OCR MEI M3 2009 January — Question 4

Exam BoardOCR MEI
ModuleM3 (Mechanics 3)
Year2009
SessionJanuary
TopicCentre of Mass 2
4
  1. The region bounded by the \(x\)-axis and the semicircle \(y = \sqrt { a ^ { 2 } - x ^ { 2 } }\) for \(- a \leqslant x \leqslant a\) is occupied by a uniform lamina with area \(\frac { 1 } { 2 } \pi a ^ { 2 }\). Show by integration that the \(y\)-coordinate of the centre of mass of this lamina is \(\frac { 4 a } { 3 \pi }\).
  2. A uniform solid cone is formed by rotating the region between the \(x\)-axis and the line \(y = m x\), for \(0 \leqslant x \leqslant h\), through \(2 \pi\) radians about the \(x\)-axis.
    1. Show that the \(x\)-coordinate of the centre of mass of this cone is \(\frac { 3 } { 4 } h\).
      [0pt] [You may use the formula \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]
      From such a uniform solid cone with radius 0.7 m and height 2.4 m , a cone of material is removed. The cone removed has radius 0.4 m and height 1.1 m ; the centre of its base coincides with the centre of the base of the original cone, and its axis of symmetry is also the axis of symmetry of the original cone. Fig. 4 shows the resulting object; the vertex of the original cone is V, and A is a point on the circumference of its base. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{b8573ee2-771c-4a93-88d9-346a9da94494-5_716_1228_1027_497} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
    2. Find the distance of the centre of mass of this object from V . This object is suspended by a string attached to a point Q on the line VA, and hangs in equilibrium with VA horizontal.
    3. Find the distance VQ.
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