# Oscillatory Motion and Waves

## Hooke’s Law: Stress and Strain Revisited


### Learning Objectives

By the end of this section, you will be able to:
1. Explain Newton’s third law of motion with respect to stress and deformation.
2. Describe the restoration of force and displacement.
3. Calculate the energy in Hooke’s Law of deformation, and the stored energy in a spring.
Newton’s first law implies that an object oscillating back and forth is experiencing forces. Without force, the object would move in a straight line at a constant speed rather than oscillate. Consider, for example, plucking a plastic ruler to the left as shown in . The deformation of the ruler creates a force in the opposite direction, known as a restoring force. Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero. However, by the time the ruler gets there, it gains momentum and continues to move to the right, producing the opposite deformation. It is then forced to the left, back through equilibrium, and the process is repeated until dissipative forces dampen the motion. These forces remove mechanical energy from the system, gradually reducing the motion until the ruler comes to rest.
The simplest oscillations occur when the restoring force is directly proportional to displacement. When stress and strain were covered in Newton’s Third Law of Motion, the name was given to this relationship between force and displacement was Hooke’s law:
Here,  is the restoring force,  is the displacement from equilibrium or deformation, and  is a constant related to the difficulty in deforming the system. The minus sign indicates the restoring force is in the direction opposite to the displacement. 
The force constant  is related to the rigidity (or stiffness) of a system—the larger the force constant, the greater the restoring force, and the stiffer the system. The units of  are newtons per meter (N/m). For example,  is directly related to Young’s modulus when we stretch a string.  shows a graph of the absolute value of the restoring force versus the displacement for a system that can be described by Hooke’s law—a simple spring in this case. The slope of the graph equals the force constant  in newtons per meter. A common physics laboratory exercise is to measure restoring forces created by springs, determine if they follow Hooke’s law, and calculate their force constants if they do.


### Energy in Hooke’s Law of Deformation

In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation, and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. The potential energy stored in a spring is . Here, we generalize the idea to elastic potential energy for a deformation of any system that can be described by Hooke’s law. Hence,
where  is the elastic potential energy stored in any deformed system that obeys Hooke’s law and has a displacement  from equilibrium and a force constant .
It is possible to find the work done in deforming a system in order to find the energy stored. This work is performed by an applied force . The applied force is exactly opposite to the restoring force (action-reaction), and so .  shows a graph of the applied force versus deformation  for a system that can be described by Hooke’s law. Work done on the system is force multiplied by distance, which equals the area under the curve or (Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to , so that the average force is , the distance moved is , and thus  (Method B in the figure).

### Test Prep for AP Courses


### Section Summary

1. An oscillation is a back and forth motion of an object between two points of deformation.
2. An oscillation may create a wave, which is a disturbance that propagates from where it was created.
3. The simplest type of oscillations and waves are related to systems that can be described by Hooke’s law:
    
    where 
        
4. Elastic potential energy  stored in the deformation of a system that can be described by Hooke’s law is given by
    

### Conceptual Questions


### Problems & Exercises
