Serway & Jewett — Rotation of a Rigid Object about a Fixed Axis: Problem 48

A cord is wrapped around a pulley that is shaped like a disk of mass \(m\) and radius \(r\). The cord’s free end is connected to a block of mass \(M\). The block starts from rest and then slides down an incline that makes an angle \(\theta\) with the horizontal as shown in Figure P10.48. The coefficient of kinetic friction between block and incline is \(\mu\). (a) Use energy methods to show that the block’s speed as a function of position \(d\) down the incline is \[v = \sqrt{\frac{4Mgd(\sin\theta - \mu\cos\theta)}{m + 2M}}\] (b) Find the magnitude of the acceleration of the block in terms of \(\mu\), \(m\), \(M\), \(g\), and \(\theta\).