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# Understanding Oscillatory Motion

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Describe, in detail, oscillatory motion with specific reference to simple harmonic motion.

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Understanding Oscillatory Motion

1. Periodic motion:

When an object undergoes a certain motion repeatedly and is at the same position after lapse of a fixed time, we say the object is executing a periodic motion. Examples : rotation of the planets around the sun, rotation of the earth around its axis, the to and fro motion of a pendulum, the motion of the hands of a clock etc.. In all the above examples, the object is repeating the same cycle of motion again and again, and the time taken to complete one cycle (counted from any arbitrary position of the object till it comes back to the same position again) is fixed. The time taken to complete one cycle is known as "Time period" of the periodic motion.

2. Periodic oscillatory motion:

In the examples of periodic motion cited above, what is the difference between the motion of the earth going around the sun and motion of a pendulum? Unlike the motion of earth around sun, the bob of the pendulum moves to and fro between two extreme positions, around a central mean position. This type of motion is a special form of periodic motion called a periodic oscillatory motion. Hence, while all oscillatory motions are periodic, all periodic motions are not oscillatory. Examples: A mass hanging from a spring pulled a little and released starts oscillating up and down, a cylinder floating in a liquid pushed a little vertically into the liquid and released starts executing up and down oscillations.

3. Simple harmonic motion (oscillation):

An example of oscillatory motion is represented graphically hereunder.

y

A

0 T/4 T/2 3T/4 T t

-A

As can be seen, at t=0, the displacement y (from a mean position represented by y=0) is 0, at t=T/4, y=A, at t=T/2, y=0, at t=3T/4, y= -A and at t=T, y=0. One cycle is completed over a time period T. Thus, T is the time period of the oscillatory motion. The maximum displacement A on either side of the mean position is known as the amplitude. As can be seen, the values of displacement indicated above shall repeat for each subsequent time period T.

The most familiar example of an oscillatory motion is one represented by a sinusoidal function graphically represented hereunder. To distinguish it from other forms of periodic oscillations, a sinusoidal oscillation is known as a simple harmonic oscillation.

y

0 T/2 T t

The study of harmonic oscillations is basic to the study of other forms of periodic oscillations because it can be shown that any periodic function can be broken up into a unique set of sine and/or cosine functions (Fourier theorem). Thus, every periodic motion is in essence a combination of one or more sine (and/or cosine) functions.

There is another simple way to understand a harmonic oscillation. Let us consider a particle executing a circular motion with a constant angular velocity ω, in a circular path of radius A, as shown in the fig. below.

Y
ω
Q P (t = t)
A
y Φ (t = 0)
O Φ0 X

Let us drop a perpendicular on the Y axis, from the position P of the particle. Let the point at which the perpendicular meets the Y axis be Q. Let us consider the motion of the point Q as the particle P goes round the circular path. We shall call the angle made by the position vector OP with X axis at any instant t as the phase Φ at that instant. Let the initial time t=0 be chosen such that the phase at that instant is Φ0 (initial phase). After lapse of time t, the angle swept by the vector OP will be ωt. The displacement y of point Q at instant t i.e. OQ will be given by A sin (ωt+Φ0). Thus, we can write in general for the displacement of point Q measured from the center of the circle O, at any instant t as:

y = A sin (ωt+Φ0) ...(1)
.... (Please see the attached file for the complete solution).

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