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.