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MHT-CET : Physics Entrance Exam

MHT - CET : Physics - Stationary Waves Page 3

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4.

Laws of Vibrating Strings

 

The fundamental frequency of vibrations of a stretched string is given by

n =

1

2l

 

where,

 

 

 

l:

vibrating length

 

m:

mass per unit length of string

 

T:

tension applied to string.

The three laws of vibrating string are:

 

Law of Length: If T and m are constant, the fundamental frequency of vibrating string is inversely proportional to its vibrating length.

 

n

1

l

 

T, m constant

 

 

Law of Tension: If l and m are constant, the fundamental frequency of vibrating string is directly proportional to the square root of the tension T in the string.

 

n

 

l, m constant

 

 

Law of Mass Per Unit Length: If l and T are constant, the fundamental frequency of the vibrating string is inversely proportional to the square root of mass per unit length of the wire.

 

n

1

 

T, l constant

 

Mass per unit length =

Total mass of the wire

Length of the wire

 

m =

Vr

l

=

Al r

l

= Ar

 

\ m = p r2 r

 

V = volume, d = density of the material of the wire, r = radius of the wire.

It can be further shown that the fundamental frequency of the vibrating frequency is inversely proportional to the radius of the wire.

 

i.e.

n

1

r

 

 

and the square root of the density of the material of the wire

 

i.e. n

1

 

 

if other quantities are constant.

 

 

 

5.

Free and Forced Vibrations

 

  • The vibrations performed by a body, when disturbed from its equilibrium position and released, under the action of restoring force, are called free vibrations. The body then vibrates with its natural frequency.
  • When a body is set into free vibrations, the amplitude of vibrations goes on decreasing and the body comes to rest after some time.
  • Examples:
    1. The free vibrations performed by a stretched string when plucked.
    2. The vibrations produced in an empty bottle when air is flown gently across its mouth.
    3. The vibrations of the prongs of a tuning fork hammered on a rubber pad.
  • If a body is made to vibrate by an external periodic force at a frequency different from its natural frequency, the vibrations executed by it are known as forced vibrations. The applied periodic force is called the driving force.
  • The amplitude of the forced vibrations is constant. It depends upon the difference between the natural frequency and the frequency of the external force. If this difference is large, the amplitude is small and vice versa.
  • Examples:
    1. The vibrations performed by a table top when a vibrating tuning fork is held in its contact.
    2. The vibrations performed by the sounding board of a violin when its string is set into vibrations.
    3. The vibrations of the air column in a resonance tube when a vibrating tuning fork is held near its mouth.

Resonance: If a body is made to vibrate by an external periodic force, with a frequency which is the same as the natural frequency of the body, the body vibrates with maximum amplitude. This phenomenon is called resonance.

  • For resonance to occur, the frequency of the external periodic force should be equal to the natural frequency of the body.
  • At resonance, the amplitude of vibrations is maximum.
  • Resonance also occurs if bodies of the same natural frequency are coupled together and one of them is set into vibrations. In that case, the second body starts vibrating by taking energy from the first.

Examples:
1. In a sonometer, resonance occurs when the natural frequency of the vibrating length of the wire equals the frequency of the tuning fork.
2. Resonance tube experiment: If a vibrating tuning fork is held near the mouth of a tube containing an air column, the length of the air column is adjusted such that the natural frequency of the air column equals the frequency of the vibrating tuning fork. A loud sound is heard due to resonance.

Advantages of Resonance

  • Used to determine frequencies of vibrating bodies.
  • Used to determine the velocity of sound in a medium.
  • Used in string instruments like violin, sitar, etc. to produce pleasing notes.
  • The principle used to tune a radio receiver to a particular frequency.

Disadvantages of Resonance:

  • Bodies of vehicles vibrate when they move. If for a certain speed, this frequency equals the natural frequency of the engine, the vehicle starts vibrating violently due to resonance.
  • If the natural frequency of a building equals the frequency of earthquake waves, the building vibrates violently due to resonance, thereby damaging itself.
  • Marching soldiers are asked to break steps while marching on suspension bridges so as to avoid damage to the bridges due to resonance.

 

 

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