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)

$V_{1}=\sqrt{\frac{\frac{K \ell / 3}{M}}{\ell+\ell / 3}}$
$=\sqrt{\frac{K \ell \times 4 \ell}{3 \times 3 M}}$
$t _{1}=\frac{\ell+\frac{\ell}{3}}{ V _{1}}=\frac{4 \ell}{3 V _{1}}$
$=\frac{4 \ell}{3} \times \sqrt{\frac{3 \times 3 M }{ K \ell \times 4 \ell}}$
$=2 \cdot \sqrt{\frac{ M }{ K }}$
$V_{2}=\sqrt{\frac{K \times \frac{\ell}{2}}{\frac{M}{\ell+\ell / 2}}}$
$=\sqrt{\frac{K \ell \times 3 \ell}{2 \times 2 M}}$
$t_{2}=\frac{\ell+\frac{\ell}{2}}{V_{2}}=\frac{3 \ell}{2 V_{2}}$
$=\frac{3 \ell}{2} \times \sqrt{\frac{2 \times 2 M}{K \ell \times 3 \ell}}$
$=\sqrt{\frac{3 M}{K}}$
$\frac{t_{1}}{t_{2}}=2 \sqrt{\frac{M}{K}} \times \sqrt{\frac{K}{3 M}}$
$=\frac{2}{\sqrt{3}}$