Section 21–3 Harmonic motion and circular motion

(Acceleration-displacement / Angular frequency & speed / Relationship to circles)

In this section, we can understand oscillatory motion from the perspectives of “acceleration-displacement relation,” “angular frequency-angular speed relation” and the relation between oscillatory motion and circular motion.

1. Acceleration-displacement:

“… when a particle is moving in a circle, the horizontal component of its motion has an acceleration which is proportional to the horizontal displacement from the center (Feynman et al., 1963, section 21–3 Harmonic motion and circular motion).”

Feynman explains that it is the magnitude of the acceleration times the cosine of the projection angle with a minus sign because it is toward the center: a_x = −a cos θ = −ω₀²x. In addition, when a particle is moving circularly, the horizontal component of its motion has an acceleration which is proportional to the horizontal displacement from the center. We should be more organized by stating three hallmarks of simple harmonic motion: (1) sign: the acceleration is anti-phase (p rad) with respect to the displacement and the minus sign is due to the restoring force of Hooke’s law, (2) magnitude: the acceleration is proportional to the displacement, and (3) gradient: the slope of the acceleration-displacement graph is equal to the square of the angular frequency ω² (or the quotient of the spring constant and the mass of the spring).

Feynman suggests devising an experiment to show how the to-and-fro motion of a mass on a spring is related to a point going around in a circle. Specifically, we can use an arc light projected on a screen to cast shadows of a crank pin on a shaft and of a vertically oscillating mass, side by side. However, a real-life example is the apparent simple harmonic motion of a Jupiter’s moon that is actually a uniform circular (or elliptical) motion. In 1610, Galileo discovered four principal moons of Jupiter using his refracting telescope (French, 1971). Each moon appears to be oscillating relative to Jupiter, but it may disappear behind the planet or cast its shadow on the planet. In essence, the oscillatory motion of each moon is equivalent to the projection of circular motion on a diameter of a circle.

2. Angular frequency & speed:

“If we let go of the mass at the right time from the right place, and if the shaft speed is carefully adjusted so that the frequencies match, each should follow the other exactly (Feynman et al., 1963, section 21–3 Harmonic motion and circular motion).”

Feynman says that the displacement of a mass on a spring will be proportional to cos ω₀t, and it will be exactly the same motion as the observed x-component of the position of an object rotating in a circle with angular velocity ω₀. He adds that if the shaft speed is carefully adjusted so that the “frequencies match,” then each should appear to move together. In a sense, Feynman could have said that the two motions are in phase if the angular speed matches the angular frequency (the same symbol w is used for both quantities). Note that the period of the circular motion and simple harmonic motion are both expressed as T = 2p/w and they are expected to be the same. We may also explain that the angular frequency of the simple harmonic motion matches the rotational frequency of the circular motion.

Feynman elaborates that if a particle moves circularly with a constant speed v, the radius vector from the center of the circle to the particle turns through an angle θ whose size is proportional to the time. He states a formula for the angle θ = vt/R and the angular speed dθ/dt = ω₀ = v/R. As a suggestion, we can distinguish angles as a physical angle for the circular motion and a phase angle for the oscillatory motion. That is, the projection of an object rotating through a physical angle with respect to a reference circle matches the motion of an oscillating object. On the other hand, we can conceptualize the oscillatory motion of an object with reference to a phasor or phase angle. The term phasor was already used in physics before the invention of the Star Trek’s phaser (or phased array pulsed energy projectile weapon).

3. Relationship to circles:

“This is artificial, of course, because there is no circle actually involved in the linear motion—it just goes up and down (Feynman et al., 1963, section 21–3 Harmonic motion and circular motion).”

We can simply analyze an oscillatory motion in the x-direction if we imagine it to be a projection of an object moving in a circle. However, Feynman suggests that we may supplement the equation md²x/dt² = -kx with md²y/dt² = -ky, and put the two equations together. By having these two equations, he claims that we can analyze the one-dimensional oscillator with circular motions, which is easier than solving a differential equation. Physics teachers should clarify that the circular motion is mathematically equivalent to a superposition of two simple harmonic motions at right angles. In other words, if we connect an object to a horizontal spring and a vertical spring, the object may move circularly due to a combined effect of the horizontal spring force and the vertical spring force.

