4 In this question, \(a\) is a constant with \(a > 1\).
Fig. 4 shows the region bounded by the curve \(y = \frac { 1 } { x ^ { 2 } }\) for \(1 \leqslant x \leqslant a\), the \(x\)-axis, and the lines \(x = 1\) and \(x = a\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b209dbe7-769c-4301-a2f3-108c27c8cefb-5_447_848_543_612}
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\caption{Fig. 4}
\end{figure}
This region is occupied by a uniform lamina ABCD , where A is \(( 1,1 ) , \mathrm { B }\) is \(( 1,0 ) , \mathrm { C }\) is \(( a , 0 )\) and D is \(\left( a , \frac { 1 } { a ^ { 2 } } \right)\). The centre of mass of this lamina is \(( \bar { x } , \bar { y } )\).
- Find \(\bar { x }\) in terms of \(a\), and show that \(\bar { y } = \frac { a ^ { 3 } - 1 } { 6 \left( a ^ { 3 } - a ^ { 2 } \right) }\).
- In the case \(a = 2\), the lamina is freely suspended from the point A , and hangs in equilibrium. Find the angle which AB makes with the vertical.
The region shown in Fig. 4 is now rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.
- Find the \(x\)-coordinate of the centre of mass of this solid of revolution, in terms of \(a\), and show that it is less than 1.5.