# Oscillatory Motion and Waves

## Simple Harmonic Motion: A Special Periodic Motion


### Learning Objectives

By the end of this section, you will be able to:
1. Describe a simple harmonic oscillator.
2. Explain the link between simple harmonic motion and waves.
The oscillations of a system in which the net force can be described by Hooke’s law are of special importance, because they are very common. They are also the simplest oscillatory systems. Simple Harmonic Motion (SHM) is the name given to oscillatory motion for a system where the net force can be described by Hooke’s law, and such a system is called a simple harmonic oscillator. If the net force can be described by Hooke’s law and there is no  damping (by friction or other non-conservative forces), then a simple harmonic oscillator will oscillate with equal displacement on either side of the equilibrium position, as shown for an object on a spring in . The maximum displacement from equilibrium is called the amplitude . The units for amplitude and displacement are the same, but depend on the type of oscillation. For the object on the spring, the units of amplitude and displacement are meters; whereas for sound oscillations, they have units of pressure (and other types of oscillations have yet other units). Because amplitude is the maximum displacement, it is related to the energy in the oscillation.

      
What is so significant about simple harmonic motion? One special thing is that the period  and frequency  of a simple harmonic oscillator are independent of amplitude. The string of a guitar, for example, will oscillate with the same frequency whether plucked gently or hard. Because the period is constant, a simple harmonic oscillator can be used as a clock.
Two important factors do affect the period of a simple harmonic oscillator. The period is related to how stiff the system is. A very stiff object has a large force constant , which causes the system to have a smaller period. For example, you can adjust a diving board’s stiffness—the stiffer it is, the faster it vibrates, and the shorter its period. Period also depends on the mass of the oscillating system. The more massive the system is, the longer the period. For example, a heavy person on a diving board bounces up and down more slowly than a light one.
In fact, the mass  and the force constant  are the only factors that affect the period and frequency of simple harmonic motion. 

### The Link between Simple Harmonic Motion and Waves

If a time-exposure photograph of the bouncing car were taken as it drove by, the headlight would make a wavelike streak, as shown in . Similarly,  shows an object bouncing on a spring as it leaves a wavelike "trace" of its position on a moving strip of paper. Both waves are sine functions. All simple harmonic motion is intimately related to sine and cosine waves.
The displacement as a function of time t in any simple harmonic motion—that is, one in which the net restoring force can be described by Hooke’s law, is given by
where  is amplitude. At , the initial position is , and the displacement oscillates back and forth with a period . (When , we get  again because .). Furthermore, from this expression for , the velocity  as a function of time is given by:
where . The object has zero velocity at maximum displacement—for example,  when , and at that time . The minus sign in the first equation for  gives the correct direction for the velocity. Just after the start of the motion, for instance, the velocity is negative because the system is moving back toward the equilibrium point. Finally, we can get an expression for acceleration using Newton’s second law. [Then we have  and , the quantities needed for kinematics and a description of simple harmonic motion.] According to Newton’s second law, the acceleration is . So,  is also a cosine function:
Hence,  is directly proportional to and in the opposite direction to .
 shows the simple harmonic motion of an object on a spring and presents graphs of  and  versus time.
The most important point here is that these equations are mathematically straightforward and are valid for all simple harmonic motion. They are very useful in visualizing waves associated with simple harmonic motion, including visualizing how waves add with one another.

### Test Prep for AP Courses


### Section Summary

1. Simple harmonic motion is oscillatory motion for a system that can be described only by Hooke’s law. Such a system is also called a simple harmonic oscillator. 
2. Maximum displacement is the amplitude . The period  and frequency  of a simple harmonic oscillator are given by
    
3. Displacement in simple harmonic motion as a function of time is given by 
        
          
4. The velocity is given by

, where

.
5. The acceleration is found to be 

      
          

### Conceptual Questions


### Problems & Exercises
