Section 24–1 The energy of an oscillator

(Work done by a force / Mean stored energy / Q factor)

In this section, Feynman discusses the work done by an external force, the mean energy stored in a forced oscillator, and defines the Q factor.

1. Work done by a force:

“In our problem, of course, F(t) is a cosine function of t. Now let us analyze the situation: how much work is done by the outside force F? (Feynman et al., 1963, section 24–1 The energy of an oscillator).”

Feynman shows that the work done by an external force per second is equal to m[(dx/dt)(d²x/dt²)+ω₀²x(dx/dt)] + γm(dx/dt)². Then, he explains that the average power is equal to < γm(dx/dt)² > because the stored energy does not change in the long run, i.e., < m[(dx/dt)(d²x/dt²) + ω₀²x(dx/dt)] > is effectively zero as t approaches infinity. In defining the Q factor, he uses this result and states that the “work done per cycle” is (γmω²<x²>)(2π/ω). In other words, Feynman has shown that the work done by an external force F₀ cos ωt per second (or average power) is equal to the work done by friction per cycle after a long time. That is, all the energy transferred to an oscillator ultimately ends up in the resistive term γm(dx/dt)².

Feynman elaborates that the rate of electrical energy loss is the electrical resistance times the average square of the current: < P > = R< I² > = ½R I₀². As an analogy, the velocity dx/dt of an object corresponds to the electric current dq/dt, whereas the resistive term mγ corresponds to the electrical resistance R. One may add that the electrical or mechanical energy of an oscillator is constant in the long run, but the total energy is not conserved if we define the oscillator to be an isolated system. However, the work done on the oscillator by a driving force (or electrical source) do not always equal to the work done by the oscillator to overcome a resistive force (or electrical resistance). These two works done are effectively equal in the long run when the system reaches “a steady state” or a stable condition that does not change with time.

2. Mean stored energy:

“At any moment there is a certain amount of stored energy, so we would like to calculate the mean stored energy <E> also. (Feynman et al., 1963, section 24–1 The energy of an oscillator).”

Feynman briefly shows that the mean stored energy <E> = ½m<(dx/dt)²> + ½mω₀²<x²> = ½m(ω²+ω₀²)(½x₀²). Some students may be confused if they consider (dx/dt)² = (iω)²(x²) = -ω²x² because d(x₀e^iωt)/dt = (iω)(x₀e^iωt) = (iω)(x). However, the real part of (iω)(x) = (iω)(x₀)(cos ωt + i sin ωt) is -ωx₀ sin ωt and thus, (dx/dt)² is equal to +ω²x₀² sin² ωt instead of -ω²x². To be clearer, one may include the following steps. For a simple harmonic oscillator, we can express the average kinetic energy stored in an object as <K.E.> = ½mw₀²(½x₀²) and the average potential energy stored in a spring as <P.E.> = ½k(½x₀²). For a forced oscillator with a light damping, we have K.E. = ½mv² = ½mw²x₀²sin² wt and <K.E.> = ½mw²(½x₀²), whereas P.E. = ½kx² = ½mw₀²cos² wt and <P.E.> = ½mω₀²(½x₀²).

Feynman mentions that another interesting feature to discuss is how much energy is stored. However, there are other interesting features, for example, one may consider an oscillator’s stored energy is practically constant only after a sufficiently long time instead of t ® ¥ that seems “unrealistic.” This can be simply achieved by increasing the displacement of the oscillator as a result of an external force. Importantly, the oscillator has a maximum stored energy and maximum displacement at the resonant frequency that is slightly lesser than the natural frequency of oscillation (see section 24–2). When the oscillator is in resonance, the force and the motion of the oscillator are in phase, but the phase difference between the force and the displacement is π/2 radians (see Fig. 23-3).

3. Q factor:

“This is called the Q of the system, and Q is defined as 2π times the mean stored energy, divided by the work done per cycle (Feynman et al., 1963, section 24–1 The energy of an oscillator).”

Feynman defines the Q factor as 2π times the mean stored energy, divided by the work done per cycle. He explains that if we specify the “work done per radian” instead of per cycle, then the 2π disappears. In other words, we can avoid having “2π times the mean stored energy” in the formula by saying the work done per radian such that the term 2π appears in the denominator as “work done/2π.”. Alternatively, we can define Q as a ratio of average energy stored in the oscillator to average energy dissipated during 1 radian of motion (e.g., Kleppner & Kolenkow, 1973). This is because the average work done by an external force is equal to the average work done by friction. In essence, the Q factor is also a comparison between the “energy stored” and the “energy lost.”

According to Feynman, the efficiency of an oscillator can be measured by how much energy is stored as compared to how much work is done by the external force per cycle. Feynman also defines the Q factor as w₀/Dw in which Dw is the frequency width in section 23-2. The frequency width is directly related to the sharpness of resonance depending on the energy stored in the oscillator and the energy lost. The Q factor is sometimes known as the “storage factor” because its numerator is a measure of the energy stored during the oscillation. On the other hand, a system has a higher “quality factor” if its denominator as a measure of the degree of damping has a lower value of friction. Some may prefer the definition of Q factor as w₀/g because the stored energy is proportional to w₀ and the energy loss is dependent on g.

Questions for discussion (Feynman’s explanations were unclear?):

1. How would you explain the work done by an external force F₀ cos ωt per second is equal to the work done by friction per cycle after a long time?

2. How would you explain the mean stored energy of a forced oscillator that has a light damping?

3. Would you explain the Q factor of a system as 2π times the mean stored energy, divided by the work done per cycle?

The moral of the lesson: the Q factor of a system is a measure of the sharpness of resonance that is dependent on the energy stored in the oscillator and the energy lost.

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