(i) Use an energy method to find the magnitude of the frictional force acting on the block.

Calculate the coefficient of friction between the block and the plane.\\
(ii) Calculate the power of the tension in the string when the block has a speed of $7 \mathrm {~ms} ^ { - 1 }$.

Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is $M \mathrm {~kg}$. The sides OP and PQ have thin rigid uniform reinforcements attached with masses $0.6 M \mathrm {~kg}$ and $0.4 M \mathrm {~kg}$, respectively. Fig. 3.1 also shows coordinate axes with origin at O .

The sheet with its reinforcements is to be used as an inn sign.\\
(i) Calculate the coordinates of the centre of mass of the inn sign.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{figure}

The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces $Y _ { \mathrm { Q } } \mathrm { N }$ and $Y _ { \mathrm { R } } \mathrm { N }$ act at Q and R respectively.\\
(ii) Calculate the value of $Y _ { \mathrm { Q } }$ and show that $Y _ { \mathrm { R } } = 120$.

The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods $\mathrm { AB } , \mathrm { BC } , \mathrm { CA }$ and CD freely pin-jointed together at $\mathrm { A } , \mathrm { B }$ and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces $X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }$ acting at A and $X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }$ acting at D .

You are given that $\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }$ and $\cos \beta = \frac { 3 } { 5 }$.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{figure}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964}
\captionsetup{labelformat=empty}
\caption{Fig. 3.3}
\end{center}
\end{figure}

(iii) Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D , including those internal to the rods, when the inn sign is hanging from the framework.\\
(iv) Show that $X _ { \mathrm { D } } = 261$.\\
(v) Calculate the forces internal to the rods $\mathrm { AB } , \mathrm { BC }$ and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of $Y _ { \mathrm { D } }$ and $Y _ { \mathrm { A } }$. [Your working in this part should correspond to your diagram in part (iii).]

\hfill \mbox{\textit{OCR MEI M2 2016 Q3 [18]}}