Rod on smooth peg or cylinder

6. \begin{figure}[h] \end{figure} A uniform rod, \(A B\), of weight \(W\) and length \(8 a\), rests in equilibrium with the end \(A\) on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is \(C\), where \(A C = 7 a\), as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle \(\alpha\) with the horizontal .

1. Show that the normal reaction of the ground on the rod at \(A\) has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is \(\mu\) and that \(\cos \alpha = \frac { 3 } { \sqrt { 10 } }\)
2. find the range of possible values of \(\mu\).
   \section*{\textbackslash section*\{Question 6 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}

5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
3. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\) (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
      1. Explain why \(A O = 13 a\)

6. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of weight 40 N and length 7 m , rests with end \(A\) on rough horizontal ground. The beam rests on a smooth horizontal peg at \(C\), with \(A C = 5 \mathrm {~m}\), as shown in Figure 5.
The beam is inclined at an angle \(\theta\) to the ground, where \(\sin \theta = \frac { 3 } { 5 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg.
The normal reaction between the beam and the peg at \(C\) has magnitude \(P\) newtons.
Using the model,

1. show that \(P = 22.4\)
2. find the magnitude of the resultant force acting on the beam at \(A\).

7. \begin{figure}[h] \end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).

5. \begin{figure}[h] \end{figure} A plank rests in equilibrium against a fixed horizontal pole. The plank is modelled as a uniform rod \(A B\) and the pole as a smooth horizontal peg perpendicular to the vertical plane containing \(A B\). The rod has length \(3 a\) and weight \(W\) and rests on the peg at \(C\), where \(A C = 2 a\). The end \(A\) of the rod rests on rough horizontal ground and \(A B\) makes an angle \(\alpha\) with the ground, as shown in Figure 2.

1. Show that the normal reaction on the rod at \(A\) is \(\frac { 1 } { 4 } \left( 4 - 3 \cos ^ { 2 } \alpha \right) W\). Given that the rod is in limiting equilibrium and that \(\cos \alpha = \frac { 2 } { 3 }\),
2. find the coefficient of friction between the rod and the ground.

3 \includegraphics[max width=\textwidth, alt={}, center]{5addd79d-d502-455c-936f-27005483164e-3_483_787_260_641} A uniform rod \(A B\) of mass 10 kg and length 2.4 m rests with \(A\) on rough horizontal ground. The rod makes an angle of \(60 ^ { \circ }\) with the horizontal and is supported by a fixed smooth peg \(P\). The distance \(A P\) is 1.6 m (see diagram).

1. Calculate the magnitude of the force exerted by the peg on the rod.
2. Find the least value of the coefficient of friction between the rod and the ground needed to maintain equilibrium.

4. \begin{figure}[h] \end{figure} A ramp, \(A B\), of length 8 m and mass 20 kg , rests in equilibrium with the end \(A\) on rough horizontal ground. The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as \(A\). The point of contact between the ramp and the drum is \(C\), where \(A C = 5 \mathrm {~m}\), as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) The ramp is modelled as a uniform rod.

1. Explain why the reaction from the drum on the ramp at point \(C\) acts in a direction which is perpendicular to the ramp.
2. Find the magnitude of the resultant force acting on the ramp at \(A\). The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of mass of the ramp is assumed to be closer to \(A\) than to \(B\),
3. state how this would affect the magnitude of the normal reaction between the ramp and the drum at \(C\).

7 \begin{figure}[h] \end{figure} A uniform solid cone of height 0.8 m and semi-vertical angle \(60 ^ { \circ }\) lies with its curved surface on a horizontal plane. The point \(P\) on the circumference of the base is in contact with the plane. \(V\) is the vertex of the cone and \(P Q\) is a diameter of its base. The weight of the cone is 550 N . A force of magnitude \(F \mathrm {~N}\) and line of action \(P Q\) is applied to the base of the cone (see Fig. 1). The cone topples about \(V\) without sliding.

1. Calculate the least possible value of \(F\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65c47bd2-eace-4fec-b1e6-a0c904c4ec3f-4_528_1143_1302_500} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The force of magnitude \(F \mathrm {~N}\) is removed and an increasing force of magnitude \(T \mathrm {~N}\) acting upwards in the vertical plane of symmetry of the cone and perpendicular to \(P Q\) is applied to the cone at \(Q\) (see Fig. 2). The coefficient of friction between the cone and the horizontal plane is \(\mu\).
2. Given that the cone slides before it topples about \(P\), calculate the greatest possible value for \(\mu\).

