question_answer

A uniform cylinder of mass M lies on a fixed plane inclined at an angle \[\theta \] with horizontal. A flight string is tied to the cylinder at the rightmost point, and a mass m hangs from the string, as shown. Assume that the coefficient of friction between the cylinder and the incline plane is sufficiently large to prevent slipping. For the cylinder the remain static, the value of m is

A) \[\frac{M\,\,\sin \,\,\theta }{1-\,\sin \,\,\theta }\]     

B) \[\frac{M\,\,\cos \,\,\theta }{1+\,\,\sin \,\,\theta }\]

C) \[\frac{M\,\,\sin \,\,\theta }{1+\,\,\sin \,\,\theta }\]             

D) \[\frac{M\,\,\cos \,\,\theta }{1-\,\,\sin \,\,\theta }\]

Correct Answer: A

Solution :

\[T=mg\] \[f=Mg\,\sin \text{ }\theta \,\,+\,\,T\,\sin \,\theta \] \[\Rightarrow \,\,f=Mg\,\sin \text{ }\theta \,\,+\,\,mg\,\sin \,\theta \] Balancing torque about \[\operatorname{C}:Tr =fr\] \[\Rightarrow \,\,\,\operatorname{mg} = Mg\,\,sin\theta  + mg\,sin\,\theta \] \[m=\,\,\frac{M\,\,\sin \,\theta }{1-\sin \,\theta }\]