8 A ladder has length 4 metres and mass 20 kg The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall. In this position the ladder is on the point of slipping.
The top of the wall is 1.5 metres above the horizontal surface.
The angle between the ladder and the horizontal surface is \(\alpha\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593} The coefficient of friction between the ladder and the wall is 0.5
The coefficient of friction between the ladder and the ground is also 0.5
Show that $$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$ stating clearly any assumptions you make.
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}