Question

A string of length 2L, obeying Hooke's Law, is stretched so that its extension is L. The speed of the transverse was traveling on the string is v. If the string is further stretched so that the extension in the string becomes 4L. The speed transverse wave traveling on the string will be.

• A. $\sqrt{2}v$
• B. $v$
• C. $2\sqrt{2}v$
• D. $2v$

Answer 1

The correct option is C $2\sqrt{2}v$

The phase velocity in the string $v=\sqrt{\frac{T}{\rho }}$ where T is the Tension in the string.

In $1^{st}$ case

$T=k\Delta l=kl$

$\rho =\frac{m}{2l+l}=\frac{m}{3l}$

$\Rightarrow v=\sqrt{\frac{3k{l}^2}{m}}$

In $2^{nd}$ case

$T=k\Delta l=k\left(4l\right)$

$\rho =\frac{m}{2l+4l}=\frac{m}{6l}$

$\Rightarrow v^{\prime }=\sqrt{\frac{24k{l}^2}{m}}$

So $\frac{v^{\prime }}{v}=\sqrt{\frac{24}{3}}=\sqrt{8}$

$v^{\prime }=2\sqrt{2}v$.