2 Two heavy rods AB and BC are freely jointed together at B and to a wall at A . AB has weight 90 N and centre of mass at \(\mathrm { P } ; \mathrm { BC }\) has weight 75 N and centre of mass at Q . The lengths of the rods and the positions of P and Q are shown in Fig. 2.1, with the lengths in metres.
Initially, AB and BC are horizontal. There is a support at R , as shown in Fig. 2.1. The system is held in equilibrium by a vertical force acting at C .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_381_703_584_680}
\captionsetup{labelformat=empty}
\caption{Fig. 2.1}
\end{figure}
- Draw diagrams showing all the forces acting on \(\operatorname { rod } \mathrm { AB }\) and on \(\operatorname { rod } \mathrm { BC }\).
Calculate the force exerted on AB by the hinge at B and hence the force required at C .
The rods are now set up as shown in Fig. 2.2. AB and BC are each inclined at \(60 ^ { \circ }\) to the vertical and C rests on a rough horizontal table. Fig. 2.3 shows all the forces acting on AB , including the forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) due to the hinge at A and the forces \(U \mathrm {~N}\) and \(V \mathrm {~N}\) in the hinge at B . The rods are in equilibrium.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_393_661_1615_429}
\captionsetup{labelformat=empty}
\caption{Fig. 2.2}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_355_438_1530_1178}
\captionsetup{labelformat=empty}
\caption{Fig. 2.3}
\end{figure} - By considering the equilibrium of \(\operatorname { rod } \mathrm { AB }\), show that \(60 \sqrt { 3 } = U + V \sqrt { 3 }\).
- Draw a diagram showing all the forces acting on rod BC .
- Find a further equation connecting \(U\) and \(V\) and hence find their values. Find also the frictional force at C .