1 A thin uniform rigid rod JK of length 1.2 m and weight 30 N is resting on a rough circular cylinder which is fixed to a floor. The axis of symmetry of the cylinder is horizontal and at all times the rod is perpendicular to this axis. Initially, the rod is horizontal and its point of contact with the cylinder is 0.4 m from K . It is held in equilibrium by resting on a small peg at J . This situation is shown in Fig. 1.1.

1. Calculate the force exerted by the peg on the rod and also the force exerted by the cylinder on the rod. A small object of weight \(W \mathrm {~N}\) is attached to the rod at K .
2. Find the greatest value of \(W\) for which the rod maintains its contact at J . The object at K is removed. Fig. 1.2 shows the rod resting on the cylinder with its end J on the floor, which is smooth and horizontal. The point of contact of the rod with the cylinder is 0.3 m from K. Fig. 1.2 also shows the normal reaction, \(S \mathrm {~N}\), of the floor on the rod, the normal reaction, \(R \mathrm {~N}\), of the cylinder on the rod and the frictional force \(F \mathrm {~N}\) between the cylinder and the rod. Suppose the rod is in equilibrium at an angle of \(\theta ^ { \circ }\) to the horizontal, where \(\theta < 90\).
   [diagram]
3. Find \(S\). Find also expressions in terms of \(\theta\) for \(R\) and \(F\). The coefficient of friction between the cylinder and the rod is \(\mu\).
4. Determine a relationship between \(\mu\) and \(\theta\).

4 Fig. 4 shows a thin rigid non-uniform rod PQ of length 0.5 m . End P rests on a rough circular peg. A force of \(T \mathrm {~N}\) acts at the end Q at \(60 ^ { \circ }\) to QP . The weight of the rod is 40 N and its centre of mass is 0.3 m from P . \begin{figure}[h] \end{figure} The rod does not slip on the peg and is in equilibrium with PQ horizontal.

1. Show that the vertical component of \(T\) is 24 N .
2. \(F\) is the contact force at P between the rod and the peg. Find
   • the vertical component of \(F\),
   • the horizontal component of \(F\).
   • Given that the rod is about to slip on the peg, find the coefficient of friction between the rod and the peg.

1. The diagram shows a uniform rod \(A B\), of length 8 m and mass 23 kg , in limiting equilibrium with its end \(A\) on rough horizontal ground and point \(C\) resting against a smooth fixed cylinder. The rod is inclined at an angle of \(30 ^ { \circ }\) to the ground. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is \(\frac { 2 } { 3 }\).

1. Calculate the magnitude of the normal reaction at \(C\) and the magnitude of the normal reaction to the ground at \(A\).
2. Find the length \(A C\).
3. Suppose instead that the rod is non-uniform with its centre of mass closer to \(A\) than to \(B\). Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.

9 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-08_302_992_260_539} The diagram shows a plank of wood \(A B\), of mass 10 kg and length 6 m , resting with its end \(A\) on rough horizontal ground and its end \(B\) in contact with a fixed cylindrical oil drum. The plank is in a vertical plane perpendicular to the axis of the drum, and the line \(A B\) is a tangent to the circular cross-section of the drum, with the point of contact at \(B\). The plank is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The plank is modelled as a uniform rod and the oil drum is modelled as being smooth.

1. Find, in terms of \(g\), the normal contact force between the drum and the plank.
2. Given that the plank is in limiting equilibrium, find the coefficient of friction between the plank and the ground.

6. \begin{figure}[h] \end{figure} A uniform beam \(A B\), of weight \(5 W\) and length \(12 a\), rests with end \(A\) on rough horizontal ground.
A package of weight \(W\) is attached to the beam at \(B\).
The beam rests in equilibrium on a smooth horizontal peg at \(C\), with \(A C = 9 a\), as shown in Figure 5.
The beam is inclined at an angle \(\theta\) to the ground, where \(\tan \theta = \frac { 5 } { 12 }\) The beam is modelled as a rod that lies in a vertical plane perpendicular to the peg. The package is modelled as a particle. The normal reaction between the beam and the peg at \(C\) has magnitude \(k W\) Using the model,

1. show that \(k = \frac { 56 } { 13 }\) The coefficient of friction between \(A\) and the ground is \(\mu\) Given that the beam is resting in limiting equilibrium,
2. find the value of \(\mu\)

\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{5}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]

\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{3}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]

\includegraphics{figure_4} A hemispherical bowl of radius \(r\) is fixed with its rim horizontal. A thin uniform rod rests in equilibrium on the rim of the bowl with one end resting on the inner surface of the bowl at \(A\), as shown in the diagram. The rod has length \(2a\) and weight \(W\). The point of contact between the rod and the rim is \(B\), and the rim has centre \(C\). The rod is in a vertical plane containing \(C\). The rod is inclined at \(\theta\) to the horizontal and the line \(AC\) is inclined at \(2\theta\) to the horizontal. The contacts at \(A\) and \(B\) are smooth. In any order, show that

1. the contact force acting on the rod at \(A\) has magnitude \(W\tan\theta\),
2. the contact force acting on the rod at \(B\) has magnitude \(\frac{W\cos 2\theta}{\cos\theta}\),
3. \(2r\cos 2\theta = a\cos\theta\).

