15.16 A steel rod 100 cm long is clamped at its middle. The fundamental frequency of longitudinal vibrations of the rod are given to be 2.53 kHz. What is the speed of sound in steel?

Length of the steel rod, l = 100 cm = 1 m

Fundamental frequency of vibration,  $\nu$ = 2.53 kHz = 2.53 $\times {10}^3$ Hz

When the rod is plucked at its middle, an antinode (A) is formed at the centre, and the nodes (N) are formed at its two ends, as shown in the figure.

The distance between two successive nodes is $\lambda$ $/2$

l = $\lambda /2$ $\lambda$ = 2l = 2 m

The speed of sound in steel is given by the relation

v = $\nu \lambda$ = 2.53 $\times {10}^3\times 2$ = 5.06 $\times {10}^3$ m/s = 5.06 km/s

<p>Length of the steel rod, l = 100 cm = 1 m</p><p>Fundamental frequency of vibration, &nbsp;<span title="Click to copy mathml"><math><mi>ν</mi></math></span>&nbsp;= 2.53 kHz = 2.53&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></math></span>&nbsp;Hz</p><p>When the rod is plucked at its middle, an antinode (A) is formed at the centre, and the nodes (N) are formed at its two ends, as shown in the figure.</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728043772php108D9X_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728043772php108D9X.jpeg" alt="" width="289" height="147"></picture></div><div><p>The distance between two successive nodes is&nbsp;<span title="Click to copy mathml"><math><mi>λ</mi></math></span>&nbsp;<span title="Click to copy mathml"><math><mo>/</mo><mn>2</mn></math></span></p><p>l =&nbsp;<span title="Click to copy mathml"><math><mi>λ</mi><mo>/</mo><mn>2</mn></math></span>&nbsp;<span title="Click to copy mathml"><math><mi>λ</mi><mi></mi></math></span>&nbsp;= 2l = 2 m</p><p>The speed of sound in steel is given by the relation</p><p>v =&nbsp;<span title="Click to copy mathml"><math><mi>ν</mi><mi>λ</mi></math></span>&nbsp;= 2.53&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup><mi></mi><mo>×</mo><mn>2</mn></math></span>&nbsp;= 5.06&nbsp;<span title="Click to copy mathml"><math><mo>×</mo><msup><mrow><mrow><mn>10</mn></mrow></mrow><mrow><mrow><mn>3</mn></mrow></mrow></msup></math></span>&nbsp;m/s = 5.06 km/s</p></div></div>

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