Question

A wire stretched between two points with a tension $200{\text{N}}$, Whose linear density is $5 \times {10^{ - 3}}{\text{kg}}{{\text{m}}^{ - 1}}$. The wire resonates at a frequency of $170{\text{Hz}}$. The next higher frequency at which the same wire resonates is $210{\text{Hz}}$. What is the length of the wire?

Answer 1

Here we need to find the length of the wire and we have given frequency, tension and linear density. We need to use the relation between frequency, tension, and length for both frequencies then we substitute the known value in the equation to get the final result.

Complete step by step solution:
Let us consider that the wire vibrates at $170{\text{Hz}}$ in its ${n^{th}}$ harmonic and at $210{\text{Hz}}$ in its ${(n + 1)^{th}}$ harmonic.
Now let us use the relation between frequency, density and tension
$f = \dfrac{n}{{2L}}\sqrt {\dfrac{F}{\mu }}$
Where $f$ is the frequency, $n$ His the number of harmonic, $L$ is the length of the wire, $F$ is the tension and $\mu$ is the linear density,
Substituting the values of frequency for both the harmonics separately we get two equations
$\Rightarrow 170 = \dfrac{n}{{2L}}\sqrt {\dfrac{F}{\mu }} - - - (1)$
$\Rightarrow 210 = \dfrac{{(n + 1)}}{{2L}}\sqrt {\dfrac{F}{\mu }} - - - (2)$
Comparing both the equations we get,
$\dfrac{{210}}{{170}} = \dfrac{{(n + 1)}}{n}$
$\Rightarrow 1.23n = n + 1$
$\therefore n = \dfrac{1}{{0.23}} = 4.3$
Putting the value of $n$ and also substituting the other given values in equation 1, we get
$170 = \dfrac{{4.3}}{{2L}}\sqrt {\dfrac{{200}}{{5 \times {{10}^{ - 3}}}}}$
$\Rightarrow L = \dfrac{{4.3}}{{2 \times 170}}\sqrt {40,000}$
$\Rightarrow L = 0.0126 \times 200$
$\therefore L = 2.5{\text{m}}$

Additional Information: For standing waves on a string the ends are fixed and the string does not move. Places, where the string is not vibrating, are called nodes. This limits the wavelengths that are possible which in turn determines the frequencies. The lowest frequency is called the fundamental or first harmonic. For a string, higher frequencies are all multiples of the fundamental and are called harmonics. The more general term overtone is used to indicate frequencies greater than the fundamental which may or may not be harmonic.

Note: We need to find the value of the number of harmonics in which the wire is in resonant condition to find the length of the wire. Linear density is the measure of a quantity of any characteristic value per unit of length.