Ladder on smooth wall and rough ground

2. \begin{figure}[h] \end{figure} Figure 1 shows a ladder \(A B\), of mass 25 kg and length 4 m , resting in equilibrium with one end \(A\) on rough horizontal ground and the other end \(B\) against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is \(\frac { 11 } { 25 }\). The ladder makes an angle \(\beta\) with the ground. When Reece, who has mass 75 kg , stands at the point \(C\) on the ladder, where \(A C = 2.8 \mathrm {~m}\), the ladder is on the point of slipping. The ladder is modelled as a uniform rod and Reece is modelled as a particle.

1. Find the magnitude of the frictional force of the ground on the ladder.
2. Find, to the nearest degree, the value of \(\beta\).
3. State how you have used the modelling assumption that Reece is a particle.

6. A uniform ladder \(A B\), of mass \(m\) and length \(2 a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is 0.6 . The other end \(B\) of the ladder rests against a smooth vertical wall. A builder of mass 10 m stands at the top of the ladder. To prevent the ladder from slipping, the builder's friend pushes the bottom of the ladder horizontally towards the wall with a force of magnitude \(P\). This force acts in a direction 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, where \(\tan \alpha = \frac { 3 } { 2 }\).

1. Show that the reaction of the wall on the ladder has magnitude 7 mg .
2. Find, in terms of \(m\) and \(g\), the range of values of \(P\) for which the ladder remains in equilibrium.

4. \section*{Figure 1} A uniform ladder, of mass \(m\) and length \(2 a\), has one end on rough horizontal ground. The other end rests against a smooth vertical wall. A man of mass \(3 m\) stands at the top of the ladder and the ladder is in equilibrium. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\), and the ladder makes an angle \(\alpha\) with the vertical, as shown in Fig. 1. The ladder is in a vertical plane perpendicular to the wall. Show that \(\tan \alpha \leq \frac { 2 } { 7 }\).

3 A uniform ladder \(P Q\), of length 8 metres and mass 28 kg , rests in equilibrium with its foot, \(P\), on a rough horizontal floor and its top, \(Q\), 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 \(69 ^ { \circ }\). A man, of mass 72 kg , is standing at the point \(C\) on the ladder so that the distance \(P C\) is 6 metres. The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-3_679_679_685_678}

1. Draw a diagram to show the forces acting on the ladder.
2. With the man standing at the point \(C\), the ladder is on the point of slipping.
   1. Show that the magnitude of the reaction between the ladder and the vertical wall is 256 N , correct to three significant figures.
   2. Find the coefficient of friction between the ladder and the horizontal floor.

6 When a car, of mass 1200 kg , travels at a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it experiences a resistance force of magnitude \(30 v\) newtons. The car has a maximum constant speed of \(48 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a straight horizontal road.

1. Show that the maximum power of the car is 69120 watts.
2. The car is travelling along a straight horizontal road. Find the maximum possible acceleration of the car when it is travelling at a speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The car starts to descend a hill on a straight road which is inclined at an angle of \(3 ^ { \circ }\) to the horizontal. Find the maximum possible constant speed of the car as it travels on this road down the hill. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-13_2484_1709_223_153} \(7 \quad\) A uniform rod \(A B\), of length 4 m and mass 6 kg , rests in equilibrium with one end, \(A\), on smooth horizontal ground. The rod rests on a rough horizontal peg at the point \(C\), where \(A C\) is 3 m . The rod is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-14_422_984_447_529}

3. \begin{figure}[h] \end{figure} Figure 1 shows a uniform ladder of mass 15 kg and length 8 m which rests against a smooth vertical wall at \(A\) with its lower end on rough horizontal ground at \(B\). The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 3 }\) and the ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 2\). A man of mass 75 kg ascends the ladder until he reaches a point \(P\). The ladder is then on the point of slipping.

1. Write down suitable models for
   1. the ladder,
   2. the man.
2. Find the distance \(A P\).

6. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-7_606_506_365_781} A uniform ladder \(A B\), of mass 10 kg and length 5 m , rests with one end \(A\) against a smooth vertical wall and the other end \(B\) on rough horizontal ground. The ladder is inclined at an angle \(\theta\) to the horizontal. A woman of mass 75 kg stands on the ladder so that her weight acts at a distance \(x \mathrm {~m}\) from \(B\).

1. Show that the frictional force, \(F \mathrm {~N}\), between the ladder and the horizontal ground is given by $$F = 5 g \cot \theta ( 1 + 3 x ) .$$ For safety reasons, it is recommended that \(\theta\) is chosen such that the ratio \(C B : C A\) is \(1 : 4\).
2. Determine the least value of the coefficient of friction such that the ladder will not slip however high the woman climbs.
3. State one modelling assumption that you have made in your solution.

10 A uniform ladder \(A B\) of length 5 m and mass 8 kg is placed at an angle \(\theta\) to the horizontal, with \(A\) on rough horizontal ground and \(B\) against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .

1. By taking moments, find the smallest value of \(\theta\) for which the ladder is in equilibrium.
2. A man of mass 75 kg stands on the ladder when \(\theta = 60 ^ { \circ }\). Find the greatest distance from \(A\) that he can stand without the ladder slipping.

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° 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 \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}

1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
2. 1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
   2. 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. [4]