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A massless spool of inner radius \( r \), outer radius \( R \) is placed against vertical wall and tilted split floor as shown. A light inextensible thread is tightly wound around the spool through which a mass \( m \) is hanging. There exists no friction at point \( A \), while the coefficient of friction between spool and point \( B \) is \( \mu \). The angle between two surfaces is \( \theta \).
(1) the magnitude of force on the spool at \( B \) in order to maintain equilibrium is \( m g \sqrt{\left(\frac{r}{R}\right)^{2}+\left(1-\frac{r}{R}\right)^{2} \frac{1}{\tan ^{2} \theta}} \)
(2) the magnitude of force on the spool at \( B \) in order to maintain equilibrium is \( m g\left(1-\frac{r}{R}\right) \frac{1}{\tan \theta} \)
(3) the minimum value of \( \mu \) for the system to remain in equilibrium is \( \frac{\cot \theta}{(R / r)-1} \)
(4) the minimum value of \( \mu \) for the system to remain in equilibrium is \( \frac{\tan \theta}{(R / r)-1} \)
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