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26. (AIEEE 2006)

A string is stretched between fixed points separated by $75.0$ $c m .$ It is observed to have resonant frequencies of $420$ $H z$ and $315$ $H z$ . There are no other resonant frequencies between these two. Then, the lowest resonant frequency for this string is

A. 105 Hz

B. 1.05 Hz

C. 1050 Hz

D. 10.5 Hz

Correct Answer is Option (A)

Given $\frac{n v}{2 ℓ} = 315$ and $(n + 1) \frac{v}{2 ℓ} = 420$

$\Rightarrow \frac{n + 1}{n} = \frac{420}{315} \Rightarrow n = 3$

Hence $3 \times \frac{v}{2 ℓ} = 315 \Rightarrow \frac{v}{2 ℓ} = 105 H z$

Lowest resonant frequency is when $n = 1$

Therefore lowest resonant frequency $= 105 H z .$

27. (AIEEE 2003)

A metal wire of linear mass density of $9.8$ $g / m$ is stretched with a tension of $10$ $k g$ - $w t$ between two rigid supports $1$ metre apart. The wire passes at its middle point between the poles of a permanent magnet, and it vibrates in resonance when carrying an alternating current of frequency $n .$ The frequency $n$ of the alternating source is :

A. 50 Hz

B. 100 Hz

C. 200 Hz

D. 25 Hz

Correct Answer is Option (A)

KEY CONCEPT : For a string vibrating between two rigid support, the fundamental frequency is given by

$n = \frac{1}{2 ℓ} \sqrt{\frac{T}{μ}} = \frac{1}{2 \times} \sqrt{\frac{10 \times 9.8}{9.8 \times 10^{- 3}}} = 50 H z$

As the string is vibrating in resonance to a.c of frequency $n,$ therefore both the frequencies are same.

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