1.1 Simple harmonic motion  (Page 2/4)

Examples of harmonic motion are :

• $x=A{e}^{-\omega t}\sin \omega t$
• $x=A\sin \omega t$
• $x=A\omega \cos \omega t$
• $x=A\sin \omega t+B\cos \omega t$

Of these, last three examples are simple harmonic oscillations.

Note that we can reduce fourth example, sum of two trigonometric functions, into a single trigonometric function with appropriate substitutions. As a matter of fact, we shall illustrate such reduction in appropriate context.

The simple harmonic oscillation is popularly known as simple harmonic motion (SHM). The important things to reemphasize here is that SHM denotes an oscillation, which does not involve change in amplitude. We shall learn that this represents a system in which energy is not dissipated. It means that mechanical energy of a system in SHM is conserved.

Non-harmonic oscillation

A non-harmonic oscillation is one, which is not harmonic motion. We can consider combination of two or more harmonic motions of different frequencies as an illustration of non-harmonic function.

$$x=A\sin \omega t+B\sin 2\omega t$$

We can not reduce this sum into a single trigonometric sine or cosine function and as such, motion described by the function is non-harmonic.

Simple harmonic motion

A simple harmonic motion can be conceived as a “to and fro” motion along an axis (say x-axis). In order to simplify the matter, we choose origin of the reference as the point about which particle oscillates. If we start our observation from positive extreme of the motion, then displacement of the particle “x” at a time “t” is given by :

$$x=A\cos \omega t$$

where “ω” is angular frequency and “t” is the time. The figure here shows the positions of the particle executing SHM at an interval of “T/8”. The important thing to note here is that displacements in different intervals are not equal, suggesting that velocity of the particle is not uniform. This also follows from the nature of cosine function. The values of cosine function are not equally spaced with respect to angles.

Amplitude

We know that value of cosine function lies between “-1” and “1”. Hence, value of “x” varies between “-A” and “A”. If we plot the function describing displacement, then the plot is similar to that of cosine function except that its range of values lies between “-A” and “A”.

The value “A” denotes the maximum displacement in either direction. The scalar value of maximum displacement from the mean position is known as the amplitude of oscillation. If we consider pendulum, we can observe that farther is the point from which pendulum bob (within the permissible limit in which the bob executes SHM) is released, greater is the amplitude of oscillation. Similarly, greater is the stretch or compression in the spring executing SHM, greater is the amplitude. Alternatively, we can say that greater is the force causing motion, greater is the amplitude. In the nutshell, amplitude of SHM depends on the initial conditions of motion - force and displacement.