Problem source

The diagram shows two particles, $A$ of mass $5m$ and $B$ of mass $3m$, connected by a light  inextensible string which passes over two smooth, light, fixed pulleys, $Q$ and  $R$, and under a smooth pulley $P$ which has mass $M$ and is free to move vertically.
Particles $A$ and $B$ lie on fixed rough planes inclined to the horizontal at angles of $\arctan \frac 7{24}$ and $\arctan\frac43$ respectively.
The segments $AQ$ and $RB$ of the string are 
parallel to their respective planes, and segments $QP$ and $PR$ are vertical. 
The coefficient of friction between each particle and its plane is $\mu$.
\begin{center}
    \begin{tikzpicture}[scale=2]
        \coordinate (Ax) at (0,0);
        \coordinate (O) at ({24/5},0);
        \coordinate (Q) at ({24/5}, {7/5});
        \coordinate (R) at ({24/5+0.5},{7/5});
        \coordinate (D) at ({24/5+0.5},0);
        \coordinate (Bx) at ({24/5+0.5+7/5*3/4}, {0});
        \coordinate (P) at ({24/5 + 0.25}, {7/10});
        \coordinate (A) at ($0.7*(Q) + 0.3*(Ax) + (0,0.1)$);
        \coordinate (B) at ($0.5*(R) + 0.5*(Bx) + (0.07,0.07)$);
    
        \draw[black,fill=blue, opacity=0.1] (O) -- (Ax) -- (Q) -- cycle;
        \draw[fill=blue, opacity=0.1] (R) -- (D) -- (Bx) -- cycle;

        \draw[fill=yellow] (Q) circle (0.1);
        \draw[fill=yellow] (R) circle (0.1);
        \filldraw (P) circle (0.15);
        \draw ($(P)+(-0.15,0)$) -- ($(Q)+(0.1,0)$);
        \draw ($(P)+(+0.15,0)$) -- ($(R)+(-0.1,0)$);
    
        \filldraw (A) circle (0.1);
        \filldraw (B) circle (0.1);
        \draw (A) -- ($(Q)+(-0.03,0.09)$);
        \draw (B) -- ($(R)+(0.04,0.10)$);
        \node at ($(A)+(0,0.1)$) [above] {$A$};
        \node at ($(B)+(0,0.1)$) [above] {$B$};
        \node at ($(Q)+(0,0.1)$) [above] {$Q$};
        \node at ($(R)+(0,0.15)$) [above] {$R$};
        \node at ($(P)-(0,0.2)$) [below] {$P$};
        
    \end{tikzpicture}
\end{center}
\begin{questionparts}
\item Given that the system is in equilibrium, with both $A$ and $B$ on the point of moving up their    planes, determine the value of $\mu$  and show that $M = 6m$.
\item In the case when $M = 9m$, determine the 
initial accelerations of $A$, $B$ and $P$ in terms of $g$.
\end{questionparts}

Solution source

\begin{center}
    \begin{tikzpicture}[scale=2]

        \coordinate (Ax) at (0,0);
        \coordinate (O) at ({24/5},0);
        \coordinate (Q) at ({24/5}, {7/5});
        \coordinate (R) at ({24/5+0.5},{7/5});
        \coordinate (D) at ({24/5+0.5},0);
        \coordinate (Bx) at ({24/5+0.5+7/5*3/4}, {0});
        \coordinate (P) at ({24/5 + 0.25}, {7/10});

        \coordinate (A) at ($0.7*(Q) + 0.3*(Ax) + (0,0.1)$);
        \coordinate (B) at ($0.5*(R) + 0.5*(Bx) + (0.07,0.07)$);
    
        \draw[black,fill=blue, opacity=0.1] (O) -- (Ax) -- (Q) -- cycle;
        \draw[fill=blue, opacity=0.1] (R) -- (D) -- (Bx) -- cycle;


        \draw[fill=yellow] (Q) circle (0.1);
        \draw[fill=yellow] (R) circle (0.1);
        \filldraw (P) circle (0.15);
        \draw ($(P)+(-0.15,0)$) -- ($(Q)+(0.1,0)$);
        \draw ($(P)+(+0.15,0)$) -- ($(R)+(-0.1,0)$);
    
