Serway & Jewett — Rotation of a Rigid Object about a Fixed Axis: Problem 46

To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance \(h\). Show that the radius \(r\) of a given portion of the track is given by \(r = r_i + \frac{h\theta}{2\pi}\) where \(r_i\) is the radius of the innermost portion of the track and \(\theta\) is the angle through which the disc turns to arrive at the location of the track of radius \(r\). (b) Show that the rate of change of the angle \(\theta\) is given by \(\frac{d\theta}{dt} = \frac{v}{r_i + (h\theta/2\pi)}\) where \(v\) is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle \(\theta\) as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.