# Simple Harmonic Oscillations

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.

## Simple Harmonic Oscillation - the motion

The easiest way to make a simple harmonic oscillator is to attach a mass to the end of a spring and then set it into motion. What happens? The mass executes repetitive motion, moving back and forth between two points. What can we do to describe its motion in more detail?

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 maximum displacement of the mass in both the positive and negative directions. We find that both of these points are the same distance from the equilibrium position. This quantity is called the amplitude of the motion.

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 trigonometric functions. You might need to review them. One caution: when using your calculator to find the position (or velocity, see below) versus time, you will need to make sure that the little "D" or "DEG" symbol is not visible in your calculator's display. Otherwise, you would be specifying an angle for the "sin" or "cos" function in degrees. In the above formula, the "angle" for the "sin" function has no units. Why? It's because both t and the period (T) have units of seconds; this means their ratio has no units. An angle specified like this is said to be given in radians (even though that's not really a physical unit). You need to make sure you see "R" or "RAD" in your calculator's display. Bring it to my office if you need help setting it up! Oh yeah, there is another units conversion factor to convert an angle in radians to one in degrees.

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 figure The maximum velocity (vmax) is related to the amplitude of the oscillation by We'll see why this is so, when we talk about energy considerations for a simple harmonic oscillator.

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.

## What causes the motion?

The motion of the mass is caused by the force exerted on it by the spring. Before relating that force to the motion of the oscillator, let's understand the force exerted by a spring when there is no motion. We can do that by hanging different masses onto the the end of a spring mounted vertically, and measuring where the equilibrium position is for each mass. When the mass is at its equilibrium position, the force of gravity (the mass's weight, W = Mg) is balanced by the force exerted on it by the spring. With careful measurements, we find a linear relationship between these equilibrium positions and the mass attached to the spring. From this, we deduce that a stretched spring exerts a force

F = -ky

Here, y is the amount the spring is stretched (or compressed) from its relaxed position and k is a constant that characterizes the "stiffness" of the spring, known as the spring constant. It's units are N/m or kg/s2. The minus sign simply is telling us the direction of the force. The spring exerts a force on the mass in the opposite direction to the displacement from equilibrium. If we stretch a spring, the force tries to pull the mass back to equilibrium (i.e., unstretch it). If we compress the spring, the force tries to push the mass back to equilibrium.

That 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/s2. That means if we take the ratio k/M, we have something with units of 1/s2. 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.