Question

A string of length $2x$ is stretched by $0.1 \, x$ and the velocity of a transverse wave along it is $v$ . When it is stretched by $0.4x$ , the velocity of the wave is

• A. $\sqrt{\frac{5}{6}}v$
• B. $\sqrt{\frac{11}{7}}v$
• C. $\sqrt{\frac{32}{7}}v$
• D. $\sqrt{\frac{27}{6}}v$

Answer 1

Answer: (C)

Wave speed $v=\sqrt{\frac{T}{\mu }}=\sqrt{\frac{T L}{M}}$
$\frac{v_{2}}{v_{1}}=\sqrt{\frac{T_{2} L_{2}}{T_{1} L_{1}}}=\sqrt{\frac{\left(k x_{2}\right) \left(l_{0} + x_{2}\right)}{\left(k x_{1}\right) \left(l_{0} + x_{1}\right)}}$
$\frac{v_{2}}{v_{1}}=\sqrt{\frac{\left(0.4 x\right) \left(2 x + 0.4 x\right)}{\left(0.1 x\right) \left(2 x + 0.1 x\right)}}$
$\frac{v_{2}}{v}=\sqrt{\frac{4 \left(24\right)}{\left(1\right) \left(21\right)}}=\sqrt{\frac{32}{7}}$
$v_{2}=\sqrt{\frac{32}{7}}v$