According to Feynman, the fact that cosines are involved in the solution of md²x/dt² = -kx indicates that there might be a relationship between oscillatory motions to circles. He explains that the relationship is artificial because there is no circle actually involved in the oscillatory motion. Although Feynman has chosen an example that has an artificial relationship to circles, the simple harmonic motion of a Jupiter’s moon observed a telescope is actually moving circularly. Thus, it is possible to solve an oscillatory motion problem by having a generous heart: to conceptualize a one-dimensional problem using higher dimensions. In the real world, the apparent one-dimensional oscillatory motion of a Jupiter’s moon is really a two-dimensional circular motion.

Questions for discussion:

1. How would you explain the relationship between the acceleration and displacement of an oscillating object?

2. How would you explain the relationship between the angular frequency of an oscillating object and the angular speed of a rotating object?  

3. Is there a relationship between oscillatory motions to circles?

The moral of the lesson: we can solve simple harmonic motion problem by having a bigger heart, that is, to conceptualize a one-dimensional oscillatory problem using two-dimensional circular motion.

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.

Section 21–2 The harmonic oscillator

(Dynamical aspects / Kinematical aspects / Mathematical properties)

In this section, Feynman discusses dynamical aspects, kinematical aspects, and mathematical properties of a simple harmonic oscillator.

1. Dynamical aspects:

“… first the spring stretches to balance the gravity; once it is balanced, we then discuss the vertical displacement of the mass from its equilibrium position (Feynman et al., 1963, section 21–2 The harmonic oscillator).”

Feynman suggests a simplest mechanical system whose motion follows a linear differential equation. It refers to a vertical spring that is stretched to balance the gravity. (A simpler motion is a uniform motion that can be represented as dx/dt = c.) Perhaps Feynman should have specified that the spring is massless such that it is not further stretched because of its mass. More important, it is simpler to conceptualize a horizontal spring-mass system that is attached to an object instead of the vertical spring-mass system. An example involving a horizontal spring is better because we need not discuss how the spring force balances the gravity. Thus, the gravitational field may be disregarded since the motion of the object is horizontal and we do not need to determine the gravitational potential energy of the system.

Feynman assumes the spring is perfectly linear in the sense that the restoring force is linearly proportional to the amount of stretch. Mathematically, the restoring force is –kx (with a minus sign to means pulls back) and thus, we have ma = –kx. As a suggestion, we may consider an object of mass m is attached to an ideal spring on a frictionless table and it oscillates in a vacuum. Specifically, the equation F = –kx is valid as long as the spring is stretched or compressed by a relatively short distance within its elastic limit. In the real world, the stiffness of a spring is non-linear, that is, the spring constant k is not definitely constant but it depends on the amount of stretch.

2. Kinematical aspects:

“As an example, we could write the solution this way: x = a cos ω₀(t − t₁), where t₁ is some constant. This also corresponds to shifting the origin of time to some new instant (Feynman et al., 1963, section 21–2 The harmonic oscillator).”

Feynman shows three possible solutions: (a) x = a cos ω₀(t − t₁), (b) x = a cos (ω₀t + Δ), and (c) x = Acos ω₀t + Bsin ω₀t. To clarify that x = cos ω₀t is only a possible solution, he asks what if we were to walk into the room at another time? In a sense, he was shocked that there is an infinite number of solutions that have different amplitudes. However, Landau prefers using exponential functions: “[t]he use of exponential factors is mathematically simpler than that of trigonometrical ones because they are unchanged in form by differentiation (Landau, & Lifshitz, 1976, p. 59).” In the next chapter, Feynman calls Euler’s exponential function “our jewel” and mentions that “in our study of oscillating systems, we shall have occasion to use one of the most remarkable, almost astounding, formulas in all of mathematics.”

