5 Fig. 5.1 shows a solid L-shaped ornament, of uniform density. The ornament is 3 cm thick. The \(x , y\) and \(z\) axes are shown, along with the dimensions of the ornament. The measurements are in centimetres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_556_887_406_244}
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\caption{Fig. 5.1}
\end{figure}
- Determine, with reference to the axes shown, the coordinates of the ornament's centre of mass.
Fig. 5.2 shows the ornament placed so that the shaded face (indicated in Fig. 5.1) is in contact with a plane inclined at \(\theta ^ { \circ }\) to the horizontal, with the 4 cm edge parallel to a line of greatest slope. The surface of the plane is sufficiently rough so that the ornament will not slip down the plane.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-6_646_844_1452_242}
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\caption{Fig. 5.2}
\end{figure} - Determine the minimum and maximum possible values of \(\theta\) for which the ornament does not topple.
The ornament is now placed with its shaded face in contact with a rough horizontal surface. A force of magnitude \(P\) N, acting parallel to the planes of the L -shaped faces, is applied to one of the edges of the ornament, as shown in Fig. 5.3. The force is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the ornament and the surface is \(\mu\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-7_524_680_452_246}
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\caption{Fig. 5.3}
\end{figure}
The value of \(P\) is gradually increased until the ornament is on the point of toppling but does not slide. - Determine the minimum value of \(\mu\).
- Explain how your answer to part (c) would change if the angle between \(P\) and the horizontal was less than \(30 ^ { \circ }\).