[9]

\includegraphics{figure_4} A uniform rod \(AB\) of length \(2a\) and weight \(W\) rests against a smooth horizontal peg at a point \(C\) on the rod, where \(AC = x\). The lower end \(A\) of the rod rests on rough horizontal ground. The rod is in equilibrium inclined at an angle of \(45°\) to the horizontal (see diagram). The coefficient of friction between the rod and the ground is \(\mu\). The rod is about to slip at \(A\). \begin{enumerate}[label=(\roman*)] \item Find an expression for \(x\) in terms of \(a\) and \(\mu\). [5] \item Hence show that \(\mu \geqslant \frac{1}{3}\). [2] \item Given that \(x = \frac{5}{3}a\), find the value of \(\mu\) and the magnitude of the resultant force on the rod at \(A\). [4] \end{enumerate]

\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).

1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
2. Given that the lamina is about to slip, find the value of \(\mu\). [3]

\includegraphics{figure_5} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).

1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]

\includegraphics{figure_3} A uniform semicircular lamina of radius \(0.7\) m and weight \(14\) N has diameter \(AB\). The lamina is in a vertical plane with \(A\) freely pivoted at a fixed point. The straight edge \(AB\) rests against a small smooth peg \(P\) above the level of \(A\). The angle between \(AB\) and the horizontal is \(30°\) and \(AP = 0.9\) m (see diagram).

1. Show that the magnitude of the force exerted by the peg on the lamina is \(7.12\) N, correct to 3 significant figures. [4]
2. Find the angle with the horizontal of the force exerted by the pivot on the lamina at \(A\). [3]

\includegraphics{figure_2} Figure 2 A uniform rod \(AB\) has length \(4a\) and weight \(W\). A particle of weight \(kW\), \(k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(AC = 3a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{1}{3}\). The peg is perpendicular to the vertical plane containing \(AB\).

1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod. [1]
2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac{\sqrt{10}}{5}W(1 + 2k)$$ [4]

The coefficient of friction between the rod and the ground is \(\frac{3}{4}\).

1. Show that for the rod to remain in equilibrium \(k \leq \frac{2}{11}\). [7]

Figure 2 \includegraphics{figure_2} A wooden plank \(AB\) has mass \(4m\) and length \(4a\). The end \(A\) of the plank lies on rough horizontal ground. A small stone of mass \(m\) is attached to the plank at \(B\). The plank is resting on a small smooth horizontal peg \(C\), where \(BC = a\), as shown in Figure 2. The plank is in equilibrium making an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between the plank and the ground is \(\mu\). The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle. Show that

1. the reaction of the peg on the plank has magnitude \(\frac{16}{5}mg\), [3]

1. \(\mu \geq \frac{48}{61}\). [6]

1. State how you have used the information that the peg is smooth. [1]

A rough circular cylinder of radius \(4a\) is fixed to a rough horizontal plane with its axis horizontal. A uniform rod \(AB\), of weight \(W\) and length \(6a\sqrt{3}\), rests with its lower end \(A\) on the plane and a point \(C\) of the rod against the cylinder. The vertical plane through the rod is perpendicular to the axis of the cylinder. The rod is inclined at 60° to the horizontal, as shown in Figure 1. \includegraphics{figure_1}

1. Show that \(AC = 4a\sqrt{3}\) [2]

The coefficient of friction between the rod and the cylinder is \(\frac{\sqrt{3}}{3}\) and the coefficient of friction between the rod and the plane is \(\mu\). Given that friction is limiting at both \(A\) and \(C\),

1. find the value of \(\mu\). [9]

\includegraphics{figure_3} A uniform rod \(AB\) has weight 30 N and length 3 m. The rod rests in equilibrium on a rough horizontal peg \(P\) with its end \(A\) on smooth horizontal ground. The rod is in a vertical plane perpendicular to the peg. The rod is inclined at 15° to the ground and the point of contact between the peg and the rod is 45 cm above the ground, as shown in Figure 3.

1. Show that the normal reaction at \(P\) has magnitude 25 N. [4]
2. Find the magnitude of the force on the rod at \(A\). [4]

The coefficient of friction between the rod and the peg is \(\mu\).

1. Find the range of possible values of \(\mu\). [4]