Chapter 3, Problem 3.12P

Instead of using the system shown in Figure 3.2.6a to raise the mass $m_2$ . An engineer proposes to use two simple machines, the pulley and the inclined plane, to reduce the weight required to lift $m_2$ . The proposed design is shown in Figure P3.12. The pulley inertias are negligible. The available horizontal space limits the angle of the inclined plane to no less than 30°.

1. Suppose that the friction between the plane and the mass $m_2$ is negligible. Determine the smallest value $m_1$ can have to lift $m_2$ . Your answer should be a function of $m_2$ and $\theta$ .

In practice, the coefficient of dynamic friction ${\mu }_d$ between the plane and the mass $m_2$ is not known precisely. Assume that the system can be started to overcome static friction. For the value of $m_1=m_2/2$ , how large can ${\mu }_d$ be before $m_1$ cannot lift $m_2$?