Problem source

A 
painter of weight $kW$ uses a ladder to reach the guttering on the
outside wall of  a house. The wall is vertical and the ground
is horizontal.
The ladder is modelled as a uniform rod of weight $W$ and length $6a$.

The ladder is not long enough, so the painter
stands the ladder on a uniform table. The table has weight $2W$ and a square 
top of side $\frac12 a$ with  a leg of length $a$ at each corner.
The foot of the ladder is at the centre of the table top and the ladder
is inclined at an angle $\arctan 2$ to the horizontal.
The edge of the table nearest the wall is parallel to the wall.
 The coefficient of friction
between the foot of the ladder and the table top is $\frac12$.
The contact between the ladder and the wall is sufficiently smooth
for the effects of friction to be ignored.
\begin{questionparts}
\item Show that, if the legs of the table are fixed to the ground,
 the ladder does not slip on the table
however high the painter stands on the ladder.
\item It is given that $k=9$  and
that the
coefficient of friction between each table leg 
and the ground is $\frac13$. If the legs of the table are not fixed 
to the ground, 
so that  the table can tilt or slip, determine which 
occurs first when
 the painter slowly climbs the ladder.
\end{questionparts}
[Note: $\arctan 2$ is another notation for $\tan^{-1}2$.]