1 A thin uniform rigid rod JK of length 1.2 m and weight 30 N is resting on a rough circular cylinder which is fixed to a floor. The axis of symmetry of the cylinder is horizontal and at all times the rod is perpendicular to this axis.

Initially, the rod is horizontal and its point of contact with the cylinder is 0.4 m from K . It is held in equilibrium by resting on a small peg at J . This situation is shown in Fig. 1.1.

\begin{tikzpicture}[>=stealth, scale=1]
  % Floor
  \fill[pattern=north east lines] (-0.5,0) rectangle (6.5,-0.3);
  \draw (-0.5,0) -- (6.5,0);
  
  % Cylinder
  \draw (4.4,0.8) circle (0.8);
  
  % Rod JK horizontal
  \draw[thick] (1.2,1.6) -- (5.8,1.6);
  
  % Peg at J
  \draw (1.2,1.6) circle (0.12);
  
  % Labels
  \node[above left] at (1.2,1.6) {J};
  \node[above right] at (5.8,1.6) {K};
  
  % Dimension: 0.4m from contact to K
  \draw[<->] (4.28,2.0) -- node[above] {0.4\,m} (5.8,2.0);
\end{tikzpicture}

(i) Calculate the force exerted by the peg on the rod and also the force exerted by the cylinder on the rod.

A small object of weight $W \mathrm {~N}$ is attached to the rod at K .\\
(ii) Find the greatest value of $W$ for which the rod maintains its contact at J .

The object at K is removed. Fig. 1.2 shows the rod resting on the cylinder with its end J on the floor, which is smooth and horizontal. The point of contact of the rod with the cylinder is 0.3 m from K. Fig. 1.2 also shows the normal reaction, $S \mathrm {~N}$, of the floor on the rod, the normal reaction, $R \mathrm {~N}$, of the cylinder on the rod and the frictional force $F \mathrm {~N}$ between the cylinder and the rod.

Suppose the rod is in equilibrium at an angle of $\theta ^ { \circ }$ to the horizontal, where $\theta < 90$.

\begin{tikzpicture}[>=stealth, scale=1]
  % Floor
  \fill[pattern=north east lines] (-1.5,0) rectangle (7,-0.3);
  \draw (-1.5,0) -- (7,0);
  
  % Cylinder
  \draw (4.0,0.9) circle (0.9);
  
  % Rod angle
  \def\thetaDeg{22}
  \pgfmathsetmacro{\ux}{cos(\thetaDeg)}
  \pgfmathsetmacro{\uy}{sin(\thetaDeg)}
  \def\rodLen{5.2}
  
  \pgfmathsetmacro{\Kx}{0.5 + \rodLen*\ux}
  \pgfmathsetmacro{\Ky}{\rodLen*\uy}
  \pgfmathsetmacro{\midX}{0.5 + 0.5*\rodLen*\ux}
  \pgfmathsetmacro{\midY}{0.5*\rodLen*\uy}
  \pgfmathsetmacro{\Cx}{0.5 + 0.75*\rodLen*\ux}
  \pgfmathsetmacro{\Cy}{0.75*\rodLen*\uy}
  \pgfmathsetmacro{\nx}{-\uy}
  \pgfmathsetmacro{\ny}{\ux}
  
  % Draw rod
  \draw[thick] (0.5,0) -- (\Kx,\Ky);
  
  % Labels
  \node[below left] at (0.5,0) {J};
  \node[above right] at (\Kx,\Ky) {K};
  
  % Angle theta at J
  \draw ({0.5+1.0},0) arc[start angle=0, end angle=\thetaDeg, radius=1.0];
  \node[right] at ({0.5+1.1},0.22) {$\theta°$};
  
  % S arrow at J
  \draw[->, thick] (0.5,0) -- (0.5,1.3) node[above left] {$S$\,N};
  
  % Weight at centre
  \draw[->, thick] (\midX,\midY) -- (\midX,{\midY-1.3}) node[right] {30\,N};
  
  % R arrow perpendicular to rod at contact
  \draw[->, thick] (\Cx,\Cy) -- ({\Cx+1.2*\nx},{\Cy+1.2*\ny}) node[above] {$R$\,N};
  
  % F arrow along rod toward K
  \draw[->, thick] (\Cx,\Cy) -- ({\Cx+0.9*\ux},{\Cy+0.9*\uy}) node[above right] {$F$\,N};
  
  % 0.3 m label
  \draw[<->] ({\Cx-0.4*\nx},{\Cy-0.4*\ny}) -- ({\Kx-0.4*\nx},{\Ky-0.4*\ny}) node[midway, below right] {0.3\,m};
\end{tikzpicture}

(iii) Find $S$. Find also expressions in terms of $\theta$ for $R$ and $F$.

The coefficient of friction between the cylinder and the rod is $\mu$.\\
(iv) Determine a relationship between $\mu$ and $\theta$.

\hfill \mbox{\textit{OCR MEI M2 2015 Q1 [16]}}