All Problems The Fundamental Equation of Dynamics
Problem 1.93
A fixed pulley carries a weightless thread with masses \(m_{1}\) and \(m_{2}\) at its ends. There is friction between the thread and the pulley. It is such that the thread starts slipping when the ratio \(m_{2} / m_{1}\) \(=\eta_{0} .\) Find: (a) the friction coefficient; (b) the acceleration of the masses when \(m_{2} / m_{1}=\eta>\eta_{0}\).
1.93. (a) Let us examine a small element of the thread in contact with the pulley (Fig. 5). Since the element is weightless, \(d T=\) \(=d F_{f r}=k d F_{n}\) and \(d F_{n}=T d \alpha .\) Hence, \(d T / T=k d \alpha .\) Integrat- \begin{aligned} &\text { ing this equation, we obtain } k=\left(\ln \eta_{0}\right) / \pi\\ &\text { (b) } w=g\left(\eta-\eta_{0}\right) /\left(\eta+\eta_{0}\right) \text { . } \end{aligned}