        \filldraw (A) circle (0.1);
        \filldraw (B) circle (0.1);

        \draw (A) -- ($(Q)+(-0.03,0.09)$);
        \draw (B) -- ($(R)+(0.04,0.10)$);

        \node at ($(A)+(0,0.1)$) [above] {$A$};
        \node at ($(B)+(0,0.1)$) [above] {$B$};
        \node at ($(Q)+(0,0.1)$) [above] {$Q$};
        \node at ($(R)+(0,0.15)$) [above] {$R$};
        \node at ($(P)-(0,0.2)$) [below] {$P$};
        

        \draw[-latex, blue, ultra thick] (A) -- ++(0,-0.5) node[below] {$5mg$};
        \draw[-latex, blue, ultra thick] (A) -- ++({24/50},{7/50}) node[above] {$T$};
        \draw[-latex, blue, ultra thick] (A) -- ++({-7/50},{24/50}) node[above] {$R_A$};
        \draw[-latex, blue, ultra thick] (A) -- ++({-24/50},{-7/50}) node[left] {$\mu R_A$};
        \draw[-latex, blue, ultra thick] (B) -- ++(0,-0.5) node[below] {$3mg$};
        \draw[-latex, blue, ultra thick] (B) -- ++({-3/10},{4/10}) node[above] {$T$};
        \draw[-latex, blue, ultra thick] (B) -- ++({4/10},{3/10}) node[above] {$R_B$};
        \draw[-latex, blue, ultra thick] (B) -- ++({3/10},{-4/10}) node[right] {$\mu R_B$};
        \draw[-latex, blue, ultra thick] (P) -- ++(0,-0.5) node[below] {$Mg$};

        \draw[-latex, blue, ultra thick] ($(P)+(-0.15,0)$) -- ++({0},{0.3}) node[above] {$T$};
        \draw[-latex, blue, ultra thick] ($(P)+(0.15,0)$) -- ++({0},{0.3}) node[above] {$T$};
        
    \end{tikzpicture}
\end{center}

First note our triangles are 7-24-25 and 3-4-5 triangles, so we can easily calculate $\sin$ and $\cos$ of our angles.

\begin{questionparts}
\item 
\begin{align*}
\text{N2}(\uparrow, P): && 2T - Mg &= 0 \\
\\
\text{N2}(\perp AQ): && R_A - 5mg \cdot \frac{24}{25} &= 0 \\
\Rightarrow && R_A &= \frac{24}{5}mg \\
\text{N2}(\parallel AQ): && T - \mu R_A - 5mg \cdot \frac{7}{25} &= 0 \\
\Rightarrow && T &= \frac15 mg \l 7+24 \mu \r
\\
\text{N2}(\perp BR): && R_B - 3mg \cdot \frac{3}{5}&= 0 \\
\Rightarrow && R_B &= \frac{9}{5}mg \\
\text{N2}(\parallel AQ): && T - \mu R_B - 3mg \cdot \frac{4}{5} &= 0 \\
\Rightarrow && T &= \frac15 mg \l 12+9 \mu \r \\
\\
\Rightarrow && 12 + 9 \mu &= 7 + 24 \mu \\
\Rightarrow && \mu &= \frac{5}{15} = \frac13 \\
\\
\Rightarrow && Mg &= 2 \cdot \frac15 \cdot mg \cdot (7 + 24 \cdot \frac13) \\
&&&= 6mg \\
\Rightarrow && M &= 6m
\end{align*}

\item Assuming $\mu = \frac13$ 
\begin{align*}
&&9m \ddot{p} &= 9mg - 2T \\
&&5m \ddot{a} &= T - 3mg  \\
&&3m \ddot{b} &= T - 3mg \\
&&2\ddot{p} &=  \ddot{a}+\ddot{b} \\
\Rightarrow &&30m\ddot{p} &= 8T - 24mg \\
&&9m\ddot{p} &= 9mg - 2T \\
\Rightarrow && 66m \ddot{p} &=12mg \\
\Rightarrow && \ddot{p} &= \frac{2}{11}g \\
&& T &= 9mg - 9m \frac{2}{11} g = \frac{9^2}{11}mg\\
&& \ddot{a} &= \frac{3}{22} g \\
&& \ddot{b} &= \frac{5}{22}g

\end{align*}