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 $450N$. The wire resonates at a frequency of $420Hz$. The next higher frequency at which the same wire resonates is $480Hz$. 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 $420Hz$ in its nth harmonic and at $490Hz$ in its $(n+1)$ th harmonic.
$420=\frac{n}{2L}\sqrt{\frac{T}{\mu }}\dots (i)$
and $490=\frac{(n+1)}{2L}\sqrt{\frac{T}{\mu }}\dots \text{ (ii) }$
Substracting equation (i) from (ii)
$\frac{1}{2L}\sqrt{\frac{T}{\mu }}=60$
$\Rightarrow L=\frac{1}{2\times 60}\sqrt{\frac{T}{\mu }}$
$\therefore L=\frac{1}{2\times 60}\sqrt{\frac{450}{5\times 10^{-3}}}=2.5m$