A wire whose mass per unit length is 10^ 3\,kg/m is stretched by a load of 4kg over the two bridges of a sonometer wire 1m apart. It is struck at its middle point, what would be the wavelength and…

Solution

Step 1: Determine the tension $T$ in the wire

The wire is stretched by a load of mass $m=4\mathrm{kg}$, so the tension is $$T=mg=4\mathrm{kg}\times 9.8m/{\mathrm{s}}^2=39.2\mathrm{N}.$$

Step 2: Find the linear mass density $\mu$

Given the mass per unit length of the wire is $$\mu =10^{-3}kg/m.$$

Step 3: Calculate the velocity $v$ of transverse waves on the wire

The velocity of transverse waves in a stretched wire is $$v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{39.2}{10^{-3}}}=\sqrt{39200}\approx 198m/s.$$

Step 4: Find the wavelength $\lambda$ of the fundamental mode

The wire is fixed at both ends (distance between bridges $L=1\mathrm{m}$). For the fundamental mode (first harmonic, $n=1$), the wavelength is $$\lambda =2L=2\times 1=2\mathrm{m}.$$

Step 5: Calculate the fundamental frequency $f$

The frequency of the fundamental mode is given by $$f=\frac{v}{\lambda }=\frac{198}{2}=99\mathrm{Hz}.$$

Final Answer

• Wavelength of the fundamental vibration: 2 m
• Frequency of the fundamental vibration: 99 Hz