3 A thin rigid non-uniform beam AB of length 6 m has weight 800 N . Its centre of mass, G , is 2 m from B .
Initially the beam is horizontal and in equilibrium when supported by a small round peg at \(\mathrm { C } , 1 \mathrm {~m}\) from A , and a light vertical wire at B . This situation is shown in Fig. 3.1 where the lengths are in metres.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-4_259_460_438_431}
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\caption{Fig. 3.1}
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\includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-4_257_586_447_1046}
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\caption{Fig. 3.2}
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- Calculate the tension in the wire and the normal reaction of the peg on the beam.
The beam is now held horizontal and in equilibrium with the wire at \(70 ^ { \circ }\) to the horizontal, as shown in Fig. 3.2. The peg at C is rough and still supports the beam 1 m from A. The beam is on the point of slipping.
- Calculate the new tension in the wire.
Calculate also the coefficient of friction between the peg and the beam.
The beam is now held in equilibrium at \(30 ^ { \circ }\) to the vertical with the wire at \(\theta ^ { \circ }\) to the beam, as shown in Fig. 3.3. A new small smooth peg now makes contact with the beam at C, still 1 m from A. The tension in the wire is now \(T \mathrm {~N}\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-4_456_353_1484_861}
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\caption{Fig. 3.3}
\end{figure} - By taking moments about C , resolving in a suitable direction and obtaining two equations in terms of \(\theta\) and \(T\), or otherwise, calculate \(\theta\) and \(T\).