A uniform solid cylinder of mass $m$ rests on two horizontal planks. A thread is wound on the cylind

Question

A uniform solid cylinder of mass $m$ rests on two horizontal planks. A thread is wound on the cylinder. The hanging end of the thread is pulled vertically down with a constant force $F$. The coefficient of friction between the cylinder and the planks is $k$. Assume the cylinder rolls without slipping initially. Which of the following statements is/are correct?

Professor PC Thomas Classes · Answer

Normal force $N = mg + F$ (A is correct). Equations of motion: Horizontal $f = ma$, Torque $FR - fR = I\alpha = (\frac{1}{2}mR^2)(a/R) \Rightarrow F - f = \frac{1}{2}ma$. Solving gives $f = \frac{2}{3}F$ and $a = \frac{2F}{3m}$ (B is correct). To prevent sliding, $f \le kN \Rightarrow \frac{2}{3}F \le k(mg + F) \Rightarrow F(2-3k) \le 3kmg$. If $k < 2/3$, $F_{max} = \frac{3kmg}{2-3k}$ (C is correct). If $2-3k \le 0$ ($k \ge 2/3$), the inequality holds for any positive $F$, so it never slides (D is correct).