Edexcel M3 2022 January — Question 4

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
Year2022
SessionJanuary
TopicCentre of Mass 2

  1. A uniform solid hemisphere \(H\) has radius \(r\) and centre \(O\)
    1. Show that the centre of mass of \(H\) is \(\frac { 3 r } { 8 }\) from \(O\)
    $$\left[ \text { You may assume that the volume of } H \text { is } \frac { 2 \pi r ^ { 3 } } { 3 } \right]$$ A uniform solid \(S\), shown below in Figure 3, is formed by attaching a uniform solid right circular cylinder of height \(h\) and radius \(r\) to \(H\), so that one end of the cylinder coincides with the plane face of \(H\). The point \(A\) is the point on \(H\) such that \(O A = r\) and \(O A\) is perpendicular to the plane face of \(H\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-12_592_791_909_660} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
  2. Show that the distance of the centre of mass of \(S\) from \(A\) is $$\frac { 5 r ^ { 2 } + 12 r h + 6 h ^ { 2 } } { 8 r + 12 h }$$ The solid \(S\) can rest in equilibrium on a horizontal plane with any point of the curved surface of the hemisphere in contact with the plane.
  3. Find \(r\) in terms of \(h\).