Question

A string of length l is elongated by $\frac{\ell }{3}$ the time taken by the transverse wave to cover the string is $t_1$. If the string is elongated by a distance $\frac{\ell }{2},$ the time taken by the transverse wave to cover the string is $t_2$. Then, $\frac{t_1}{t_2}=$ (Assume that the string obey's Hook's law)

• A. $\frac{4}{3}$
• B. $\frac{4}{\sqrt{3}}$
• C. $\frac{4}{3\sqrt{3}}$
• D. $\frac{2}{\sqrt{3}}$

Answer 1

Answer: (D)

$V_1=\sqrt{\frac{\frac{K\ell /3}{M}}{\ell +\ell /3}}$
$=\sqrt{\frac{K\ell \times 4\ell }{3\times 3M}}$
$t_1=\frac{\ell +\frac{\ell }{3}}{V_1}=\frac{4\ell }{3V_1}$
$=\frac{4\ell }{3}\times \sqrt{\frac{3\times 3M}{K\ell \times 4\ell }}$
$=2\cdot \sqrt{\frac{M}{K}}$
$V_2=\sqrt{\frac{K\times \frac{\ell }{2}}{\frac{M}{\ell +\ell /2}}}$
$=\sqrt{\frac{K\ell \times 3\ell }{2\times 2M}}$
$t_2=\frac{\ell +\frac{\ell }{2}}{V_2}=\frac{3\ell }{2V_2}$
$=\frac{3\ell }{2}\times \sqrt{\frac{2\times 2M}{K\ell \times 3\ell }}$
$=\sqrt{\frac{3M}{K}}$
$\frac{t_1}{t_2}=2\sqrt{\frac{M}{K}}\times \sqrt{\frac{K}{3M}}$
$=\frac{2}{\sqrt{3}}$