7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-24_590_469_292_484}
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\caption{Figure 5}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b070338-1de4-4c33-be29-d37ac06c9fed-24_415_554_383_1025}
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\caption{Figure 6}
\end{figure}
The shaded region shown in Figure 5 is bounded by the line with equation \(x = a\) and the curve with equation \(x ^ { 2 } + y ^ { 2 } = 4 a ^ { 2 }\)
This shaded region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis to form a solid of revolution.
This solid is used to model a dome with height \(a\) metres and base radius \(\sqrt { 3 } a\) metres.
The dome is modelled as being non-uniform with the mass per unit volume of the dome at the point \(( x , y , z )\) equal to \(\frac { \lambda } { x ^ { 2 } } \mathrm {~kg} \mathrm {~m} ^ { - 3 }\), where \(a \leqslant x \leqslant 2 a\) and \(\lambda\) is a constant.
- Show that the distance of the centre of mass of the dome from the centre of its plane face is \(\left( 4 \ln 2 - \frac { 5 } { 2 } \right) a\) metres.
A solid uniform right circular cone has base radius \(\sqrt { 3 } a\) metres and perpendicular height \(4 a\) metres. A toy is formed by attaching the plane surface of the dome to the plane surface of the cone, as shown in Figure 6.
The weight of the cone is \(k W\) and the weight of the dome is \(2 W\)
The centre of mass of the toy is a distance \(d\) metres from the plane face of the dome. - Show that \(d = \frac { | k + 5 - 8 \ln 2 | } { 2 + k } a\)
The toy is suspended from a point on the circumference of the plane face of the dome and hangs freely in equilibrium with the plane face of the dome at an angle \(\alpha\) to the downward vertical.
Given that \(\tan \alpha = \frac { 1 } { 2 \sqrt { 3 } }\) - find the exact value of \(k\).