Oscillations are of central importance to the study of sound and to nearly all areas of physics. Sound itself correponds to oscillations of the air pressure. Sound is produced by many different kinds of oscillators that share similar properties to those we will describe today. We turn first to the study of simple harmonic oscillators, so that we can more fully develop a description of an oscillating system, including understanding the motion in detail, the forces that cause that motion and then look at the energy exchanges.

Looking at the system before we set it into motion, we see the mass
at rest at a position known as its *equilibrium position*. We'll use that position as the origin of our coordinate system for quantatively describing its motion. If we tap on the mass while it is in its equilibrium position, the oscillations begin. In words, the mass first moves away from equilibrium in one direction (we'll call that the positive direction), reaches a maximum displacement from equilibrium where it changes its direction of motion (instantaneously coming to rest), speeds up as it moves back towards the equilibrium position (going in the opposite direction compared to when we tapped it), slows down as it passes the equilibrium position until it reaches its maximum negative displacement (the same distance from the origin as the maximum positive displacement) and then heads back to the origin. What we've described is *one cycle* of its oscillation. The oscillation cycles repeat.

Quantitatively we can measure *the time* to complete one cycle. This is called the *period* of the motion (generally abbreviated as T). We could also *count* the number of cycles that occur in each second. That number, in general, will be a fraction: say, 5 cycles in 2 seconds. This measure is called the *frequency* of the motion (abbreviated as f). These two measures of the motion are clearly interrelated: f = 1/T and T = 1/f. The units of f are cycles per second. In honor of Heinrich Hertz, we use the units of Hertz (abbreviated Hz):

1 Hz = 1 cycle per second.

We can also easily measure the
There is a simple correspondence between the terms we've used to
describe simple harmonic oscillations and those we use to describe sound. The *frequency* of oscillations is related to the *pitch* of sound. The *amplitude* of oscillation is related to the *loudness* of sound. We'll discuss this in more detail later in the semester. Sound generally involves the *superposition* of many different pitches, corresponding to describing general oscillations as a *superposition* of simple oscillations at different frequencies. The motion of a simple harmonic oscillator is related to a *pure tone* (single frequency) in sound.

We can quantitatively measure the position of the mass versus time.
Graphically, the position versus time looks like

When working with the equation describing position versus time, we will end up dealing with

2 pi radians = 360 degrees

We can analyze the position versus time graph using the tangent method and find the velocity of the mass at each instant in time. That is shown in the next figureThe

We'll see why this is so, when we talk about energy considerations for a

We can analyze the velocity versus time graph using techniques we discussed earlier, to deduce the *acceleration* of the mass at each instant in time. See if you can convince yourself of the following:

- the acceleration is zero when the mass is at its equilibrium position, where it has its greatest speed.
- the acceleration is largest at the maximum displacement where the mass is instantaneously at rest.

F = -ky

Here, y is the amount the spring is stretched (or compressed) from itsThat minus sign in our mathematical equation for the force is actually the critical thing that leads to simple harmonic oscillation. Whenever the mass is displaced from the equilibrium, the spring tries to pull it back. The problem is that it pulls to hard, the mass overshoots the equilibrium position and the oscillations continue on.

How do we relate properties of the spring (its spring constant, k) and the mass (M) to quantities of the motion (the frequency and amplitude of oscillation)? We can try to use a *dimensional analysis*, based only on the units of the quantities involved, to guide our observations. We've already said that k has units of kg/s^{2}. That means if we take the ratio k/M, we have something with units of 1/s^{2}. If we then take the square root of k/M, we end up with something with units of 1/s. That is exactly the units of frequency (1/s is the same thing as cycles per second). We could then go and make careful measurements and we would find

This means that the frequency of oscillation grows as the stiffness of the spring increases. The oscillation frequency also grows if we reduce the mass attached to the spring.

What about the amplitude of the motion? Careful observations show
that the frequency of oscillation does not depend on the amplitude. That's
why we call it a *simple* oscillator. The harmonic part comes from the fact that the position varies as the "sin" of a constant multiplied by the time.

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Last updated: `10 Sep 1999
`Comments: bland@indiana.edu