3 A bracket is being made from a sheet of uniform thin metal. Firstly, a plate is cut from a square of the sheet metal in the shape OABCDEFHJK, shown shaded in Fig. 3.1. The dimensions shown in the figure are in centimetres; axes $\mathrm { O } x$ and $\mathrm { O } y$ are also shown.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_561_569_429_788}
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\caption{Fig. 3.1}
\end{center}
\end{figure}

(i) Show that, referred to the axes given in Fig. 3.1, the centre of mass of the plate OABCDEFHJK has coordinates (0.8, 2.5).

The plate is hung using light vertical strings attached to $\mathbf { J }$ and $\mathbf { H }$. The edge $\mathbf { J H }$ is horizontal and the plate is in equilibrium. The weight of the plate is 3.2 N .\\
(ii) Calculate the tensions in each of the strings.

The plate is now bent to form the bracket. This is shown in Fig. 3.2: the rectangle IJKO is folded along the line IA so that it is perpendicular to the plane ABCGHI ; the rectangle DEFG is folded along the line DG so it is also perpendicular to the plane ABCGHI but on the other side of it. Fig. 3.2 also shows the axes $\mathrm { O } x , \mathrm { O } y$ and $\mathrm { O } z$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-4_611_782_1713_678}
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\caption{Fig. 3.2}
\end{center}
\end{figure}

(iii) Show that, referred to the axes given in Fig. 3.2, the centre of mass of the bracket has coordinates ( $1,2.7,0$ ).

The bracket is now hung freely in equilibrium from a string attached to O .\\
(iv) Calculate the angle between the edge OI and the vertical.

\hfill \mbox{\textit{OCR MEI M2 2011 Q3 [18]}}