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The systems in which the particles or considered objects oscillate about their corresponding mean positions are called oscillatory systems. The examples are: (i) the movement of a simple pendulum hanging from the roof, (ii) the oscillation of a block attached to a spring, (iii) the swinging of a child on the playground swing, (iv) the motion of the molecules of a solid and (v) vibration of a stringed musical instrument.
Simple Harmonic Motion
The simple harmonic motions are those oscillatory motions in which, the acceleration (a) of the object is directly proportional to the displacement (x) from the equilibrium position and is oppositely directed. Thus, the acceleration (a) of the object undergoing simple harmonic motion is given as: a = -k’x.
In general, an object moving along the x-axis exhibits simple harmonic motion when the displacement (x) of the object from the equilibrium position varies in time according to the relation given by,
where A, ω and ¢ are constants and are called amplitude, angular displacement, and phase angle respectively. The amplitude of an object is the maximum displacement either in the positive or negative x-direction from the mean position. The angular frequency (ω) has units of radians per second. The phase angle is always measured with respect to the initial position. Basically, the phase angle is determined by the initial displacement and velocity of the object. The graph of displacement ~ time and velocity ~ time are shown as below:
The time period T of the motion is the time it takes for the particle to go through one full cycle. Then we say that the particle has made one oscillation. And the time period of simple harmonic motion is given by,
The inverse of the time period is called frequency f of the motion. The frequency of motion is defined as the number of oscillations the particle makes per unit time:
The unit of f is cycles/sec or hertz (Hz). Now, the angular frequency is given as:
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Waves and Optics