Simple Harmonic Motion - Part I

We have seen the idea of motion in earlier chapters. Motion can take any path. Linear motion is the easiest motion to study and we have seen how equations of motion can be used to calculate velocities, displacements and accelerations. In circular motion, a body is under a central force (centripetal force) which is constant in magnitude. We have seen in the chapter on Wave Motion and Sound that there is another type of motion called oscillatory motion. This motion is a periodic in nature and is also called simple harmonic motion. Here, a force that varies in magnitude and direction, acts upon a body.  

What we will study in this chapter
1. Simple harmonic motion
2. Simple pendulum and its uses  

1. Simple harmonic motion
Consider an object, say a steel ball, hanging at the end of a spring. The spring is held firmly at the other end.  

Let the spring along with the steel ball be kept steady at an equilibrium position. Now pull the ball down, displace it by a distance x and release it. You will notice that the steel ball is performing an up and down motion. This type of motion is called a simple harmonic motion (more precisely, it is called linear simple harmonic motion). The word harmonic implies that the motion is periodic in nature. Periodicity shows that the ball returns to the same position after an interval of a definite time.

If we analyse the motion carefully, we will see that when the steel ball is pulled down by distance x, the spring is stretched by a distance x. Because the spring is elastic in nature, it will try to restore its
                                          
original shape or position. Let F be the restoring force. It is easy to understand that the restoring force
                                                 
is proportional to the displacement x and acts in the opposite direction. This means that larger is the displacement, larger will be the restoring force, which will try and bring back the steel ball to its equilibrium position.
                      
Thus       F       x  

                     
Thus     F   = - k  x

The constant of proportionality k is called the force constant. Its value depends on the elastic properties of the spring, friction with the air molecules, etc. The force constant is equal to the restoring force F per unit displacement x. The negative sign indicates that direction in which F and x act are opposite to each other.

When the steel ball is released, the restoring force will accelerate it towards its original equilibrium position. If m is the mass of the steel ball, then the acceleration can be calculated from the equation  

          
  F  =   m. a

                                              
                     F                  - k  x
Thus      a  =            =              
                         m                      m

Thus the acceleration of the object is proportional to its displacement but its direction is opposite to that of the direction of the displacement. The acceleration is directed towards the mean or the equilibrium position.

Once the steel ball is pulled and released, the restoring force will cause the ball to move back towards its original position. As this is happening, the acceleration will decrease continuously as x is decreasing continuously. When the steel ball reaches its original position, x = 0, both the force and the acceleration is zero. But the steel ball does not stop here!! Due to its momentum, the steel ball over shoots its original equilibrium or mean position. Again a restoring force will act on the steel ball. The restoring force acts to stop the steel ball and bring it back to its mean position. Again at the mean position, the momentum of the ball will make to go down beyond its mean position. In this way the steel ball will bob up and down continuously, about its mean position. This type of motion is called a simple harmonic motion. (The simple harmonic motion of the steel ball attached to the spring can be compared to a ball falling from a height h as seen in the chapter on Work, Energy, Power).

A linear simple harmonic motion can be defined as the linear periodic motion of a body, where the restoring force is directed towards the mean position and its magnitude is proportional to the displacement from the mean position. 

It has to be remembered that in a simple harmonic motion, the force that is acting on the body, varies both in magnitude and direction, and is periodic in nature.

Examples of simple harmonic motion in our everyday lives are a) motion of a pendulum of a wall clock,  b) up and down motion of  the needle in a sewing machine. 

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