6.
Figure 3
Figure 3 shows a uniform equilateral triangular lamina \(P R T\) with sides of length \(2 a\).
- Using calculus, prove that the centre of mass of \(P R T\) is at a distance \(\frac { 2 \sqrt { } 3 } { 3 } a\) from \(R\). (6)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2c7ac0e1-14fd-4e50-935a-d82e7127c2f8-11_545_588_1121_678}
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\caption{Figure 4}
\end{figure}
The circular sector \(P Q U\), of radius \(a\) and centre \(P\), and the circular sector TUS, of radius \(a\) and centre \(T\), are removed from \(P R T\) to form the uniform lamina \(Q R S U\) shown in Figure 4. - Show that the distance of the centre of mass of QRSU from \(U\) is \(\frac { 2 a } { 3 \sqrt { 3 } - \pi }\)