A pipe 20 cm long is closed at one end. Which harmonic mode of the pipe is resonantly excited by a source of 1237.5 Hz ?(sound velocity in air = 330 m/s)

This is a short answer type question as classified in NCERT Exemplar

length of pipe

L=20cm =20 $\times {10}^{-2}m$

$v_{close}=\frac{v}{4L}=\frac{330}{4\times 20\times {10}^{-2}}$

$v_{close}=330\times \frac{100}{80}=412.5Hz$

$\frac{v_{given}}{v_{closed}}=1237.5/412.5$ =3

Hence 3^rd harmonic node of the pipe is resonantly excited by the source of the given frequency.

<p><span data-teams="true">This is a short answer type question as classified in NCERT Exemplar</span></p><p>length of pipe</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1745488023php59Uxuo_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1745488023php59Uxuo.jpeg" alt="" width="191" height="104"></picture></div></div><p>L=20cm =20<span contenteditable="false"> <math> <mo>×</mo> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </mrow> </msup> <mi>m</mi> </math> </span></p><p><span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> </mrow> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <mi>L</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mn>330</mn> </mrow> </mrow> <mrow> <mrow> <mn>4</mn> <mo>×</mo> <mn>20</mn> <mo>×</mo> <msup> <mrow> <mrow> <mn>10</mn> </mrow> </mrow> <mrow> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> </mfrac> </math> </span></p><p><span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>330</mn> <mo>×</mo> <mfrac> <mrow> <mrow> <mn>100</mn> </mrow> </mrow> <mrow> <mrow> <mn>80</mn> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>412.5</mn> <mi>H</mi> <mi>z</mi> </math> </span></p><p><span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>g</mi> <mi>i</mi> <mi>v</mi> <mi>e</mi> <mi>n</mi> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>v</mi> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> <mi>l</mi> <mi>o</mi> <mi>s</mi> <mi>e</mi> <mi>d</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>1237.5</mn> <mo>/</mo> <mn>412.5</mn> <mi> </mi> </math> </span>=3</p><p>Hence 3^rd harmonic node of the pipe is resonantly excited by the source of the given frequency.</p>

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