Wave Motion 4 Question 31
29. A sonometer wire under tension of $64 N$ vibrating in its fundamental mode is in resonance with a vibrating tuning fork. The vibrating portion of the sonometer wire has a length of $10 cm$ and mass of $1 g$. The vibrating tuning fork is now moved away from the vibrating wire with a constant speed and an observer standing near the sonometer hears one beat per second. Calculate the speed with which the tuning fork is moved, if the speed of sound in air is $300 m / s$.
(1983, 6M)
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Answer:
Correct Answer: 29. $0.75 m / s$
Solution:
- Fundamental frequency of sonometer wire,
$$ \begin{aligned} f=\frac{v}{2 l} & =\frac{\sqrt{T / \mu}}{2 l}=\frac{1}{2 \times 0.1} \sqrt{\frac{64 \times 0.1}{10^{-3}}} \\ & =400 Hz \end{aligned} $$
Given beat frequency, $f _b=f-f^{\prime}=1 Hz$
$$ \begin{array}{lll} \therefore & f^{\prime}=399 Hz \\ \text { Using, } & f^{\prime}=f\left(\frac{v}{v+v _s}\right) \\ \text { or } & 399=400\left(\frac{300}{300+v _s}\right) \\ \therefore & v _s=0.75 m / s \end{array} $$
