3 A box is to be assembled in the shape of the cuboid shown in Fig. 3.1. The lengths are in centimetres. All the faces are made of the same uniform, rigid and thin material. All coordinates refer to the axes shown in this figure. \begin{figure}[h] \end{figure}

1. The four vertical faces \(\mathrm { OAED } , \mathrm { ABFE } , \mathrm { FGCB }\) and CODG are assembled first to make an open box without a base or a top. Write down the coordinates of the centre of mass of this open box. The base OABC is added to the vertical faces.
2. Write down the \(x\) - and \(y\)-coordinates of the centre of mass of the box now. Show that the \(z\)-coordinate is now 1.875 . The top face FGDE is now added. This is a lid hinged to the rest of the box along the line FG. The lid is open so that it hangs in a vertical plane touching the face FGCB.
3. Show that the coordinates of the centre of mass of the box in this situation are (10, 2.4, 2.1). The box, with the lid still touching face FGCB, is now put on a sloping plane with the edge OA horizontal and the base inclined at \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26d30179-589e-462a-a38f-f9e4e5dec4f4-5_610_1091_370_477} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} The weight of the box is 40 N . A force \(P \mathrm {~N}\) acts parallel to the plane and is applied to the mid-point of FG at \(90 ^ { \circ }\) to FG. This force tends to push the box down the plane. The box does not slip and is on the point of toppling about the edge AO .
4. Show that the clockwise moment of the weight of the box about the edge AO is about 0.411 Nm .
5. Calculate the value of \(P\).