3
\begin{enumerate}[label=(\alph*)]
\item Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods $\mathrm { AB } , \mathrm { BC } , \mathrm { AD }$ and BD . The rods are freely pin-jointed to each other at $\mathrm { A } , \mathrm { B }$ and D and to a vertical wall at C and D. There are vertical loads of $L \mathrm {~N}$ at A and $3 L \mathrm {~N}$ at B . Angle DAB is $30 ^ { \circ }$, angle DBC is $60 ^ { \circ }$ and ABC is a straight, horizontal line.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_538_617_497_804}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\roman*)]
\item Draw a diagram showing the loads and the internal forces in the four rods.
\item Find the internal forces in the rods in terms of $L$, stating whether each rod is in tension or in thrust (compression). [You may leave answers in surd form. Note that you are not required to find the external forces acting at C and at D.]
\end{enumerate}\item Fig. 3.2 shows uniform beams PQ and QR , each of length 2 lm and of weight $W \mathrm {~N}$. The beams are freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at $60 ^ { \circ }$ to the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal. This force, $U \mathrm {~N}$, is shown in Fig. 3.3.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_428_566_1699_536}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-4_296_282_1699_1407}
\captionsetup{labelformat=empty}
\caption{Fig. 3.3}
\end{center}
\end{figure}

Show that the frictional force between the floor and each beam is $\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}$.\\
(a)

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-5_641_885_269_671}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}

A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing line of length 2 m . The other end of the line, A , is fixed and the line can swing freely.

The sphere swings with the line taut from a point where the line is at an angle of $40 ^ { \circ }$ with the vertical, as shown in Fig. 4.
\begin{enumerate}[label=(\roman*)]
\item Explain why no work is done on the sphere by the tension in the line.
\item Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the lowest point of its swing and calculate the amount of gravitational potential energy lost when it is at this point.
\item Assuming that there is no air resistance and that the sphere swings from rest from the position shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing.
\item Now consider the case where

\begin{itemize}
\end{enumerate}\item there is a force opposing the motion that results in an energy loss of 0.6 J for every metre travelled by the sphere,
  \item the sphere is given an initial speed of $2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ (and it is descending) with AB at $40 ^ { \circ }$ to the vertical.
\end{itemize}

Calculate the speed of the sphere at the lowest point of its swing.\\
(b) A block of mass 3 kg slides down a uniform, rough slope that is at an angle of $30 ^ { \circ }$ to the horizontal. The acceleration of the block is $\frac { 1 } { 8 } g$.

Show that the coefficient of friction between the block and the slope is $\frac { 1 } { 4 } \sqrt { 3 }$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI M2 2008 Q3 [18]}}