Question 4 - A-Level Maths

The region bounded by the curve \(y = x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 2\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8]
The region bounded by the circular arc \(y = \sqrt { 4 - x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 1\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution, as shown in Fig. 4.1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_627_499_593_785} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
\end{figure}
1. Show that the \(x\)-coordinate of the centre of mass of this solid of revolution is 1.35 . This solid is placed on a rough horizontal surface, with its flat face in a vertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig. 4.2. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_573_613_1662_728} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
  \end{figure}
2. Find the least possible coefficient of friction between the solid and the horizontal surface.