Section 18–4 Conservation of angular momentum

(External & internal torques / Constant angular momentum / Moment of inertia)

In this section, Feynman discusses the change of angular momentum due to external and internal torques, constant angular momentum, and moment of inertia of a system of particles. This section could be titled as “angular momentum of a system of particles” because he has discussed the conservation of angular momentum in the previous section.

1. External and internal torques:

“… internal torques balance out pair by pair, and so we have the remarkable theorem that the rate of change of the total angular momentum about any axis is equal to the external torque about that axis (Feynman et al., 1963, section 18–4 Conservation of angular momentum).”

Feynman explains that the total angular momentum of a system of particles is the sum of the angular momenta of all the parts. Thus, the rate of change of the total angular momentum about an axis of rotation is equal to the external torque about the axis. This theorem of angular momentum is applicable to any system of objects whether they form a rigid body or not. In the next chapter, Feynman discusses why the torque is equal to the rate of change of the angular momentum about an axis through the center of mass (CM) of an object that is accelerating. To be precise, French (1971) writes that “[r]egardless of any acceleration that the center of mass of a system of particles may have as a result of a net external force exerted on the system, the rate of change of internal angular momentum about the CM is equal to the total torque of the external forces about the CM (p. 641).”

According to Feynman, if Newton’s third law means that the action and reaction are equal, and they are directed in opposite directions exactly along the same line, then the two torques on two interacting objects are equal and opposite because the lever arms for any axis are equal. Note that he did not specify whether both action and reaction pass through the two objects. Interestingly, Kleppner and Kolenkow (1973) explain that “there is no way to prove from Newton’s laws that the internal torques add to zero. Nevertheless, it is an experimental fact that internal torques always cancel because the angular momentum of an isolated system has never been observed to change (p. 253).” Importantly, action and reaction are forces at a distance that do not necessarily lie on the straight line that connects the two objects.

Note: There are at least two forms of Newton’s third law: 1. Strong form means that action and reaction must act along the line joining the two particles. 2. Weak form means that action and reaction need not act along the line joining the two particles.

2. Constant angular momentum:

“…the law of conservation of angular momentum: if no external torques act upon a system of particles, the angular momentum remains constant (Feynman et al., 1963, section 18–4 Conservation of angular momentum).”

Feynman states the law of conservation of angular momentum as “if no external torques act upon a system of particles, the angular momentum remains constant.” A special case is that of a rigid body in which it has a definite shape while it is rotating around an axis. One may add that the conservation of angular momentum is dependent on the absence of an external torque, but the kinetic energy of a body may not be constant. For example, a planet orbiting about the Sun may increase its speed when it is moving nearer to the Sun. Feynman adds that we should consider a body that is fixed in its geometrical dimensions and is rotating about a fixed axis. However, a “fixed axis” means that the axis is fixed relative to the body and fixed in direction relative to an inertial frame, but it is not necessarily fixed in space.

One may expect Feynman to discuss Kepler’s second law of planetary motion in this section that is titled conservation of angular momentum. Historically, Kepler formulates the law of constant areal velocity as “a line drawn from the sun to a planet sweeps out equal areas in equal time intervals.” In other words, the areal velocity of a planet, orbiting the sun as a focal point, is always constant. Newton was the first physicist to recognize the physical significance of Kepler’s second law that is related to a radial force. Importantly, the law of conservation of angular momentum is applicable to a radial force that is not necessarily inversely proportional to the square of the distance. Therefore, physics teachers should clarify that “no external torque” does not mean that there is no external force.

3. Moment of inertia:

“Velocity is replaced by angular velocity, and we see that the mass is replaced by a new thing which we call the moment of inertia I, which is analogous to the mass (Feynman et al., 1963, section 18–4 Conservation of angular momentum).”

Feynman says that a body has inertia for turning which depends on the masses of its parts and how far away they are from the axis. He adds that the mass of an object never changes, but its moment of inertia can be changed; this is in contrast to his concept of relativistic mass. In Tips on Physics, Feynman elaborates that “for any rigid body, there is an axis through the body’s center of mass about which the moment of inertia is maximal, there is another axis through the body’s center of mass about which the moment of inertia is minimal, and these are always at right angles (Feynman et al., 2006, p. 122).” In short, the momentum of inertia of a rigid body is also dependent on the axis of rotation.

The law of conservation of angular momentum may be rephrased as “if the external torque is zero, then the angular momentum, the moment of inertia (I) times angular velocity (w), remains constant.” Feynman mentions that an important difference between mass and moment of inertia is very dramatic: if we stand on a frictionless rotatable stand with our arms outstretched, we may change our moment of inertia by drawing our arms in, but our mass does not change. However, one may expect Feynman to describe the spinning of a ice-skater that can be dramatic. A more dramatic example would be how a cat can rotate itself in the air after dropped vertically from an upside-down position and can become upright on its feet.

Questions for discussion:

1. How would you state the law of rate of change of angular momentum that is in terms of an external torque?

2. How would you state the law of conservation of angular momentum that is in the absence of an external torque?

3. How would you state the law of conservation of angular momentum that is in terms of moment of inertia?

The moral of the lesson: the angular momentum of a system of particles is constant if there is no external torque, or equivalently, the moment of inertia of an object times its angular velocity remains constant.

References:

1. Feynman, R. P., Gottlieb, & M. A., Leighton, R. (2006). Feynman’s tips on physics: reflections, advice, insights, practice: a problem-solving supplement to the Feynman lectures on physics. San Francisco: Pearson Addison-Wesley.

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

4. Kleppner, D., & Kolenkow, R. (1973). An Introduction to Mechanics. Singapore: McGraw-Hill.