1.1 Simple harmonic motion

Oscillation is a periodic motion, which repeats after certain time interval. Simple harmonic motion is a special type of oscillation. In real time, all oscillatory motion dies out due to friction, if left unattended. We, therefore, need to replenish energy of the oscillatory motion to continue oscillating. However, we shall generally consider an ideal situation in which mechanical energy of the oscillating system is conserved. The object oscillates indefinitely. This is the reference case.

Though, we refer an object or body to describe oscillation, but it need not be. We can associate oscillation to energy, pattern and anything which varies about some value in a periodic manner. The oscillation, therefore, is a general concept. We shall, however, limit ourselves to physical oscillation, unless otherwise mentioned.

Further, study of oscillation has two distinct perspectives. One is the description of motion i.e. the kinematics of the motion. Second is the study of the cause of oscillation i.e. dynamics of the motion. In this module, we shall deal with the first perspective.

Oscillation

Oscillation is a periodic, to and fro, bounded motion about a reference, usually the position of equilibrium.

The characteristics of oscillation are enumerated here :

• It is a periodic motion that repeats itself after certain time interval.
• The motion is about a point, which is often the position of equilibrium.
• The motion is bounded.

Note that revolution of second hand in the wrist watch is not an oscillation as the concept of “to and fro” motion about a point is missing. Thus, this is a periodic motion, but not an oscillatory motion. On the other hand, periodic swinging of pendulum in mechanical watch is an oscillatory motion.

Description of oscillation

We need a mathematical model to describe oscillation. We often use trigonometric functions. However, we can not use all of them. It is essentially because many of them are not bounded. Recall the plot of tangent function. It extends from minus infinity to plus infinity - periodically. Actually, only the sine and cosine trigonometric functions are bounded.

The plot of tangent function is shown here. Note that value of function extends from minus infinity to plus infinity.

The plots of sine and cosine functions are shown here. Note that value of function lies between "-1" and "1".

Harmonic oscillation

Harmonic oscillation and simple harmonic oscillation both are described by a single bounded trigonometric function like sine or cosine function having single frequency (it is the number of times a motion is repeated in 1 second). The difference is only that simple harmonic function has constant amplitude over all time (amplitude represents maximum displacement from central or mean position of the periodic motion) as a result of which mechanical energy of the oscillating system is conserved.