3.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b6e93edf-1b9f-4ea9-bb41-f46f380bc623-06_990_985_244_539}
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\caption{Figure 1}
\end{figure}
A uniform circular disc \(C\) has centre \(X\) and radius \(R\).
A disc with centre \(Y\) and radius \(r\), where \(0 < r < R\) and \(X Y = R - r\), is removed from \(C\) to form the template shown shaded in Figure 1.
The centre of mass of the template is a distance \(k r\) from \(X\).
- Show that \(r = \frac { k } { 1 - k } R\)
- Hence find the range of possible values of \(k\).
The point \(P\) is on the outer edge of the template and \(P X\) is perpendicular to \(X Y\).
The template is freely suspended from \(P\) and hangs in equilibrium.
Given that \(k = \frac { 4 } { 9 }\) - find the angle that \(X Y\) makes with the vertical.
The mass of the template is \(M\).
- Find, in terms of \(M\), the mass of the lightest particle that could be attached to the template so that it would hang in equilibrium from \(P\) with \(X Y\) horizontal.