\begin{tikzpicture}[scale=1.2, thick]
  % Compute positions
  \pgfmathsetmacro{\Wx}{2*3.5*cos(30)}
  \pgfmathsetmacro{\By}{2*3.5*sin(30)}

  \coordinate (A) at (0,0);
  \coordinate (W) at (\Wx,0);
  \coordinate (B) at (\Wx,\By);
  \coordinate (C) at ({0.35*\Wx},{0.35*\By});
  \coordinate (midAB) at ({0.5*\Wx},{0.5*\By});
  \coordinate (midAC) at ({0.175*\Wx},{0.175*\By});

  % Ground
  \draw (-.5,0) -- ({\Wx+1},0);
  \foreach \x in {-0.3,0.0,...,6.3} {
    \draw (\x,0) -- ++(-.2,-.2);
  }

  % Wall
  \fill[gray!60] (\Wx,0) rectangle ++(0.15,{\By+0.8});
  \draw (\Wx,0) -- (\Wx,{\By+0.8});

  % Ladder
  \draw (A) -- (B);

  % Angle arc at A
  \draw (0.9,0) arc[start angle=0, end angle=30, radius=0.9];
  \node at (1.15,0.22) {$30^\circ$};

  % Height h label
  \draw[<->] ({\Wx+0.6},0) -- ({\Wx+0.6},\By);
  \node[right] at ({\Wx+0.6},{0.5*\By}) {$h$};

  % Label 2a along ladder
  \node[above left] at (midAB) {$2a$};

  % Label d from A to C
  \node[above left] at (midAC) {$d$};

  % Point C dot and label
  \fill (C) circle (2pt);
  \node[below right] at (C) {$C$};

  % Labels for A and B
  \node[below left] at (A) {$A$};
  \node[right] at (B) {$B$};
\end{tikzpicture}

The diagram shows a ladder $AB$, of length $2a$ and mass $m$, resting in equilibrium on a vertical wall of height $h$. The ladder is inclined at an angle of $30°$ to the horizontal. The end $A$ is in contact with horizontal ground. An object of mass $2m$ is placed on the ladder at a point $C$ where $AC = d$.

The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.

\begin{enumerate}[label=(\alph*)]
\item Show that the normal contact force between the ladder and the wall is $\frac{mg(a + 2d)\sqrt{3}}{4h}$. [4]
\end{enumerate}

It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is $\frac{1}{3}\sqrt{3}$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumii}{1}
\item Show that $h = k(a + 2d)$, where $k$ is a constant to be determined. [7]

\item Hence find, in terms of $a$, the greatest possible value of $d$. [2]

\item State one improvement that could be made to the model. [1]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2019 Q11 [14]}}