A uniform chain is at rest partially on the incline and partially hanging vertically. Coefficient of friction between chain and incline is $\mu = \frac{1}{{2\sqrt 3 }}$. The ratio of $\frac{{{L_{\max }}}}{{{L_{\min }}}}$ is $(L_{max} =$ maximum length of chain kept on inclined so that chain remains at rest, $L_{min} =$ minimum length of chain kept on incline so that chain remains at rest)
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For $\mathrm{L}_{\mathrm{max}}$
$(\lambda)$ $(L_ {max} )$ g $\sin 30^{\circ}$
$=(\mu)(\lambda)\left(L_{\max }\right) g \sin 30^{\circ}+(\lambda)\left(\mathrm{L}-\mathrm{L}_{\max }\right) g$
For $L_{\min }$
$\left((\lambda)\left(L_{\min }\right) g+(\mu)(\lambda) L_{\min } g\right) \sin 30^{\circ}$
$=(\lambda)\left(L-L_{\min }\right) g$
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