\(\ \)\vspace{-1.5cm}
\noindent
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\par
The above diagram illustrates a makeshift stepladder, made from two
equal light planks \(AB\) and \(CD\), each of length \(2l\). The plank
\(AB\) is smoothly hinged to the ground at \(A\) and makes an angle
of \(\alpha\) with the horizontal. The other plank \(CD\) has its bottom
end \(C\) resting on the same horizontal ground and makes an angle
\(\beta\) with the horizontal. It is pivoted smoothly to \(B\) at a
point distance \(2x\) from \(C\). The coefficient of friction between
\(CD\) and the ground is \(\mu.\) A painter of mass \(M\) stands on \(CD\),
half between \(C\) and \(B\). Show that, for equilibrium to be possible,
\[
\mu\geqslant\frac{\cot\alpha\cot\beta}{2\cot\alpha+\cot\beta}.
\]
Suppose now that \(B\) coincides with \(D\). Show that, as \(\alpha\)
varies, the maximum distance from \(A\) at which the painter will be
standing is
\[
l\sqrt{\frac{1+81\mu^{2}}{1+9\mu^{2}}}.
\]