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Q.
A block of mass $M$ hangs from the lowest a end of the string of mass m and length $l. A$ kink is generated at P. Find the time taken by the kink to reach $Q.$
Waves
Solution:
$V_{p}=\sqrt{\frac{M g}{m / \ell}}=\sqrt{\frac{M g \ell}{m}}$
$V _{ Q }=\sqrt{\frac{( M + m ) q }{ m / \ell}}=\sqrt{\frac{( M + m ) g \ell}{ m }}$
$V_{\text {average }}=\frac{V_{P}+V_{Q}}{2}$
$=\frac{\sqrt{\frac{M g \ell}{m}}+\sqrt{\frac{(M+m) g \ell}{m}}}{2 \sqrt{m}}$
$=\frac{\sqrt{ Mg \ell}+\sqrt{( M + m ) g \ell}}{2 \sqrt{ m }}$
$t=\frac{\ell}{V_{\text {average }}}$
$=\frac{\ell \times 2 \sqrt{m}}{\sqrt{g \ell}(\sqrt{M+m}+\sqrt{m})}$
$=2 \sqrt{\frac{\ell}{ g }} \times \frac{\sqrt{ m }}{(\sqrt{ M + m }+\sqrt{ M })}$
Rationalise,
$=2 \sqrt{\frac{\ell}{ g }} \cdot \frac{\sqrt{ m }}{(\sqrt{ M + m }+\sqrt{ M })}$
$+\frac{\sqrt{ M + m }-\sqrt{ M }}{\sqrt{\sqrt{ M }+ m }-\sqrt{ M }}$
$=2 \sqrt{\frac{\ell}{ g }} \cdot\left(\frac{\sqrt{ M + m }-\sqrt{ M }}{\sqrt{ m }}\right)$
$=2 \sqrt{\frac{\ell}{ mg }} \cdot(\sqrt{ M + m }-\sqrt{ M })$
