Standard +0.3 This is a standard M2 centre of mass question with routine calculations: part (i) uses the given formula for a semicircle, part (ii) applies composite body techniques with straightforward coordinate geometry, and part (iii) involves basic moment equilibrium about a pivot. All steps follow textbook methods with no novel problem-solving required, making it slightly easier than average.
A uniform semicircular lamina has radius 4 cm . Show that the distance from its centre to its centre of mass is 1.70 cm , correct to 3 significant figures.
\end{figure}
A model bridge is made from a uniform rectangular board, \(A B C D\), with a semicircular section, \(E F G\), removed. \(O\) is the mid-point of \(E G\). \(A B = 8 \mathrm {~cm} , B C = 20 \mathrm {~cm} , A O = 12 \mathrm {~cm}\) and the radius of the semicircle is 4 cm (see Fig. 1).
Show that the distance from \(A B\) to the centre of mass of the model is 9.63 cm , correct to 3 significant figures.
Calculate the distance from \(A D\) to the centre of mass of the model.
\end{figure}
The model bridge is smoothly pivoted at \(A\) and is supported in equilibrium by a vertical wire attached to \(D\). The weight of the model is 15 N and \(A D\) makes an angle of \(10 ^ { \circ }\) with the horizontal (see Fig. 2). Calculate the tension in the wire.
8 (i) A uniform semicircular lamina has radius 4 cm . Show that the distance from its centre to its centre of mass is 1.70 cm , correct to 3 significant figures.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_429_947_405_640}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
A model bridge is made from a uniform rectangular board, $A B C D$, with a semicircular section, $E F G$, removed. $O$ is the mid-point of $E G$. $A B = 8 \mathrm {~cm} , B C = 20 \mathrm {~cm} , A O = 12 \mathrm {~cm}$ and the radius of the semicircle is 4 cm (see Fig. 1).
\begin{enumerate}[label=(\alph*)]
\item Show that the distance from $A B$ to the centre of mass of the model is 9.63 cm , correct to 3 significant figures.
\item Calculate the distance from $A D$ to the centre of mass of the model.\\
(iii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-4_572_945_1416_641}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
The model bridge is smoothly pivoted at $A$ and is supported in equilibrium by a vertical wire attached to $D$. The weight of the model is 15 N and $A D$ makes an angle of $10 ^ { \circ }$ with the horizontal (see Fig. 2). Calculate the tension in the wire.
\end{enumerate}
\hfill \mbox{\textit{OCR M2 2008 Q8 [16]}}