6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_257_1079_447_246}
\end{figure}
When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.
- Determine the value of \(x\).
P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2).
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
\includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
\end{figure}
The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4.
A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\).
As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
- \(\quad \mathrm { T } _ { \mathrm { L } }\) is the value of \(T\) at which the beam would start lifting, assuming that is not already sliding.
- \(\quad \mathrm { T } _ { \mathrm { S } }\) is the value of \(T\) at which the beam would start sliding, assuming that it has not already lifted.
- Show that \(\mathrm { T } _ { \mathrm { L } } = 49.1\), correct to 3 significant figures.
- Determine whether the beam will first slide or lift.
\section*{END OF QUESTION PAPER}