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A string 25cm long and having a mass of ...

A string `25cm` long and having a mass of `2.5 gm` is under tension. A pipe closed at one end is `40cm` long. When the string is set vibrating in its first overtone and the air in the pipe in its fundamental frequency, `8` beats per second are heard. It is observed that decreasing the tension in the string decreases beat frequency. If the speed of sound in air is `320m//s`, find the tension in the string.

Text Solution

Verified by Experts

The correct Answer is:
27.04
For the first overtone of string
Linear mass density of the string= Mass of string per unit
Length `=(2.5xx10^(-3))/(0.25)=0.01 kg//m`
`f_s=1/(lambda)sqrt((T)/(m))=1/(0.25)sqrt((T)/(0.01))`…..(i)
Fundamenta frequency of closed organ
`therefore(lambda_T)/4=0.4implieslambda_T=1.6m, v=f_Tlambda_T`
`f_T=(C)/(lambda_T)=320/1.6=200Hz`.....(II)
In this case 8 beats/second is heard. The beat frequency decreases with the decreasing tension.This means that beat frequency decreass with decreasing `f_s`. This mean that beat frequency is guven by the expression `f_(beat)=f_s-f_T`
`therefore 8=1/0.25sqrt((T)/(0.01))-200 implies sqrt((T)/(0.01))=208xx0.25=52 implies T=0.01xx52^(2)=27.04N`
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