Physics 136 / Music 126 Duke University Fall 2012 Handout 2

The Simple Harmonic Oscillator

We compare here four specific examples of simple harmonic oscillators, emphasizing how an understanding of the simplest of them -- the mass on a spring -- can be applied to other more complicated cases.

Mass on a Spring

The mass m is free to move back and forth along the frictionless surface shown in the diagram. We define the variable x to be displacement from the equilibrium position x = 0, with displacement to the right being considered positive.

(1) The spring provides a restoring force that accelerates the mass M, the relationship among force, mass, and acceleration being described by the famous Newtonian equation F = Ma.

(2) The force provided by the spring is proportional to the displacement of the mass from its equilibrium position: F = -kx, where the spring constant k is a measure of the spring's stiffness and the negative sign indicates that the force is always in the restoring direction -- opposite to the direction of displacement. Notice that a spring constant has dimensions of (mass)/(time²).

(3) The period of oscillation T of a mass on a spring is given by the equation

We note that k is independent of M, and that T depends on M but is independent of the amplitude of the oscillations and independent of any gravitational acceleration (the latter turns out to be true even if the mass is hanging vertically from the spring). The frequency of oscillation, of course, is just f = 1/T.

Pendulum

The point mass m is suspended by a massless string of length l (for a real mass of finite size, l is the distance of the center of mass from the pendulum's point of support). Its equilibrium position corresponds to the string being vertical in the Earth's gravitational field. Again we make an arbitrary choice that displacement from x = 0 to the right will be a positive quantity. The angle of displacement is also defined to be positive for positive x.

(1) Again the obvious mass m is the mass being accelerated, but this time by gravity rather than a mechanical spring.

(2) In general, the restoring force varies as the sine of the displacement angle, from a tiny fraction of the weight of the mass for small displacements to the full weight when the pendulum is horizontal.

Here g is the acceleration due to gravity. Again, the negative sign indicates a restoring force. In the second part of the above equation we assume that the amplitudes of displacement will be very small. In the limit of small angular displacements of the pendulum, the displacement of the mass becomes horizontal, and the ratio x/l equals the sine of the angle. In its final form, this equation can be written as F = -"k"x, where "k" is an "effective spring constant" defined as

(3) Having identified "k", we can substitute it in the equation we used before for the period of oscillation:

Now "k" depends on both g and m because the acceleration of gravity supplies the restoring force. For the same reason, the T of a pendulum would be different on another planet, while the period of a mass on a spring would be unaltered. That T is independent of m is a highly significant physical finding, evidence of the equivalence of inertial mass and gravitational mass. Again, so long as the amplitudes of oscillation are small enough to ensure a restoring force proportional to displacement from equilibrium, T is independent of the amplitude.

Fluid Oscillating in a U-shaped Tube

Flow of the fluid within the tube is assumed to be frictionless (this can be achieved experimentally, using superfluid helium at very low temperatures). Here the x = 0 equilibrium position is the common level of the fluid on both sides when the fluid is at rest. The arbitrary choice indicated on the diagram is that displacement be positive for upward movement of the fluid level in the right arm of the tube. The variable a describes the cross sectional area of the bore of the tube; L is the length of the "slug" of fluid in the tube. Let the density of the liquid (mass per unit volume) be

(1) In this case, the mass on which the restoring force acts is the entire mass of the "slug" of fluid, calculated by multiplying the volume of the fluid (the product of a and L) by its density.

(2) But, while the source of the restoring force again is gravitation, only part of the fluid is involved in determining the magnitude of that force. There is a net gravitational restoring force acting on the part of the fluid that is above the lower fluid level. If the displacement is x, then the volume of that fluid is 2xa and the restoring force may be written:

and in this case our effective spring constant is

(3) Using that expression, we find the period of oscillation to be

"k" depends on g and the density of the fluid because of the gravitational origin of the restoring force. T is planet-dependent again for the same reason, but is independent of fluid density and tube cross section, as well as oscillation amplitude.

Helmholtz Resonator

Our final example is much more subtle and less intuitive, but highly relevant to musical acoustics. A relatively large volume V of air ends in a tubular opening that is d long and a in cross sectional area. The density of the air is, and the velocity of sound in the air.

(1) The mass being accelerated by the restoring force here is the mass of the air contained in the exit "lip" of the resonator, which oscillates back and forth like a piston:

(2) The restoring force for the resonator is the springiness of the air contained in the main volume. This turns out to result in an effective spring constant given by

While we might expect that a smaller volume V would correspond to a stiffer air-spring, some of the other dependencies are less intuitive in this case.

(3) In any event, the period of oscillation once again can be obtained by substituting "k" into the equation for the mass on a spring:

Notice that T depends on the structural dimensions of the resonator and on the velocity of sound in the gas (the period would be quite different for a resonator filled with, say, helium). It is, as always for a simple harmonic oscillator, independent of the amplitude of oscillation.