15.15 A meter-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.

Frequency of the tuning fork,  $\nu$ = 340 Hz

Since the given pipe is attached with a piston at one end, it will behave as a pipe with one end closed and the other end open, as shown in this figure.

$l_1$ = $\frac{\lambda }{4},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}$ of the pipe $l_1$ = 25.5 cm = 0.255 m

$\lambda =4l_1$ = 4 $\times 0.255$ m = 1.02 m

The speed of sound is given by the relation, v = $\nu \lambda$ = 340 $\times 1.02$ = 346.8 m/s

<p>Frequency of the tuning fork, &nbsp;<span title="Click to copy mathml"><math><mi>ν</mi></math></span>&nbsp;= 340 Hz</p><p>Since the given pipe is attached with a piston at one end, it will behave as a pipe with one end closed and the other end open, as shown in this figure.</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728043553php6aT7fC_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728043553php6aT7fC.jpeg" alt="" width="384" height="216"></picture></div></div><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>l</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;=&nbsp;<span title="Click to copy mathml"><math><mfrac><mrow><mrow><mi>λ</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></mfrac><mo>, </mo><mi></mi><mi mathvariant="normal">w</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></math></span>&nbsp;of the pipe&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>l</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;= 25.5 cm = 0.255 m</p><p><span title="Click to copy mathml"><math><mi>λ</mi><mo>=</mo><mn>4</mn><msub><mrow><mrow><mi>l</mi></mrow></mrow><mrow><mrow><mn>1</mn></mrow></mrow></msub></math></span>&nbsp;= 4&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>0.255</mn></math></span>&nbsp;m = 1.02 m</p><p>The speed of sound is given by the relation, v =&nbsp;<span title="Click to copy mathml"><math><mi>ν</mi><mi>λ</mi></math></span>&nbsp;= 340&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><mn>1.02</mn></math></span>&nbsp;= 346.8 m/s</p>

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