Question

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 $Q.$

• A. $t=\sqrt{\frac{2l}{mg}}(\sqrt{M+m}+\sqrt{M})$
• B. $t=2\sqrt{\frac{l}{mg}}(\sqrt{M+m}+\sqrt{M})$
• C. $t=\sqrt{\frac{2l}{mg}}(\sqrt{M+m}-\sqrt{M})$
• D. $t=2\sqrt{\frac{l}{mg}}(\sqrt{M+m}-\sqrt{M})$

Answer 1

Answer: (D)

$V_p=\sqrt{\frac{Mg}{m/\ell }}=\sqrt{\frac{Mg\ell }{m}}$
$V_Q=\sqrt{\frac{(M+m)q}{m/\ell }}=\sqrt{\frac{(M+m)g\ell }{m}}$
$V_{\text{average }}=\frac{V_P+V_Q}{2}$
$=\frac{\sqrt{\frac{Mg\ell }{m}}+\sqrt{\frac{(M+m)g\ell }{m}}}{2\sqrt{m}}$
$=\frac{\sqrt{Mg\ell }+\sqrt{(M+m)g\ell }}{2\sqrt{m}}$
$t=\frac{\ell }{V_{\text{average }}}$
$=\frac{\ell \times 2\sqrt{m}}{\sqrt{g\ell }(\sqrt{M+m}+\sqrt{m})}$
$=2\sqrt{\frac{\ell }{g}}\times \frac{\sqrt{m}}{(\sqrt{M+m}+\sqrt{M})}$
Rationalise,
$=2\sqrt{\frac{\ell }{g}}\cdot \frac{\sqrt{m}}{(\sqrt{M+m}+\sqrt{M})}$
$+\frac{\sqrt{M+m}-\sqrt{M}}{\sqrt{\sqrt{M}+m}-\sqrt{M}}$
$=2\sqrt{\frac{\ell }{g}}\cdot \left(\frac{\sqrt{M+m}-\sqrt{M}}{\sqrt{m}}\right)$
$=2\sqrt{\frac{\ell }{mg}}\cdot (\sqrt{M+m}-\sqrt{M})$