Question

# Two strings (fixed at both ends) P and Q of equal length, made of same material and vibrating in same mode produces beats of frequency 5 Hz. String Q is vibrating at a frequency 460 Hz. If the tension in string P increases, the beat frequency increases to 7 Hz. The initial frequency of vibration for string P is :

A
455 Hz
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B
465 Hz
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C
453 Hz
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D
462 Hz
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Solution

## The correct option is B 465 HzFrequency of string Q is 460 Hz Given the beat frequency for string P and Q, fB=5 Hz ⇒|fQ−fP|=5 ⇒460−fP=±5 Therefore the initial frequency of P is either 455 Hz or 465 Hz. Since, both the strings(fixed at both ends) are made of same material and vibrating in same mode, hence λ and μ remains fixed. v=fλ ⇒v∝f ...(i) and v=√Tμ ⇒v∝√T ...(ii) From Eq.(i), (ii) we get: f∝√T It clearly indicates that increase in tension in string P will lead to increase in its vibrating frequency. Also after TP↑, fB=7 Hz ⇒fQ−f′P=±7 ⇒f′P=460±7 ∴f′P=453 Hz Or 467 Hz Hence initial frequency of string must be 465 Hz, so that it increases to 467 Hz after increasing tension.

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