Problem source

A uniform ladder of length $l$ and mass $m$ rests with one end in contact with a smooth ramp inclined at an angle of $\pi/6$ to the vertical. The foot of the ladder rests, on horizontal ground, at a distance $l/\sqrt{3}$ from the foot of the ramp, and the coefficient of friction between the ladder and the ground is $\mu.$ The ladder is inclined at an angle $\pi/6$ to the horizontal, in the vertical plane containing a line of greatest slope of the ramp. A labourer of mass $m$ intends to climb slowly to the top of the ladder. 
\begin{center}
    \begin{tikzpicture}[scale=1.2]
    % Filled triangle with opacity
    \fill[black, opacity=0.1] (0,5) -- (0,0) -- (3,0) -- cycle;
    
    % Lines
    \coordinate (O) at (0,0);
    \coordinate (Ltop) at (0,5);
    \coordinate (Lbot) at (3,0);
    \coordinate (Lbott) at (3,1);
    \coordinate (Lmet) at ($(Ltop)!0.4!(Lbot)$);
    \coordinate (Lend) at (7,0);
    
    \draw (O) -- (8,0);
    \draw (Lbot) -- (Ltop);
    \draw (Lmet) -- (Lend);
    \draw (Lbot) -- (Lbott);

    \pic [draw, angle radius=1cm, "$\frac{\pi}{6}$"] {angle = Lbott--Lbot--Ltop};
    \pic [draw, angle radius=1cm, "$\frac{\pi}{6}$"] {angle = Lmet--Lend--Lbot};
    \node at (0.5,2.32) {$\text{ramp}$};
    
    % Double arrow
    \draw[<->] ($(Lbot)+(0,-0.5)$) -- ($(Lend)+(0,-0.5)$) ;
    % Label for the double arrow
    \node at ($0.5*(Lbot)+0.5*(Lend)+(0,-0.5)$) [below] {$l\sqrt{3}$};
\end{tikzpicture}
\end{center}
\begin{questionparts}
\item Find the value of $\mu$ if the ladder slips as soon as the labourer reaches the midpoint. 
\item Find the minimum value of $\mu$ which will ensure that the labourer can reach the top of the ladder. 
\end{questionparts}

Solution source

\begin{center}
    \begin{tikzpicture}[scale=1.2]
    % Filled triangle with opacity
    \fill[black, opacity=0.1] (0,5) -- (0,0) -- (3,0) -- cycle;
    
    % Lines
    \coordinate (O) at (0,0);
    \coordinate (Ltop) at (0,5);
    \coordinate (Lbot) at (3,0);
    \coordinate (Lbott) at (3,1);
    \coordinate (Lmet) at ($(Ltop)!0.4!(Lbot)$);
    \coordinate (Lend) at (7,0);
    
    \draw (O) -- (8,0);
    \draw (Lbot) -- (Ltop);
    \draw (Lmet) -- (Lend);
    \draw (Lbot) -- (Lbott);


    \pic [draw, angle radius=1cm, "$\frac{\pi}{6}$"] {angle = Lbott--Lbot--Ltop};
    \pic [draw, angle radius=1cm, "$\frac{\pi}{6}$"] {angle = Lmet--Lend--Lbot};
    \node at (0.5,2.32) {$\text{ramp}$};
    
    % Double arrow
    \draw[<->] ($(Lbot)+(0,-0.5)$) -- ($(Lend)+(0,-0.5)$) ;
    % Label for the double arrow
    \node at ($0.5*(Lbot)+0.5*(Lend)+(0,-0.5)$) [below] {$l\sqrt{3}$};

    \draw[-latex, blue, ultra thick] ($(Lend)!0.5!(Lmet)$) -- ++({0},{-1/2}) node[below] {$mg$};
    \draw[-latex, blue, ultra thick] ($(Lend)!0.4!(Lmet)$) -- ++({0},{-1/2}) node[below] {$mg$};
    \draw[-latex, blue, ultra thick] ($(Lmet)$) -- ++({5/10},{3/10}) node[right] {$R_2$};
    \draw[-latex, blue, ultra thick] ($(Lend)$) -- ++({0},{5/10}) node[above] {$R_1$};
    \draw[-latex, blue, ultra thick] ($(Lend)$) -- ++({-4/10},{0}) node[left, below] {$F_r$};
\end{tikzpicture}
\end{center}

\begin{questionparts}
\item \begin{align*}
\text{N2}(\uparrow): && R_1 + R_2\sin(\frac{\pi}{6})-2mg &= 0 \\
\text{N2}(\rightarrow): && R_2 \cos (\frac{\pi}{6})-F_r &= 0 \\
\overset{\curvearrowleft}{X}: && lmg \cos \tfrac{\pi}{6} - l R_2 \cos \tfrac{\pi}{6} &= 0 \\
\\
\Rightarrow && R_2 &= mg \\
\Rightarrow && R_1 &= 2mg - \frac12mg \\
&&&=\frac32mg \\
\Rightarrow && \frac{\sqrt{3}}2mg - \mu\frac32mg &= 0 \\
\Rightarrow && \mu &= \frac{1}{\sqrt{3}}
\end{align*}

\item \begin{align*}
\text{N2}(\uparrow): && R_1 + R_2\sin(\frac{\pi}{6})-2mg &= 0 \\
\overset{\curvearrowleft}{X}: && \frac12 lmg \cos \tfrac{\pi}{6}+xmg \cos \tfrac{\pi}{6} - l R_2 \cos \tfrac{\pi}{6} &= 0 \\
\\
\Rightarrow && R_2 &= mg(\frac{1}2+\frac{x}{l}) \\
\Rightarrow && R_1 &= 2mg - \frac12mg(\frac{1}2+\frac{x}{l}) \\
&&&=(\frac74 - \frac{x}{2l})mg \\
&&&\geq \frac{5}{4}mg\\
\text{N2}(\rightarrow): && R_2 \cos (\frac{\pi}{6})-\mu R_1& \leq 0 \\
\Rightarrow && \frac{\sqrt{3}}2mg - \mu\frac54mg &\leq 0 \\
\Rightarrow && \mu &\geq \frac{2\sqrt{3}}{5}
\end{align*}

\end{questionparts}