Oscillatory Motion

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During our childhood, we all have enjoyed swinging on a swing or rocking in a cradle. In both these motions, the object moves to and fro about a mean position. These motions are repetitive. However, they are quite different from the periodic motion of a planet like that of Earth. A few more examples of to and fro periodic motions are - the pendulum of a wall clock, a boat tossing up and down in a river, the piston in a steam engine going back and forth, and so on. This kind of motion is also known as the oscillatory motion.

The concepts of oscillatory motion are required for having an adequate understanding of many physical phenomena. In musical instruments, including sitar, the violin, or the guitar, we usually come across vibrating strings producing pleasing and melodious sounds. Furthermore, the membranes in drums, diaphragms in telephones and speaker systems vibrate to and fro about their mean positions.

Periodic and Oscillatory Motions

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Let's understand the motion in Figure (2). If a child climbs up a step, comes down, and then repeats the entire process identically, one after the other, then his height above the ground would be like the one shown in Figure (2). When we bounce a ball on the ground, between our palm, and the ground, the ball's height versus time graph would look like the one shown in Figure (3). Please make a note of the fact that both the curved parts in Figure (3) are sections of a parabola, which are given by Newton’s equation of motion specified below:

h = ut + gt (downward motion)

h = ut – gt (upward motion), having different values of u in both cases. Hence, a motion that repeats itself at regular intervals of time is known as periodic motion.

Quite often, the body, which undergoes periodic motion, has an equilibrium position at least somewhere inside its path. When the body is at this position, there is no net external force acting on it. Hence, if it is left there at rest, it shall remain there itself always. If the body is given a small amount of displacement from its position, a force comes into action, which tries to bring the body back to the equilibrium point by giving rise to oscillations or vibrations. For instance, a ball placed in a bowl will be in equilibrium at the bottom. If it is displaced a little from the point, it will further perform oscillations in the bowl. It is essential to keep in mind that every oscillatory motion is periodic; however, every periodic motion may not be oscillatory. Circular motion is a periodic motion; however, it is still not oscillatory as the net force on the particle in a circular motion is never zero and is always directed towards the centre. Consequently, there is no mean position or equilibrium position; hence, it is not an example of oscillatory motion. There is no crucial difference between oscillations and vibrations. It appears that when the frequency is small, we call it oscillation (the oscillation of a branch of a tree) while when the frequency is high, we call it vibration (the vibration of a string of a musical instrument). Simple harmonic motion is indeed the simplest form of oscillatory motion. It arises when the force on the oscillating body is directly proportional to its displacement from the mean position (also known as the equilibrium position) and at any point in its oscillation; the same force is directed towards the mean position. In practice, oscillating bodies eventually come to rest at their equilibrium positions as a result of damping force due to friction and several other dissipative causes. However, they can be forced to remain oscillating through some external periodic agency. Any material medium can be viewed as a collection of a huge number of coupled oscillators. Besides, the collective oscillations of the medium's constituents manifest themselves as waves. Examples of waves include water waves, seismic waves, and electromagnetic waves.

Simple Harmonic Motion (SHM)

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Let us consider a particle oscillating back and forth between the limits A and A' that is about the origin of an x-axis, as displayed in the figure above. This oscillatory motion is referred to as simple harmonic provided the displacement x of the particle from the origin varies with time as in the equation mentioned below:

x(t) = A cos (ω t + φ)

In the above equation, the variables A, ω, and φ are constants – A is the amplitude, that is, the maximum displacement from the equilibrium position, ω = 2∏f is the angular frequency, and φ is the initial phase.

Therefore, simple harmonic motion or SHM is not periodic motion, but it is the motion in which displacement is a sinusoidal function of time. The above figure displays the positions of a particle executing Simple Harmonic Motion at a discrete value of time, with each interval of time being T/4, where T is the period of motion.

The above figure has plotted the graph of x versus t, giving the values of displacement as a continuous function of time. The quantities A, ω, and φ characterizing a given SHM have standard names, summarized in the below figure.

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Although the amplitude A is fixed for a given SHM, the state of motion, including the position and velocity of the particle at any time t is determined by the argument (ωt + φ) in the cosine function. This time-dependent quantity, (ωt + φ), is known as the phase of the motion. The value of phase at t = 0 is φ and is called the phase constant (or phase angle). If the value of the amplitude is known, then we can determine the value of φ from the displacement at t = 0. It is imperative to keep in mind that two simple harmonic motions may have the same A and ω, but different phase angle φ.

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At last, the value of ω can be seen to be corresponding to the period of motion T. Let us take φ = 0 for simplicity and substitute it in the above equations:

x(t) = A cosωt.

Now, as the motion has a period T, x (t) is equal to x (t + T), which implies:

A cosωt = A cos ω (t + T).

Now, the cosine function is periodic with period 2π, that is, it first repeats itself when the argument changes by 2π. Hence, ω (t + T) = ωt + 2π, that is, ω = 2π/ T (14.7), where ω is called the angular frequency of SHM. Its S.I. unit is radians per second. As the frequency of oscillations is 1/T, the value of ω is 2π times the frequency of oscillation. Please make a point of the fact that two simple harmonic motions may have the same value of A and φ, but a different value of ω.