OCR MEI M3 2013 June — Question 4

Exam BoardOCR MEI
ModuleM3 (Mechanics 3)
Year2013
SessionJune
TopicCentre of Mass 2
4
  1. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
    \end{figure}
    1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
    2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
  2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
    \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.
This paper (4 questions)
View full paper
Q1 Q2 Q3 Q4