A wire of density $8 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ is stretched between two clamps $0.5 \mathrm{~m}$ apart. The extension developed in the wire is $3.2 \times 10^{-4} \mathrm{~m}$. If $Y=8 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}$, the fundamental frequency of vibration in the wire will be ___________ $\mathrm{Hz}$.

<p>To determine the fundamental frequency of the vibrating wire, we need to first find the tension (T) in the wire and the wave velocity (V) in the wire.</p> <ol> <li>Tension in the wire (T): We used Young&#39;s modulus (Y) to relate the stress and strain in the wire. The formula for stress is:</li> </ol> <p>$\text{stress} = Y \times \text{strain}$</p> <p>Here, the strain is the extension ($\Delta L$) divided by the original length (L):</p> <p>$\text{strain} = \frac{\Delta L}{L}$</p> <p>Now, the tension (T) in the wire is the product of stress and cross-sectional area (A):</p> <p>$T = \text{stress} \times A$</p> <p>Combining the above equations, we get the expression for tension:</p> <p>$T = \frac{Y \Delta L}{L} \times A$</p> <ol> <li>Wave velocity in the wire (V): The linear mass density ($\mu$) of the wire is given by:</li> </ol> <p>$\mu = \frac{m}{L}$</p> <p>We need to find the ratio $\frac{T}{\mu}$, which represents the square of the wave velocity. Using the expressions for tension and linear mass density, we get:</p> <p>$$\frac{T}{\mu} = \frac{Y \Delta L}{L} \times \frac{A}{m} = \frac{Y \Delta L}{L} \times \frac{1}{\rho}$$</p> <p>Here, $\rho$ is the density of the wire material. Plugging in the given values, we find the value of $\frac{T}{\mu}$, which is:</p> <p>$\frac{T}{\mu} = 6.4 \times 10^3$</p> <p>Now, we find the wave velocity (V) by taking the square root of $\frac{T}{\mu}$:</p> <p>$V = \sqrt{T/\mu} = 80 \mathrm{~m/s}$</p> <ol> <li>Fundamental frequency (f): Finally, we find the fundamental frequency of the vibrating wire using the formula:</li> </ol> <p>$f = \frac{V}{2L}$</p> <p>Plugging in the values, we get the fundamental frequency (f) as:</p> <p>$f = 80 \mathrm{~Hz}$</p> <p>So, the fundamental frequency of vibration in the wire is 80 Hz.</p>