# Resonance

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### Free Vibrations:

• A body or a system capable of vibrating, when displaced from its position of rest, vibrates with a certain definite frequency. This frequency is characteristic of the body or the system. Such oscillations are called free oscillations or free vibrations and the frequency of such oscillations is called natural frequency of the body or the system.
• Example – 1: When a wire under tension, which is fixed at its ends, is plucked and released, it vibrates with a frequency which depends on the length of the string, its mass per unit length and tension in the string.
• Example – 2: When the bob of a simple pendulum oscillates, its frequency of oscillation depends on the length of the pendulum.
• Due to frictional force, the amplitude of oscillation decreases continuously and finally, the body stops vibrating.

#### Characteristics of Free Vibrations:

• They are produced when a body capable of vibrating is disturbed from its normal equilibrium position and then released.
• The frequency of vibration depends on the body and is called natural frequency.
• The frequency of vibration is same as the natural frequency of the body.
• The amplitude of vibration is large.
• Vibration continues for a little more time after the external force is removed.
• Example: Oscillations of bob of pendulum

### Forced Vibrations:

• Forced vibrations are the vibrations produced in a body by applying an external periodic force having a frequency, normally different from the natural frequency of the body.
• A body or a system, capable of vibrating can also be made to vibrate at any desired frequency. The body can be made to vibrate with the same frequency as the frequency of the applied periodic force.
• Suppose that the natural frequency of a metal vessel is 200 Hz.  If a tuning fork of frequency 256 Hz is set up into vibrations and its stem is placed in contact with the vessel, then the vessel will be forced to vibrate at a frequency of 256 Hz.  In such a case, the vessel is said to perform forced vibrations.
• Initially, the body tends to vibrate with its natural frequency. But very soon, the natural vibrations die out and it begins to vibrate with the frequency of the applied periodic force.
• The amplitude of  forced vibrations depends on:
• The difference in frequencies of the external force and the natural frequency of the body.
• The amplitude of the applied force.
• damping.

#### Characteristics of  Forced Vibrations:

• They are produced when an external periodic force acts on the body.
• The frequency of vibration is same as the frequency of external periodic force.
• The frequency of vibration is different from the natural frequency of the body.
• The amplitude of vibration is small.
• Amplitude becomes zero as soon as the external force is removed.
• Example: A vibrating tuning fork on a wooden box, a musical instrument having soundboard or box.

#### Resonance:

• The amplitude of the forced vibrations depends on the difference between the natural frequency of the body and the frequency of the applied periodic force.
• When the difference between the two frequencies is large, the response of the body is poor or the forced vibrations are of small amplitude. When the frequency difference becomes smaller, the body vibrates more readily or the amplitude of the forced vibrations increases.
• Finally, when the frequency (f) of the applied periodic force becomes the same as the natural frequency fo of the body, the amplitude of the forced vibrations becomes maximum and the phenomenon is known as resonance.
• If anybody is made to vibrate, by an external periodic force, with a frequency which is same as the natural frequency of the body, the body begins to vibrate with a very large amplitude. This phenomenon is called resonance.

#### Forced vibrations

• They are produced by the external periodic force of any frequency.
• The frequency of vibration is different from the natural frequency of the body.
• The amplitude of vibration is small.
• Vibration stops as soon as the external force is removed.

#### Resonance :

• They are produced by an external periodic force whose frequency is equal to the natural frequency of the body.
• The frequency of vibration is same as the natural frequency of the body.
• The amplitude of vibration is large.
• Vibration continues for a little more time after the external force is removed

#### Advantages of Resonance:

• Resonance is useful to determine an unknown frequency.
• Resonance is useful to increase the intensity of sound in musical instruments.
• Resonance is useful to tune a radio receiver to any desired frequency.
• Resonance is useful to analyse musical notes.

#### Disadvantages of Resonance:

• Soldiers are asked to break steps when crossing a bridge. It can be explained as follows
• Soldiers marching on a bridge take steps with definite frequency and force the bridge to vibrate with the frequency of the steps.
• If the forced frequency on the bridge is equal to the natural frequency of vibration of the bridge, the bridge is set into resonant vibrations.
• Due to the resonance, the bridge vibrates with higher amplitude and due to this, it may collapse.
• Due to rhythmic clapping of the audience, the roof of the stadium may collapse. It can be explained as follows
• When audience claps rhythmically they do so with a certain frequency and force the roof of a stadium to vibrate with the frequency of the clap.
• If the forced frequency on the roof of a stadium is equal to the natural frequency of vibration of the roof of a stadium, the roof of a stadium is set into resonant vibrations.
• Due to the resonance, the roof of a stadium vibrate with higher amplitude and due to this, it may collapse.
• When the speed of an aircraft increases, different parts are forced to vibrate. which is dangerous for the structure of the aircraft.

#### Resonance Tube Experiment to Determine Velocity of Sound in Air: • A metal tube, open at both the ends is immersed in a tall glass jar filled with water. An air column is thus formed between the open end of the metal tube and the surface of the water. The length of the air column, which is closed by the water surface at the lower end, can be varied by raising or lowering the metal tube.
• A tuning fork is set up into vibrations and held near the mouth of the tube so that its arms vibrate parallel to the axis of the tube. The longitudinal wave, starting from the tuning fork, travels along the length of the air column and is reflected back from the surface of the water. The incident and reflected waves interfere to produce a stationary wave.
• The molecules of air, in contact with the water surface, remain at rest. Therefore, the closed end becomes a node. The molecules of air, near the mouth of the tube, vibrate with maximum amplitude. Therefore, the open end becomes an antinode.
• The frequency of the air column can be changed by adjusting its length. When its frequency becomes the same as the frequency of the fork, resonance takes place, and a loud sound is heard.
• If the length of the air column is increased from a small value, the first resonance occurs when there is a node at the closed end and an antinode at the open end, with no other nodes or antinodes in between. Therefore, the length of the air column  l = λ/4

∴   λ = 4 l

Due to resonance the frequency of the air column is the same as that of the fork

Now velocity of sound is given by v = nλ

∴  v =   4 n l

#### End correction :

• It was shown by Regnault, that the antinode is not formed exactly at the open end but at a distance 0.3 d above the open end where d is the internal diameter of the tube. This additional distance, called the end correction, is necessary, as the wave spreads out slightly, just above the open end, and the air particles just outside the open end are also set into vibrations.

∴  corrected length = + 0.3d

Hence, the velocity of sound in air at room temperature is

V = nλ = 4n ( + 0.3d)

Hence, knowing n, l and d, the velocity of sound in air is determined.

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