7 A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of \(20 ^ { \circ }\) with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x \mathrm {~m}\) from the lower end of the ladder.
The system is in equilibrium.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{76875226-2e6c-4571-9318-ecce51ba8b9f-06_803_936_607_577}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{figure}
- Show that the frictional force between the ladder and the horizontal surface is \(F N\), where \(F = 90 ( 1 + x ) \tan 20 ^ { \circ }\).
- (A) State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases.
(B) Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping.