First slide

Stationary waves

Question

Two wires of ratio of lengths 1:2 and tensions in the ratio 3:1 have the ratio of their fundamental frequencies having ratio 2:1. Find the ratio of their masses.

Moderate

Solution

The fundamental frequency of oscillation is
$\begin{array}{l} f = \frac{1}{2 l} \sqrt{\frac{Tl}{m}} = \frac{1}{2} \sqrt{\frac{T}{ml}} \\ \frac{f_{1}}{f_{2}} = \sqrt{\frac{T_{1}}{T_{2}}} \times \sqrt{\frac{m_{2}}{m_{1}}} \times \sqrt{\frac{l_{2}}{l_{1}}} \\ \frac{m_{1}}{m_{2}} = {\{\sqrt{\frac{T_{1}}{T_{2}}} \times \sqrt{\frac{l_{2}}{l_{1}}} \times (\frac{f_{2}}{f_{1}})\}}^{2} \\ = \frac{T_{1}}{T_{2}} \frac{l_{2}}{l_{1}} {(\frac{f_{2}}{f_{1}})}^{2} = \frac{3}{1} \times \frac{2}{1} \times {(\frac{1}{2})}^{2} = 3 : 2 \end{array}$

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