Question

A string of length ' $l$ ' fixed at both ends vibrates in resonance with a tuning fork of frequency $f$ ' at two successive values of tension $T_{1}$ and $T_{2}$ in the string. Find the specific mass (mass per unit length) of the string :

• A. $\frac{T_{1} T_{2}}{f^{2} l^{2}\left(\sqrt{T_{1}}-\sqrt{T_{2}}\right)^{2}}$
• B. $\frac{2 T_{1} T_{2}}{f^{2} l^{2}\left(\sqrt{T_{1}}-\sqrt{T_{2}}\right)^{2}}$
• C. $\frac{T_{1} T_{2}}{2 f^{2} l^{2}\left(\sqrt{T_{1}}-\sqrt{T_{2}}\right)^{2}}$
• D. None of these

Answer 1

Answer: (D)

We use
$f=\frac{n}{2 l} \sqrt{\frac{T_{1}}{\mu}}$ ...(1)
$f=\frac{n +1}{2 l} \sqrt{\frac{T_{2}}{\mu}}$ ...(2)
From equation (1) and (2) eliminate ' $n$ ' to get $\mu$.
$\mu=\frac{T_{1} T_{2}}{4 f^{2} l^{2}\left(\sqrt{T_{1}}-\sqrt{T_{2}}\right)^{2}}$