Question

A wire having a linear mass density $5.0 \times 10^{3} kg / m$ is stretched between two rigid supports with a tension of $450\, N$. The wire resonates at a frequency of $420\, Hz$. The next higher frequency at which the same wire resonates is $480\, Hz$. The length of the wire is

• A. 2.0 m
• B. 2.1 m
• C. 2.5 m
• D. 3 m

Answer 1

Answer: (C)

Suppose the wire vibrates of $420 Hz$ in its nth harmonic and at $490 Hz$ in its $( n +1)$ th harmonic.
$ 420=\frac{n}{2 L} \sqrt{\frac{T}{\mu}} \ldots(i) $
and $ 490=\frac{(n+1)}{2 L} \sqrt{\frac{T}{\mu}} \ldots \text { (ii) } $
Substracting equation (i) from (ii)
$ \frac{1}{2 L} \sqrt{\frac{T}{\mu}}=60 $
$\Rightarrow L=\frac{1}{2 \times 60} \sqrt{\frac{T}{\mu}} $
$\therefore L=\frac{1}{2 \times 60} \sqrt{\frac{450}{5 \times 10^{-3}}}=2.5 m $