Feynman initially calls ω₀ the angular frequency and defines ω₀ as the number of radians by which the phase changes in a second. In addition, the quantity ω₀t is the phase of the motion and the time that changes by an amount t₀ is the period of one complete oscillation. At the end of this chapter, he also calls ω₀ the natural frequency of the harmonic oscillator, and ω the applied (forcing) frequency. One may prefer the term natural frequency that is dependent on the mass m and spring constant k of an ideal spring (and its natural motion). Physics teachers should emphasize that this property of the natural frequency is based on the assumption that the oscillations are relatively small, but it can vary with the real spring constant in the real world.

3. Mathematical properties:

“That fact illustrates one of the most important properties of linear differential equations: if we multiply a solution of the equation by any constant, it is again a solution (Feynman et al., 1963, section 21–2 The harmonic oscillator).”

A property of linear differential equations is: if x is a solution, then Ax is also a solution of the same equation (A is a constant). Feynman explains that if we pull a spring twice as far, the force, acceleration, velocity, and distance covered are also twice as great. (One may use the equation a = -ω₀²x to explain how acceleration is directly proportional to the distance.) Thus, it takes the same time for the spring to return to the origin and is independent of the initial displacement. In other words, the period of a simple harmonic oscillator is independent of its amplitude or total energy, and this property of the system is known as isochronous. Feynman’s so-called horror is not warranted because this is an ideal motion of an ideal spring that does not experience friction or air resistance and it does not heat up as it oscillates.

Feynman elaborates that the constant Δ and ω₀t+Δ are both sometimes called the phase of the oscillation, but he prefers to say that Δ is a phase shift from some defined zero to avoid confusions. He says that the constants A = a cos Δ and B = −a sin Δ are not determined by the equation, but they depend on how the motion is started. However, mathematicians may clarify that the general solution of a second-order linear differential equation can be expressed as y = Ay₁ + By₂ in which A and B are arbitrary constants. In general, it is possible to have infinite pairs of arbitrary constants that fit the equation. More importantly, the two arbitrary constants are not completely arbitrary because they depend on the initial conditions (e.g., initial position and initial velocity of an object) or how we release the spring at time t = 0.

Questions for discussion:

1. How would you explain the dynamical aspects of a simple harmonic oscillator?

2. How would you explain the kinematical aspects of a simple harmonic oscillator?

3. How is the mathematical property of a simple harmonic oscillator related to the arbitrary constants of a solution of a linear differential equation?

The moral of the lesson: we need to idealize a spring that oscillates in vacuum in order that the period of a simple harmonic motion is independent of its amplitude or total energy.

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. Landau, L. D., & Lifshitz, E. M. (1976). Mechanics (3rd ed.). Oxford: Pergamon Press.

Section 21–1 Linear differential equations

(Oscillator-like phenomena / Daily-life applications / Mathematical definition)

In this section, Feynman discusses linear differential equations from the perspective of oscillator-like phenomena, daily-life applications, and mathematical definition.

1. Oscillator-like phenomena:

“The harmonic oscillator, which we are about to study, has close analogs in many other fields; … we are really studying a certain differential equation. (Feynman et al., 1963, section 21–1 Linear differential equations).”

Feynman says that a strange thing occurs again and again: the equations which appear in different fields of physics and in other sciences are often almost exactly the same. Many phenomena are modeled using similar differential equations, for example, the propagation of sound waves is analogous to the propagation of light waves. There is a better explanation: “[t]he first is the limited imagination of physicists: when we see a new phenomenon we try to fit it into the framework we already have… (Feynman, 1985, p. 149).” That is, we idealize phenomena that can be described as oscillator-like. In essence, physicists develop models of phenomena using linear differential equations that can be solved and understood.

