# PHYSICS OF VIBRATIONS introduction

(a)  Periodic Motion

If an object repeats its motion on a definite path after a regular interval of time, its motion is said to be periodic  motion. The definite time after which it repeats its motion, is called the periodic time and it is represented  by the letter T. In periodic motion, the path of the body can be straight line, circular curve.

Examples:

(i) In a wall dock, the hour arm completes one round in 12 hours and the minute arm complete one round in 1 hour. Thus the motion of the arms of the clock is periodic in which the time period of hour arm is 12 hours and that of the minute arm is 1 hour.

(ii) The earth completes one round on its axis in 24 hours and one round around the sun in nearly 365 days. Thus the rotational motion of earth on its own axis and the circular motion around the sun are also the periodic motions in which the time period of rotational motion about its own axis is 24 hours and that of circular motion around the sun is nearly 365 days.

(iii) The orbital motion of moon around the sun is also a periodic motion, the time period of which is 27 1/3 days. 3

Similarly, the oscillations of pendulum of a dock, motion of swing, vibration of strings the musical instruments, beating of heart etc., are also the periodic motion.

At any instant t the displacement x of the particle moving periodically is a function of time t.

It can be expressed as follows

X= f (t)

where f is the function of time t.

If time period of motion is T, then obviously

f (t) = f (t + T)

It is clear that if the function/is the sine or cosine function (i.e., if x =sin w t or x cos wt), the  value of the displacement x changes between – 1 and +1. Such function is called a simple

harmonic Junction. Its time period is  T =[2π /w] since
x =cos  w t =cos w (t +2π /w) and x =cos wt = cos w [t+2π/w]

On the other hand, the tan and cot functions are only the periodic functions, and not the simple harmonic functions, the time period of which is T = π/w since

tan wt =tan w (t +π/w ) and cot wt =cot w (t  +π/w)

(b) Vibratory or Oscillatory Motion

If a body in periodic motion moves to and fro about a definite point on a single path, the motion of the body is said to be the vibratory or oscillatory motion. For example, the motion of simple pendulum, the motion of arms of a tuning fork are the vibratory motions. Obviously, the  laboratory or oscillatory motion is definitely a periodic motion.

Mean or Equilibrium Position

The point on either side of which the body vibrates is called the mean or equilibrium position of the body.

One Vibration or One Oscillation

In a vibratory motion, one vibration is said to be complete, when the vibrating body goes from its equilibrium position to one extreme after which returns back to the equilibrium position, then goes to the extreme on other side and then returns back to the same position of

equilibrium.

For example, in Figure 1.1, the vibration of the body from 0 to A, A to 0, 0 to Band then from B to 0, is said to be one complete vibration.

Period of Vibration

The time taken to complete one vibration, is called the period of vibration (7) of a body. Its unit is second (s).

Frequency

The number of complete vibrations made by the body in 1 second is called the frequency of that body. It is represented by the letter for n or v. Its unit is s-1 or Hertz (Hz).

Relationship Between the Period of Vibration and Frequency

If n be the frequency of vibratory body and T is its time period, then  number of oscillations in  T second= 1

Therefore, number of oscillations in 1 second (i.e., frequency) n =1/T

Thus, frequency =   1/ time period     or n=  1 /T

Amplitude: In vibrating motion, the maximum displacement of the body on either side of the mean position is called the amplitude. It is represented by the letter (a). Its unit is meter (m).