3 Three particles are attached to a light rectangular lamina \(O A B C\), which is fixed in a horizontal plane. Take \(O A\) and \(O C\) as the \(x\) - and \(y\)-axes, as shown. Particle \(P\) has mass 1 kg and is attached at the point \(( 25,10 )\).
Particle \(Q\) has mass 4 kg and is attached at the point ( 12,7 ).
Particle \(R\) has mass 5 kg and is attached at the point \(( 4,18 )\). \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-3_782_1033_703_482} Find the coordinates of the centre of mass of the three particles.

$\bar{X} = \frac{25 \times 1 + 12 \times 4 + 4 \times 5}{1 + 4 + 5} = \frac{93}{10}$ or 9.3 | M1 A1 | Two terms on top correct (+third) and denominator correct

$\bar{Y} = \frac{10 \times 1 + 7 \times 4 + 18 \times 5}{10} = \frac{128}{10}$ or 12.8 | M1 A1 | 4 marks

Centre of mass is at (9.3, 12.8) | | SC3 for interchanged $\bar{X}$ and $\bar{Y}$

**Total: 4 marks**

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3 Three particles are attached to a light rectangular lamina $O A B C$, which is fixed in a horizontal plane.

Take $O A$ and $O C$ as the $x$ - and $y$-axes, as shown.

Particle $P$ has mass 1 kg and is attached at the point $( 25,10 )$.\\
Particle $Q$ has mass 4 kg and is attached at the point ( 12,7 ).\\
Particle $R$ has mass 5 kg and is attached at the point $( 4,18 )$.\\
\includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-3_782_1033_703_482}

Find the coordinates of the centre of mass of the three particles.

\hfill \mbox{\textit{AQA M2 2008 Q3 [4]}}