Feynman suggests that it is best to realize how the study of a phenomenon in one field may permit an extension of our knowledge in another field using linear differential equations. However, Heisenberg (1967) clarifies that “[p]ractically every problem in theoretical physics is governed by nonlinear mathematical equations, except perhaps quantum theory, and even in quantum theory it is a rather controversial question whether it will finally be a linear or nonlinear theory. Therefore by far the largest part of theoretical physics is devoted to nonlinear problems (p. 27).” Generally speaking, we simplify physical problems by ignoring factors such as friction or assuming an object’s displacement is small. Real phenomena are often not modeled by non-linear differential equations because they are either difficult to be solved or insolvable.

2. Daily-life applications:

“… all these phenomena follow equations which are very similar to one another, and this is the reason why we study the mechanical oscillator in such detail (Feynman et al., 1963, section 21–1 Linear differential equations).”

Linear differential equations are applicable to phenomena such as the oscillations of a mass on a spring, the oscillations of charge flowing back and forth in an electrical circuit and the analogous vibrations of the electrons in an atom. Specifically, the simplest linear differential equation is the equation of uniform motion that can be represented by dx/dt = c. In Heisenberg’s (1967) words, “[a]ctually mathematical physics started 300 years ago with the law of inertia, which may be considered to be the solution of the homogeneous linear equation d²x/dt² = 0 where x is the coordinate and t is the time. In the laws of free fall of Galileo, we find that he actually had solved an inhomogeneous linear equation, the force being the inhomogeneous term (p. 28).” In short, linear differential equations are useful in the modeling of physical motions.

There are other applications of differential equations such as a thermostat adjusting a temperature, complicated interactions in chemical reactions, and foxes eating rabbits (rate of change of population). Feynman’s reason for the study of the mechanical oscillator is: these phenomena follow differential equations which are very similar to one another. One may add that “it’s not because Nature is really similar; it’s because the physicists have only been able to think of the same damn thing, over and over again (Feynman, 1985, p. 149).” As a suggestion, we may explain that the differential equations are useful because they can represent the rate of change of an independent variable with respect to a dependent variable. In general, physicists need to find out how an observable would change with respect to time or with respect to distance.

3. Mathematical definition:

“Thus a_ndⁿx/dtⁿ + a_n−1dⁿ⁻¹x/dtⁿ⁻¹ +…+ a₁dx/dt + a₀x = f(t) is called a linear differential equation of order n with constant coefficients (each a_i is constant) (Feynman et al., 1963, section 21–1 Linear differential equations).”

Feynman defines a linear differential equation as a differential equation consisting of a sum of several terms, each term being a derivative of the dependent variable with respect to the independent variable, which is multiplied by some constant. In chapter 25, he adds that “we spend so much time on linear equations: because if we understand linear equations, we are ready, in principle, to understand a lot of things.” However, one may add that a linear differential equation is homogeneous if f(t) = 0 (i.e., the right-hand side of the equation is zero). On the contrary, the linear differential equation that does not fulfill this condition is known as inhomogeneous. Feynman ends this chapter by discussing how an inhomogeneous differential equation is applicable to a forced harmonic oscillator in which f(t) = F₀cos ωt.

In defining the linear differential equation, Feynman uses the phrase “linear differential equation of order n with constant coefficients” and provides an example of the equation. Mathematicians may not be satisfied with his definition because he did not define the order and degree of a differential equation. The order of a differential equation is the order n of the highest derivative dⁿx/dtⁿ present in the equation. The degree of a differential equation is the power of the highest order derivative in the same equation. Furthermore, a linear differential equation is said to be linear if the dependent variable and its derivatives are of first degree, for example, there are no derivatives such as (dy/dx)².

Questions for discussion:

1. Why do we study oscillator-like phenomena?

2. What are the possible applications of linear differential equations?

3. Could a linear differential equation be of order zero or infinity?

The moral of the lesson: oscillator-like phenomena can be modeled by linear differential equations of order n with constant coefficients.

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

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. Heisenberg, W. (1967). Nonlinear problems in physics. Physics Today, 20(5), 27-33.

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.