Laws of Transverse Vibration of Stretched String

Velocity of a transverse wave travelling in stretched string is given by:

$ v = \sqrt{ \frac{T}{ \mu }}$

where T is the tension in the string and μ is the mass per unit length. Now, the fundamental frequency of the stretched string is given by:

$$f = \frac{1}{2l}  \sqrt{ \frac{T}{ \mu } } $$

Here, l is the resonating length.

From the above expression, we knew that there are three laws of transverse vibration of string.

  1. Law of length: The fundamental frequency is directly proportional to the resonating length (L) of the string. So,

$$f  \propto  \frac{1}{L}$$

  1. Law of Tension: The fundamental frequency is directly proportional to the square root of the tension. So, $$f  \propto  \sqrt T$$
  2. Law of mass: The fundamental frequency is inversely proportional to the square root of the mass per unit length.$$f  \propto \frac{1}{\mu}$$

The laws of transverse vibration of stretched string are verified by using sonometer. A sonometer is a hollow wooden box with the wire fixed at its one end and stretched with the load at its another end as shown in figure below. The vibration of the wire is passed by the movable bridge through the box.

Sonometer

We can verify the three laws of vibration as follows:

Law of length (f ∝ 1/L): When we take different tuning forks of different frequencies, and measure the resonating length for each of them keeping tension applied and the material of the wire as constant. The product of frequency and resonating length of one tuning fork was found equal to the frequency and resonating length of another tuning fork. Also, This verifies the law of length.

Law of tension (f ∝ √T): The resonance of different tuning forks of different frequencies was observed by varying tension keeping resonating length and wire as constant. When a graph for f vs √T is plotted, a straight line is obtained. This verifies the law of tension.

Laws of mass (f ∝ 1√μ): The resonance is observed by taking different tuning forks of different frequencies and different wires with separate mass per unit length (μ) keeping resonating length and tension applied as constant. When the graph for f vs 1√μ is plotted and the graph is found to be a straight line. So, law of mass is verified.

 

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5 Responses to “Laws of Transverse Vibration of Stretched String”

  1. Md

    Awesome site

  2. tarun tak

    Velocity of a trasverse wave along a stretched string is proportional to__________

    • surya prasath p

      its is equal to square root of T divide by m

  3. surya prasath p

    how the 1 divide by 2l comes?