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  • Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops, and 4 loops, the frequencies are in the ratio 1:2:3:4.

Show that when a string fixed at its two ends vibrates in 1 loop, 2 loops, 3 loops, and 4 loops, the frequencies are in the ratio 1:2:3:4.

Answers (1)

Let the number of loops in the string be n

The length of each loop is \frac{\lambda }{2}

 L=n\frac{\lambda}{2} or \lambda =\frac{2L}{n}

V=\frac{v}{\lambda} and \lambda =\frac{V}{v}=\frac{2L}{n}

 v=\frac{Vn}{2L }Where V in a stretched string is \sqrt{\frac{T}{m}}

 

v=\frac{n}{2L}\sqrt{\frac{T}{m}}

For n=1, v_{1}=\frac{1}{2L}\sqrt{\frac{T}{m}}=v_{0}

For n=2, v_{2}=\frac{2}{2L}\sqrt{\frac{T}{m}}=2v_{0}

Similarly for n=3 and so on

 v_{1}:v_{2}:v_{3}:v_{4}=n_{1}:n_{2}:n_{3}:n_{4}=1:2:3:4

Posted by

infoexpert24

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