3 A non-uniform beam AB has weight 85 N . The length of the beam is 5 m and its centre of mass is 3 m from A . In this question all the forces act in the same vertical plane.
Fig. 3.1 shows the beam in horizontal equilibrium, supported at its ends.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_215_828_466_660}
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\caption{Fig. 3.1}
\end{figure}
- Calculate the reactions of the supports on the beam.
Using a smooth hinge, the beam is now attached at A to a vertical wall. The beam is held in equilibrium at an angle \(\alpha\) to the horizontal by means of a horizontal force of magnitude 27.2 N acting at B . This situation is shown in Fig. 3.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_725_675_1153_347}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_732_565_1153_1231}
\captionsetup{labelformat=empty}
\caption{Fig. 3.3}
\end{figure} - Show that \(\tan \alpha = \frac { 15 } { 8 }\).
The hinge and 27.2 N force are removed. The beam now rests with B on a rough horizontal floor and A on a smooth vertical wall, as shown in Fig. 3.3. It is at the same angle \(\alpha\) to the horizontal. There is now a force of 34 N acting at right angles to the beam at its centre in the direction shown. The beam is in equilibrium and on the point of slipping.
- Draw a diagram showing the forces acting on the beam.
Show that the frictional force acting on the beam is 7.4 N .
Calculate the value of the coefficient of friction between the beam and the floor.