A) \[\frac{\ell }{\mu }\]
B) \[\frac{\ell }{\mu +1}\]
C) \[\frac{\mu \ell }{\mu +1}\]
D) \[\frac{\mu \ell }{\mu -1}\]
Correct Answer: C
Solution :
Rope of length £ has mass = m \[\operatorname{Rope} of hanging length\,{{\ell }_{1}} has mass = \frac{m}{\ell }{{\ell }_{1}}\] In equilibrium Weight of hanging part = force of friction on the part of rope \[=\frac{\operatorname{m}}{\ell }{{\ell }_{1}}\,\operatorname{g}=\frac{\mu \operatorname{m}}{\operatorname{g}}\left( \ell -{{\ell }_{1}} \right)\operatorname{g}\] \[={{\ell }_{1}}\,\operatorname{g}=\mu \operatorname{m}\operatorname{g}-\mu \ell g\] \[{{\ell }_{1}}=\left[ \frac{\mu }{1+\mu } \right]\ell \]
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