3.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-3_531_899_299_497}
\end{figure}
A uniform lamina occupies the region \(R\) bounded by the \(x\)-axis and the curve
$$y = \sin x , \quad 0 \leq x \leq \pi$$
as shown in Figure 2.
- Show, by integration, that the \(y\)-coordinate of the centre of mass of the lamina is \(\frac { \pi } { 8 }\).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{51510155-a8cc-4e70-8ffa-44ed35618261-3_652_792_1439_568}
\end{figure}
A uniform prism \(S\) has cross-section \(R\). The prism is placed with its rectangular face on a table which is inclined at an angle \(\theta ^ { \circ }\) to the horizontal. The cross-section \(R\) lies in a vertical plane as shown in Figure 3. The table is sufficiently rough to prevent \(S\) sliding. Given that \(S\) does not topple, - find the largest possible value of \(\theta\).
(3)