Shown in the figure is a block at rest having an inclined smooth groove in a vertical plane. The horizontal…

Shown in the figure is a block at rest having an inclined smooth groove in a vertical plane. The horizontal surface is smooth. A ball of the same mass as that of the block at rest is released from the top end. The time when the ball will leave groove is $\sqrt{\frac{nL}{2\sqrt{3}g}}$. Find $n$.

Solution

$–mv \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }{60}^{\mathrm{o}}\mathrm{ }+\mathrm{ }2mu=0\mathrm{ }\Rightarrow v=4u$
$\frac{1}{2}m\mathrm{ }[v^2\mathrm{ }+u^2\mathrm{ }+2uv\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{ }120°]+\frac{1}{2}m{u}^2\mathrm{ }=mgx\mathrm{ }\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{n}60°$
$\Rightarrow v^2=\frac{8}{7}\sqrt{3}\mathrm{ } gx \mathrm{ }\mathrm{ }\Rightarrow \mathrm{ }\mathrm{ } a_{\mathrm{r}}\mathrm{ }\mathrm{ }=\mathrm{ }\mathrm{ }\frac{4\sqrt{3}}{7}g$
$\therefore \mathrm{ } t=\sqrt{\frac{2L \mathrm{ }\times \mathrm{ }\mathrm{ }7}{4\sqrt{3}\mathrm{ }g}}$

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