Ladder against smooth wall in limiting equilibrium

3 A uniform ladder of length 4 metres and mass 20 kg rests in equilibrium with its foot, \(A\), on a rough horizontal floor and its top leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall and the angle between the ladder and the floor is \(60 ^ { \circ }\). A man of mass 80 kg is standing at point \(C\) on the ladder. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the floor is 0.4 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{1bc18163-b20e-4dc6-bd35-496efec8dc73-3_567_448_708_788}

1. Draw a diagram to show the forces acting on the ladder.
2. Show that the magnitude of the frictional force between the ladder and the ground is 392 N .
3. Find the distance \(A C\).

3 A uniform ladder, of length 6 metres and mass 22 kg , rests with its foot, \(A\), on a rough horizontal floor and its top, \(B\), leaning against a smooth vertical wall. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the floor is \(\theta\). A man, of mass 90 kg , is standing at point \(C\) on the ladder so that the distance \(A C\) is 5 metres. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the horizontal floor is 0.6 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-3_707_702_742_646}

1. Show that the magnitude of the frictional force between the ladder and the horizontal floor is 659 N , correct to three significant figures.
2. Find the angle \(\theta\).

9. Figure 1 A uniform ladder \(A B\), of length \(2 a\) and weight \(W\), has its end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
The end \(B\) of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
A builder of weight \(7 W\) stands at the top of the ladder.
To stop the ladder from slipping, the builder's assistant applies a horizontal force of magnitude \(P\) to the ladder at \(A\), towards the wall.
The force acts in a direction which is perpendicular to the wall.
The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 5 } { 2 }\).
The builder is modelled as a particle and the ladder is modelled as a uniform rod.

1. Show that the reaction of the wall on the ladder at \(B\) has magnitude \(3 W\).
2. Find, in terms of \(W\), the range of possible values of \(P\) for which the ladder remains in equilibrium. Often in practice, the builder's assistant will simply stand on the bottom of the ladder.
3. Explain briefly how this helps to stop the ladder from slipping.

A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]