right on a certain small scale, then it will be right on a larger scale (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

The concept of scaling is not commonly found in current physics textbooks. Feynman explains that Newton’s laws of dynamics hold for the motion of objects at a higher scale and it becomes more accurate as the scale gets larger. On the contrary, quantum mechanics for the small-scale atoms are quite different from Newtonian mechanics that are applicable to large-scale objects. Historically, Galileo (1638) writes that “if the size of a body be diminished, the strength of that body is not diminished in the same proportion; indeed the smaller the body the greater its relative strength (p. 131).” He investigated scaling properties, for example, the relationship between speed and distance of a moving body as well as the irregular shapes of bones. Feynman (1969) has also contributed to the concept of scaling in his interpretation of experimental results on deep inelastic scattering (electron-proton collisions).

Approximation: according to Feynman, we can approximate the motion of bodies at a larger scale by a certain expression in which it keeps reproducing itself on a larger and larger scale. In addition, Newton’s laws of dynamics are similar to the “tail end” of the atomic laws and they can be extrapolated to a very large scale. Interestingly, the laws of motion of particles on a small scale are very peculiar, but a large number of particles also approximately obey Newton’s laws. Currently, one may prefer Feynman to suggest the need of laws of motion for galaxies, for example, Modified Newtonian dynamics (MOND) is developed for motion of bodies at an even larger scale. It provides an alternative explanation for the motion of galaxies that do not appear to obey Newton’s laws.

3. Accelerating reference frame:

“… the theorem that torque equals the rate of change of angular momentum is true in two general cases: (1) a fixed axis in inertial space, (2) an axis through the center of mass, even though the object may be accelerating (Feynman et al., 1963, section 19–1 Properties of the center of mass).”

Feynman states two validity conditions of the theorem concerning the center of mass in which the torque is equal to the rate of change of angular momentum: (1) a fixed axis in an inertial frame, (2) an axis through the center of mass in an accelerating frame. He elaborates that an observer in an accelerating box would expect the same situation (or experience same magnitude of forces) if an object were in a uniform gravitational field whose g value is equal to the acceleration a. One may add that an inertial force acting on the object is equivalent to an apparent gravitational force based on Einstein’s principle of equivalence. In other words, the theorem involving an external torque acting on an accelerating object is equivalent to the same object that is at rest, but it is now under the influence of apparent gravitational field.

Exception (or limitation): when a small object is supported at its center of mass, there is no torque on it because of a parallel gravitational field. This is not strictly true for a large object because gravitational forces are non-uniform, and thus, the center of gravity of the large object departs slightly from its center of mass. However, Feynman could have included Chasles’ theorem that describes the motion of a body as a sum of two independent motions: a translation of the body plus a rotation about an axis. A special case of this theorem is to choose the axis at the center of mass of the body that allows the angular momentum to be split into two components: the motion of the center of mass and the motion around the center of mass. It helps to connect the discussions of translational kinetic energy and rotational kinetic energy in section 19.4.

Questions for discussion:

1. What are the symmetrical properties of the center of mass?

2. What are the scaling properties of the center of mass?

3. What are the validity conditions of the theorem concerning the center of mass in which an external torque is equal to the rate of change of angular momentum?

The moral of the lesson: the center of mass of a rigid body has symmetrical properties and scaling properties, and there are two validity conditions of the theorem concerning the center of mass in which the torque is equal to the rate of change of angular momentum: a fixed axis in an inertial reference frame and an axis through the center of mass in an accelerating reference frame.

References:

1. Feynman, R. P. (1969). Very high-energy collisions of hadrons. Physical Review Letters, 23(24), 1415-1417.

2. 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.

3. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences (Trans. by Crew, H. and de Salvio, A.). New York: Dover.