4 You are given that the centre of mass, \(G\), of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1. Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where \(\mathrm { OA } = \mathrm { AB }\), and OBCD is a rectangle. The distance OD is \(h \mathrm {~cm}\), where \(h\) can take various positive values. All coordinates refer to the axes \(\mathrm { O } x\) and Oy shown. The units of the axes are centimetres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Write down the coordinates of the centre of mass of the triangle OAB .
2. Show that the centre of mass of the region OABCD is \(\left( \frac { 12 - h ^ { 2 } } { 2 ( h + 3 ) } , 2.5 \right)\). The \(x\)-axis is horizontal.
   The prism is placed on a horizontal plane in the position shown in Fig. 4.2.
3. Find the values of \(h\) for which the prism would topple. The following questions refer to the case where \(h = 3\) with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N . You should assume that the prism does not slide.
4. Suppose that the prism is held in this position by a vertical force applied at A . Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force.
5. Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD , applied at \(30 ^ { \circ }\) below the horizontal at C , as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{680f1be3-13a2-4f75-a324-fb6aadf07607-5_215_510_2397_860} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}