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]
\includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_325_1132_365_669}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{figure}
- 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]
\includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_586_871_1016_806}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\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]
\includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-3_394_958_1937_552}
\captionsetup{labelformat=empty}
\caption{Fig. 2.3}
\end{figure} - 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]
\includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-4_339_374_388_842}
\captionsetup{labelformat=empty}
\caption{Fig. 2.4}
\end{figure} - 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 \(\alpha\). - 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 \(F\).