3.03u Static equilibrium: on rough surfaces

8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.

1. Draw and label a diagram to show the forces acting on the crate.
2. Express the normal reaction between the surface and the crate in terms of \(T\).
3. Find \(T\).

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 \(\theta\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
The two blocks, \(A\) and \(B\), are modelled as particles.
Given that

• \(\tan \alpha = \frac { 5 } { 12 }\) and \(\tan \beta = \frac { 7 } { 24 }\) and \(\tan \gamma = \frac { 3 } { 4 }\)
• the coefficient of friction, \(\mu\), is the same between each block and its plane
• one of the blocks is on the point of sliding up its plane
• the tension in the string is \(T\)
  1. determine, in terms of \(m\) and \(g\), an expression for \(T\),
  2. draw a diagram showing the forces on block \(A\), clearly labelling each of the forces acting on the block,
  3. determine the value of \(\mu\), giving a justification for your answer. \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_2266_50_312_1978}

10 A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal.
The body is held at rest on the plane by the action of a force of magnitude \(P \mathrm {~N}\).
The force is acting up the plane in a direction parallel to a line of greatest slope of the plane.
The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g \mu \cos \alpha + 5\).
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\).

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\).

A particle of mass \(m\) is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\mu\).
The particle moves up the plane with a constant deceleration of \(\frac { 4 } { 5 } \mathrm {~g}\).

1. Find the value of \(\mu\). The particle comes to rest at the point \(A\) on the plane.
2. Determine whether the particle will remain at \(A\), carefully justifying your answer.

9 Two blocks have square cross sections. One block has mass 9 kg and its cross section has sides of length 20 cm
The other block has mass 1 kg and its cross section has sides of length 4 cm
The blocks are fixed together to form the composite body shown in Figure 1. \begin{figure}[h] \end{figure} 9

1. Find the distance of the centre of mass of the composite body from \(A F\) [0pt] [2 marks]
   Question 9 continues on the next page 9
2. A uniform rod has mass 12 kg and length 1 metre. One end of the rod rests against a smooth vertical wall.
   The other end of the rod rests on the composite body at point \(B\) The composite body is on a horizontal surface.
   The coefficient of friction between the composite body and the horizontal surface is 0.3 The angle between the rod and \(A B\) is \(60 ^ { \circ }\) A particle of mass \(m \mathrm {~kg}\) is fixed to the rod at a distance of 75 cm from \(B\) The rod, particle and composite body are shown in Figure 2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-14_939_1020_1133_511}
   \end{figure} 9 (b) (i) Write down the magnitude of the vertical reaction force acting on the rod at \(B\) in terms of \(m\) and \(g\) [0pt] [1 mark] 9 (b) (ii) Show that the magnitude of the horizontal reaction force acting on the rod at \(B\) is $$\frac { g ( 6 + 0.75 m ) } { \sqrt { 3 } }$$ 9 (b) (iii) Find the maximum value of \(m\) for which the composite body does not slide or topple. Fully justify your answer.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass 5 kg lies on the surface of a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The particle is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The horizontal force acts in a vertical plane containing a line of greatest slope of the inclined plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the smallest possible value of \(H\). The horizontal force is now removed, and \(P\) starts to slide down the slope.
   In the first \(T\) seconds after \(P\) is released from rest, \(P\) slides 1.5 m down the slope.
2. Find the value of \(T\).

9 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_430_565_260_790} The diagram shows a block of wood, weighing 100 N , at rest on a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 . A force of \(P \mathrm {~N}\) acts on the block up the slope.

1. Find the maximum possible value of the friction acting on the block.
2. Given that the block is on the point of moving up the slope, find \(P\).
3. Given that the block is on the point of moving down the slope, find \(P\).

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

8 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]{a19fab61-da1c-4803-9dbc-38d618a0c58e-4_657_655_1128_705}

1. Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
2. Explain why it is impossible for the rod to be in equilibrium with one end on smooth horizontal ground and the other against a rough vertical wall.

10 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
3. Show that \(0.5 < \tan \alpha \leqslant 1\).
4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.

10 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).

1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
3. Show that \(0.5 < \tan \alpha \leqslant 1\).
4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.

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.

11 \includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-4_280_1002_1722_568} A non-uniform \(\operatorname { rod } A B\) of mass 1.6 kg and length 1.25 m has its centre of mass at \(G\) where \(A G = 0.4 \mathrm {~m}\). The rod rests on a rough horizontal table. A force \(P \mathrm {~N}\) is applied at \(B\), acting at an angle \(\alpha\) above the horizontal, such that the rod is in equilibrium but about to rotate about \(A\) (see diagram).

1. Assume that the rod is in contact with the table only at \(A\). By taking moments about \(A\), show that \(P \sin \alpha = 5.12\).
2. The coefficient of friction between the rod and the table is \(\frac { 6 } { 17 }\). Show that \(P \leqslant 6.4\).

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]{f4acd242-eb78-4124-bfa2-fdecaa188690-5_657_659_392_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.

\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 uniform rod \(AB\) of length \(4a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac{5}{2}a\) has one end attached to the point \(C\) on the rod, where \(AC = \frac{3}{2}a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the rod \(AB\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\). [10]

\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]

The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}

1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
2. Find the acceleration of the ring when \(T = 3\). [3]

\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

A particle \(P\) of mass 0.4 kg is in limiting equilibrium on a plane inclined at \(30°\) to the horizontal.

1. Show that the coefficient of friction between the particle and the plane is \(\frac{1}{3}\sqrt{3}\). [3]

A force of magnitude 7.2 N is now applied to \(P\) directly up a line of greatest slope of the plane.

1. Given that \(P\) starts from rest, find the time that it takes for \(P\) to move 1 m up the plane. [4]