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Problem Text

A small light ring is attached to the end $A$ of a uniform rod $AB$ of weight $W$ and length $2a$. The ring can slide on a rough horizontal rail.
One end of a light inextensible string of length $2a$ is attached to the rod at $B$ and the other end is attached to a point $C$ on the rail so that the rod makes an angle of $\theta$ with the rail, where $0 < \theta < 90^{\circ}$. The rod hangs in the same vertical plane as the rail.
A force of $kW$ acts vertically downwards on the rod at $B$ and the rod is in equilibrium.
\begin{questionparts}
\item You are given that the string will break if the tension $T$ is greater than $W$. Show that (assuming that the ring does not slip) the string will break if
$$2k + 1 > 4 \sin \theta.$$
\item Show that (assuming that the string does not break) the ring will slip if
$$2k + 1 > (2k + 3)\mu \tan \theta,$$
where $\mu$ is the coefficient of friction between the rail and the ring.
\item You are now given that $\mu \tan \theta < 1$.
Show that, when $k$ is increased gradually from zero, the ring will slip before the string breaks if
$$\mu < \frac{2 \cos \theta}{1 + 2 \sin \theta}.$$
\end{questionparts}

Solution (Optional)

\begin{center}
    \begin{tikzpicture}

\def\a{-2};
        \def\b{-1};
        \coordinate (A) at (\a, 0);
        \coordinate (B) at (0, \b);
        \coordinate (C) at (-\a, 0);

\draw[thin] ($(A)!-0.5!(C)$) -- ($(A)!1.5!(C)$);

\filldraw (A) circle (1.5pt) node[above left] {$A$};
        \filldraw (B) circle (1.5pt) node[above] {$B$};
        \filldraw (C) circle (1.5pt) node[above] {$C$};

\draw[ultra thick] (A) -- (B);
        \draw[thick] (B) -- (C);

\draw[-latex, ultra thick, blue] (A) -- ++(0, 1) node[above] {$R$};
        \draw[-latex, ultra thick, blue] (A) -- ++(-1, 0) node[above] {$F_A$};
        \draw[-latex, ultra thick, blue] (B) -- ++(0, -1) node[below] {$kW$};
        \draw[-latex, ultra thick, blue] (B) -- ($(B)!0.5!(C)$) node[right] {$T$};
        \draw[-latex, ultra thick, blue] ($(A)!0.5!(B)$) -- ++(0, -1.5) node[below] {$W$};

\pic [draw, angle radius=1.2cm, angle eccentricity=.6, "$\theta$"] {angle = B--A--C};
    \end{tikzpicture}
\end{center}

\begin{questionparts}
    \item $\,$
    \begin{align*}
        \overset{\curvearrowright}{A}:&& W \cos \theta \cdot a + kW \cos \theta \cdot 2a - T \cos \theta \sin \theta \cdot 2a - T \sin \theta \cos \theta \cdot 2a &= 0 \\
        && (2k+1) \cos \theta W &= T \cos \theta \cdot 4 \sin \theta \\
        \Rightarrow && T &= \frac{2k+1}{4 \sin \theta} W \\
        \Rightarrow && \text{breaks if }\quad \quad 2k+1 &> 4 \sin \theta
    \end{align*}

\item $\,$ \begin{align*}
        \text{N2}(\uparrow): && R - W - kW - T \sin \theta &= 0 \\
        \Rightarrow && R &= (k+1)W - T \sin \theta \\
        &&&= (k+1)W - \frac{2k+1}{4} W \\
        &&&= \frac{2k+3}{4}W \\
        \text{N2}(\leftarrow): && F_A - T \cos \theta &= 0 \\
        \Rightarrow && F_A &= \frac{2k+1}{4 }\cot \theta \\
        \Rightarrow && \text{slips if }\quad \quad\quad \quad\quad \quad F_A &> \mu R \\
        \Rightarrow && \text{slips if }\quad \quad \frac{2k+1}{4 }\cot \theta &> \mu \frac{2k+3}{4}W \\
        \Rightarrow && 2k+1 &> (2k+3) \mu \tan \theta
    \end{align*}

\item The condition for breaking is $k > 2\sin \theta -\frac12$. The condition for slipping is equivalent to:
    \begin{align*}
        && 2k+1 &> (2k+3) \mu \tan \theta \\
        \Leftrightarrow && 2k(1- \mu \tan \theta) &> 3 \mu \tan \theta-1 \\
        \Leftrightarrow && k &> \frac{3 \mu \tan \theta-1}{2(1- \mu \tan \theta)} 
    \end{align*}
    Therefore we will slip first if:
    \begin{align*}
        && \frac{3 \mu \tan \theta-1}{2(1- \mu \tan \theta)} &< 2 \sin \theta - \frac12 \\
        \Leftrightarrow && 3 \mu \tan \theta-1 &< 4 \sin \theta (1- \mu \tan \theta) - (1- \mu \tan \theta) \\
        &&&=4 \sin \theta - 1 + \mu \tan \theta (1-4 \sin \theta) \\
        \Leftrightarrow && 3 \mu \tan \theta &< 4 \sin \theta + \mu \tan \theta (1- 4 \sin \theta) \\
        \Leftrightarrow && \mu \tan \theta(3-1+4\sin \theta) &< 4 \sin \theta  \\
         \Leftrightarrow && \mu &< \frac{2 \cos \theta}{1+2 \sin \theta}
        \end{align*}
    
\end{questionparts}

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