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.