Section 20–4 Angular momentum of a solid body

(Angular velocity / Principal axes / Rotational kinetic energy)

In this section, Feynman relates the angular momentum of a solid body to its angular velocity, principal axes, and rotational kinetic energy. The ending of this chapter may seem abrupt without applying the equation of rotational kinetic energy, but he did give an additional lecture on gyroscope. Chapter 21 on gyroscope is omitted in Volume I possibly because it was “just for entertainment” (Feynman et al., 2006) and it is not easy for many students to understand.

1. Angular velocity:

“The main effect is that, in general, the angular momentum of a rigid body is not necessarily in the same direction as the angular velocity (Feynman et al., 1963, section 20–4 Angular momentum of a solid body).”

Feynman mentions that the main effect in three-dimensional rotations is the angular momentum of a rigid body is not necessarily in the same direction as the angular velocity. In Tips on Physics, he uses the word fact instead of effect: “the interesting fact that the angular momentum of a rigid body is not necessarily in the same direction as its angular velocity (Feynman et al., 2006, p. 122).” He explains that the moments of inertia about the two axes of a wheel are different and thus, the angular momentum components (in special axes only) are in a different ratio to the angular velocity components. However, this explanation is unclear. Using the equations L = r ´ mv and v = w ´ r, we can express the angular momentum as L = r ´ m(w ´ r). One may deduce that L is not necessarily a multiple of w (unless r is perpendicular to w).

In explaining L doesn’t point along ω, Feynman describes a wheel that is fastened onto a shaft in a lopsided fashion, and with the rotational axis through the center of gravity. Then, he asks how can there be a rate of change of angular momentum when we are only rotating the wheel about the axis? As a suggestion, we could use a simple example involving only a point mass that is rotating about the z-axis. For a point mass, it is simple to use L = r × mv to show that L is not definitely parallel to w. Alternatively, it is more illuminating to use a simple system consisting of two particles of mass m separated by a massless rod of length (Kleppner & Kolenkow, 1973). In this case, L is perpendicular to the massless rod and lies in the plane of the rod and the z-axis that is rotating at an angular velocity w.

2. Principal axes:

“These axes are called the principal axes of the body, and they have the important property… (Feynman et al., 1963, section 20–4 Angular momentum of a solid body).”

According to Feynman, “any rigid body, even an irregular one like a potato, possesses three mutually perpendicular axes through the CM…” The three axes are known as the principal axes of a rigid body in the sense that if the body is rotating about one of them, its angular momentum is in the same direction as the angular velocity. Specifically, the angular momentum of a body in any three axes can be expressed as L_x = I_xxω_x + I_xyω_y + I_xzω_z, L_y = I_yxω_x + I_yyω_y + I_yzω_z, and L_z = I_zxω_x + I_zyω_y + I_zzω_z. We can choose a co-ordinate system that rotates with the body to simplify the equations of angular momentum as L_x = I_xxω_x, L_y = I_yyω_y, and L_z = I_zzω_z. In general, L may not be parallel to ω, but L = Iω if the body is rotating about one of the principal axes of the body.

In Tips on Physics, Feynman says 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).” He adds that it is too complicated to prove the existence of principal axes that are perpendicular to each other. The proof involves a mathematical method P⁻¹AP that is known as a similarity transformation of the matrix A (P is an orthogonal matrix). This is related to the concept of Hermitian matrix that is very useful in quantum mechanics. We can use a Hermitian matrix to represent physical quantities such as position, energy, or angular momentum. It is discussed in chapter 20 of Volume III.

3. Rotational kinetic energy:

“The kinetic energy of rotation is KE = ½(Aω_x² + Bω_y² + Cω_z²) = ½ L·ω (Feynman et al., 1963, section 20–4 Angular momentum of a solid body).”

Using the x-, y-, and z-axes along the principal axes and the corresponding principal moments of inertia A, B, and C, we can find the angular momentum and the kinetic energy of rotation of a body for any angular velocity ω. Resolving ω into components ω_x, ω_y, and ω_z along the x-, y-, z-axes, we can express the angular momentum as L = Aω_xi + Bω_yj + Cω_zk and the rotational kinetic energy as KE = ½(Aω_x² + Bω_y² + Cω_z²) = ½ L·ω. Better still, we should apply the equations KE = L²/2I and L = Iω to explain why flying saucers are better than flying cigars in designing a spacecraft (Kleppner & Kolenkow, 1973). Based on L²/2I, a spinning object has higher rotational energy if its moment of inertia (I) is smaller. The motion of a flying saucer is more stable with a minimum L²/2I when it rotates with the largest moment of inertia after losing some rotational kinetic energy.

Feynman was disappointed that chapter 21 on gyroscope is excluded in Volume I of his Lectures. (It is included in chapter 4 of Tips on Physics.) In this chapter, he suggests that “[w]e picture the electron not simply as a point charge, but as a point charge that is a sort of limit of a real object that has angular momentum. It is something like an object spinning on its axis in the classical theory, but not exactly: it turns out that the electron is analogous to the simplest kind of gyro, which we imagine to have a very small moment of inertia, spinning extremely fast about its main axis (Feynman et al., 2006, pp. 127-128). However, a good ending of chapter 20 could be a mention of how he was having fun in physics without worrying about any importance.  In essence, he got the Nobel Prize by playing and thinking about the wobbling plate and relates it to spinning electrons.

Questions for discussion:

1. How would you explain the angular momentum of a rigid body is not necessarily in the same direction as the angular velocity?

2. How would you explain the property of principal axes of a rigid body?

3. Why flying saucers make better spacecraft than do flying cigars (using rotating kinetic energy)?

The moral of the lesson: the principal axes of a rigid body are fixed in the body whereby its angular momentum is in the same direction as the angular velocity if the body is rotating about one of them.

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