Question

A solid cylinder of mass m is kept in balance on a fixed incline of angle $\alpha ={37}^{\circ }$ with the help of a thread fastened to its jacket. The cylinder does not slip.
What force F is required to keep the cylinder in balance when the thread is held vertically?

• A. $\frac{mg}{2}$
• B. $\frac{3mg}{4}$
• C. $\frac{3mg}{8}$
• D. None of these

Answer 1

The correct option is C $\frac{3mg}{8}$

Rotational equilibrium:$Fr=fr⟹F=f⟹\left(1\right)$

Translational equilibrium:Horizontal direction

$Nsin\alpha =fcos\alpha ⟹N=fcos\alpha sin\alpha ⟹\left(2\right)$

Vertical direction:$F+Ncos\alpha +fsin\alpha =mg$

$F+\frac{Fcos2\alpha }{sin\alpha }+Fsin\alpha =mg$

Solving we get$F(1+\frac{1}{sin\alpha })=mg.$where $sin\alpha =\frac{3}{5}$

$\therefore F(1+\frac{5}{3})=mg⟹F=\frac{3mg}{8}$