15.18 Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?

<p>Frequency of string A, <span title="Click to copy mathml"><math><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>A</mi></mrow></mrow></msub></math></span>&nbsp;= 324 Hz, let the frequency of string B, be =&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub></math></span></p><p>Beat’s frequency, n = 6 Hz</p><p>Beat’s frequency is given as n =&nbsp;<span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>A</mi></mrow></mrow></msub><mo>±</mo><mi></mi><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub></mrow></mrow></mfenced></math></span>&nbsp;or 6 = 324&nbsp;<span title="Click to copy mathml"><math><mo>±</mo><mi></mi><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub></math></span></p><p><span title="Click to copy mathml"><math><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub><mo>=</mo><mn>318</mn><mi></mi><mi>H</mi><mi>z</mi><mi></mi><mi>o</mi><mi>r</mi><mi></mi><mn>330</mn><mi></mi><mi>H</mi><mi>z</mi></math></span></p><p>Frequency decreases with a decrease in the tension in a string. This is because frequency is directly proportional to the square root of tension, . It is given as</p><p><span title="Click to copy mathml"><math><mi>ν</mi><mi></mi><mo>∝</mo><mi></mi><mo>√</mo><mi>T</mi></math></span></p><p>Hence the beat frequency cannot be 330 Hz. So&nbsp;<span title="Click to copy mathml"><math><msub><mrow><mrow><mi>f</mi></mrow></mrow><mrow><mrow><mi>B</mi></mrow></mrow></msub><mo>=</mo><mn>318</mn><mi></mi><mi>H</mi><mi>z</mi></math></span></p>

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