OCR MEI M3 2008 June — Question 4

Exam BoardOCR MEI
ModuleM3 (Mechanics 3)
Year2008
SessionJune
TopicCentre of Mass 2

4
  1. A uniform solid of revolution is obtained by rotating through \(2 \pi\) radians about the \(y\)-axis the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the \(y\)-axis.
    1. Find the \(y\)-coordinate of the centre of mass of this solid. The solid is now placed on a rough plane inclined at an angle \(\theta\) to the horizontal. It rests in equilibrium with its circular face in contact with the plane as shown in Fig. 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-4_511_568_616_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
    2. Given that the solid is on the point of toppling, find \(\theta\).
  2. Find the \(y\)-coordinate of the centre of mass of a uniform lamina in the shape of the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(- 2 \leqslant x \leqslant 2\), and the \(x\)-axis.