A 25 cm long string is fixed at both ends. It has a mass of 2.5 g and vibrates in its first overtone under tension. Also given is a closed pipe which is 30 cm long which vibrates in its fundamental frequency. When both of these are sounded simultaneously, 8 beats per second are heard. It is also observed that decreasing the tension in the string also decreases the beat frequency. What is the tension (in N) in the string? [Given, speed of sound in air is 360ms−1]

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A 25 cm long string is fixed at both ends. It has a mass of 2.5 g and vibrates in its first overtone under tension. Also given is a closed pipe which is 30 cm long which vibrates in its fundamental frequency. When both of these are sounded simultaneously, 8 beats per second are heard. It is also observed that decreasing the tension in the string also decreases the beat frequency. What is the tension (in N) in the string? [Given, speed of sound in air is $360 {ms}^{- 1}$ ]

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59.29 N

52.11 N

56.66 N

55.28 N

answer is C.

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Detailed Solution

$μ = \frac{2 .5}{25} = 0 .1 g / cm = 10^{- 2} kg / m$

$1^{st} overtone, λ_{s} = 25 cm = 0 .25 m \Rightarrow f_{s} = \frac{1}{λ_{s}} \sqrt{\frac{T}{μ}}$
Pipe in fundamental frequency,

$\frac{λ_{p}}{4} = 0 .3 \Rightarrow λ_{p} = 1 .2 m \Rightarrow f_{p} = \frac{v_{s}}{λ_{p}} = \frac{360}{1 .2} = 300 Hz$
$∵$ By decreasing the tension, beat frequency is decreased $∴ f_{s} > f_{p}$