Rod against wall and ground

5. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of mass 15 kg and length 6 m , rests with end \(A\) on rough horizontal ground. The end \(B\) of the beam rests against a rough vertical wall. The beam is inclined at \(75 ^ { \circ }\) to the ground, as shown in Figure 2.
The coefficient of friction between the beam and the wall is 0.2
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall. The beam rests in limiting equilibrium.

1. Find the magnitude of the normal reaction between the beam and the wall at \(B\).
2. Find the value of \(\mu\)

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of mass \(m\) and length \(2 a\), rests in limiting equilibrium with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall.
The vertical plane containing the rod is at right angles to the wall.
The rod is inclined to the wall at an angle \(\alpha\), as shown in Figure 2.
The coefficient of friction between the rod and the ground is \(\frac { 1 } { 3 }\)

1. Show that \(\tan \alpha = \frac { 2 } { 3 }\) With the rod in the same position, a horizontal force of magnitude \(k m g\) is applied to the \(\operatorname { rod }\) at \(A\), towards the wall. The line of action of this force is at right angles to the wall. The rod remains in equilibrium.
2. Find the largest possible value of \(k\).

3. \begin{figure}[h] \end{figure} A non-uniform rod, \(A B\), of mass \(m\) and length 2l, rests in equilibrium with one end \(A\) on a rough horizontal floor and the other end \(B\) against a rough vertical wall. The rod is in a vertical plane perpendicular to the wall and makes an angle of \(60 ^ { \circ }\) with the floor as shown in Figure 1. The coefficient of friction between the rod and the floor is \(\frac { 1 } { 4 }\) and the coefficient of friction between the rod and the wall is \(\frac { 2 } { 3 }\). The rod is on the point of slipping at both ends.

1. Find the magnitude of the vertical component of the force exerted on the rod by the floor. The centre of mass of the rod is at \(G\).
2. Find the distance \(A G\).

1. A ladder \(A B\), of weight \(W\) and length \(2 l\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a rough vertical wall. The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\). The coefficient of friction between the ladder and the ground is \(\mu\). Friction is limiting at both \(A\) and \(B\). The ladder is at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 3 }\). The ladder is modelled as a uniform rod which lies in a vertical plane perpendicular to the wall.

Find the value of \(\mu\).

A uniform rod \(A B\) rests in limiting equilibrium in a vertical plane with the end \(A\) on rough horizontal ground and the end \(B\) against a rough vertical wall that is perpendicular to the plane of the rod. The angle between the rod and the ground is \(\theta\). The coefficient of friction between the rod and the wall is \(\mu\), and the coefficient of friction between the rod and the ground is \(2 \mu\). Show that \(\tan \theta = \frac { 1 - 2 \mu ^ { 2 } } { 4 \mu }\). Given that \(\theta \leqslant 45 ^ { \circ }\), find the set of possible values of \(\mu\).

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).

1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
   [0pt] [3]
   The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
2. Show that \(x = \frac { 4 } { 5 }\).
   The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{0eb3892f-628f-449a-b022-b38170754d89-08_462_693_301_731}
   \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).

1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
   [0pt] [3]
   The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
2. Show that \(x = \frac { 4 } { 5 }\).
   The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{c6c8e0fd-6af2-40c9-9513-6581e26e2aec-08_462_693_301_731}
   \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The angle between the rod and the horizontal is \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\). One end of a light inextensible rope is attached to a point \(C\) on the rod. The other end is attached to a point where the vertical wall and the horizontal ground meet. The rope is taut and perpendicular to the rod. The rope and rod are in a vertical plane perpendicular to the wall.

1. Show that \(A C = \frac { 18 } { 25 } a\).
   The magnitude of the frictional force at \(A\) is equal to one quarter of the magnitude of the normal reaction force at \(A\).
2. Show that the tension in the rope is \(\frac { 1 } { 4 } W\).
3. Find expressions, in terms of \(W\), for the magnitudes of the normal reaction forces at \(A\) and \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{982647bd-8514-40cf-b4ee-674f51df32c5-2_412_380_909_884} A uniform rod \(A B\), of weight 25 N and length 1.6 m , rests in equilibrium in a vertical plane with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall which is inclined at \(80 ^ { \circ }\) to the horizontal. The rod is inclined at \(60 ^ { \circ }\) to the horizontal (see diagram). Calculate the magnitude of the force acting on the rod at \(B\).

3. \begin{figure}[h] \end{figure} A beam \(A B\) has mass \(m\) and length \(2 a\).
The beam rests in equilibrium with \(A\) on rough horizontal ground and with \(B\) against a smooth vertical wall. The beam is inclined to the horizontal at an angle \(\theta\), as shown in Figure 2.
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall. Using the model,

1. show that \(\mu \geqslant \frac { 1 } { 2 } \cot \theta\) A horizontal force of magnitude \(k m g\), where \(k\) is a constant, is now applied to the beam at \(A\). This force acts in a direction that is perpendicular to the wall and towards the wall.
   Given that \(\tan \theta = \frac { 5 } { 4 } , \mu = \frac { 1 } { 2 }\) and the beam is now in limiting equilibrium,
2. use the model to find the value of \(k\).

9 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{c4bbba86-2968-4247-b300-357217cf213b-4_657_647_1923_708} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.