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  • A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340 Hz) when the tube length is 25.5 cm or 79.3 cm.

Q.15.15 A metre-long tube open at one end, with a movable piston at the other end, shows resonance with a fixed frequency source (a tuning fork of frequency 340\: Hz) when the tube length is 25.5\: cm or 79.3\: cm.  Estimate the speed of sound in air at the temperature of the experiment. The edge effects may be neglected.

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The pipe behaves as a pipe open at one end and closed at one end. Such a pipe would produce odd harmonics i.e.

\nu =(2n-1)\frac{v}{4l_{n}}

Two consecutive modes of vibration are given in the question

For l1 = 25.5 cm

\nu =(2n-1)\frac{v}{4l_{1}}

For l2 = 79.3 cm

\\\nu =(2(n+1)-1)\frac{v}{4l_{2}}\\ \nu =(2n+1)\frac{v}{4l_{2}}

Since at both these modes the system resonates with the same frequency we have

\\(2n-1)\frac{v}{4l_{1}} =(2n+1)\frac{v}{4l_{2}}\\ \frac{2n-1}{2n+1}=\frac{25.5}{79.3}\\ \frac{2n-1}{2n+1}\approx \frac{1}{3}\\ n=1(our approximation is correct since the edge effects may be neglected)

\\\nu =\frac{v}{4l_{1}}\\ v=340\times 4\times 0.255\\ v=346.8\ ms^{-1}

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