2 The position of the centre of mass, $G$, of a uniform wire bent into the shape of an arc of a circle of radius $r$ and centre C is shown in Fig. 2.1.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_325_1132_365_669}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{center}
\end{figure}

(i) Use this information to show that the centre of mass, G , of the uniform wire bent into the shape of a semi-circular arc of radius 8 shown in Fig. 2.2 has coordinates $\left( - \frac { 16 } { \pi } , 8 \right)$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_586_871_1016_806}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\end{center}
\end{figure}

A walking-stick is modelled as a uniform rigid wire. The walking-stick and coordinate axes are shown in Fig. 2.3. The section from O to A is a semi-circular arc and the section OB lies along the $x$-axis. The lengths are in centimetres.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_394_958_1937_552}
\captionsetup{labelformat=empty}
\caption{Fig. 2.3}
\end{center}
\end{figure}

(ii) Show that the coordinates of the centre of mass of the walking-stick are ( $25.37,2.07$ ), correct to two decimal places.

The walking-stick is now hung from a shelf as shown in Fig. 2.4. The only contact between the walking-stick and the shelf is at A .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-4_339_374_388_842}
\captionsetup{labelformat=empty}
\caption{Fig. 2.4}
\end{center}
\end{figure}

(iii) When the walking-stick is in equilibrium, OB is at an angle $\alpha$ to the vertical.

Draw a diagram showing the position of the centre of mass of the walking-stick in relation to A .\\
Calculate $\alpha$.\\
(iv) The walking-stick is now held in equilibrium, with OB vertical and A still resting on the shelf, by means of a vertical force, $F \mathrm {~N}$, at B . The weight of the walking-stick is 12 N . Calculate $F$.

\hfill \mbox{\textit{OCR MEI M2 2007 Q2 [17]}}