Rod or block on rough surface in limiting equilibrium (no wall)

5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).

1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\),
2. the least possible value of the coefficient of friction at \(B\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass 5 kg and length 8 m , has its end \(B\) resting on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), by a rope attached to the rod at \(C\). The distance \(A C = 1 \mathrm {~m}\). The rope is in the same vertical plane as the rod. The angle between the rope and the rod is \(\beta\) and the tension in the rope is \(T\) newtons, as shown in Figure 3. The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\). The vertical component of the force exerted on the rod at \(B\) by the ground is \(R\) newtons.

1. Find the value of \(R\).
2. Find the size of angle \(\beta\).

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), rests with its end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a light string attached to the rod at \(B\), as shown in Figure 3. The string is perpendicular to the rod and lies in the same vertical plane as the rod. The coefficient of friction between the ground and the rod is \(\mu\).
Show that \(\mu = \frac { \cos \theta \sin \theta } { 2 - \cos ^ { 2 } \theta }\)

3 \includegraphics[max width=\textwidth, alt={}, center]{f334f6e4-2a60-4647-8b37-e48937c85747-2_465_757_1146_694} Two identical uniform rods, \(A B\) and \(B C\), are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in limiting equilibrium in a vertical plane, with \(C\) resting on a rough horizontal surface. \(A B\) is horizontal, and \(B C\) is inclined at \(60 ^ { \circ }\) to the horizontal. The weight of each rod is 160 N (see diagram).

1. By taking moments for \(A B\) about \(A\), find the vertical component of the force on \(A B\) at \(B\). Hence or otherwise find the magnitude of the vertical component of the contact force on \(B C\) at \(C\). [3]
2. Calculate the magnitude of the frictional force on \(B C\) at \(C\) and state its direction.
3. Calculate the value of the coefficient of friction at \(C\).

4 Fig. 4 shows a uniform beam of length \(2 a\) and weight \(W\) leaning against a block of weight \(2 W\) which is on a rough horizontal plane. The beam is freely hinged to the plane at O and makes an angle \(\theta\) with the horizontal. The contact between the beam and the block is smooth. The beam and block are in equilibrium, and it may be assumed that the block does not topple. \begin{figure}[h] \end{figure} Let

• \(S\) be the normal contact force between the beam and the block,
• \(R\) be the normal contact force between the plane and the block,
• \(F\) be the frictional force between the plane and the block.

Partially complete force diagrams showing the beam and the block separately are given in the Printed Answer Booklet.

1. Add the forces listed above to these diagrams. It is given that \(\theta = 30 ^ { \circ }\).
2. Determine the minimum possible value of the coefficient of friction between the block and the plane.
3. In each case explain, with justification, how your answer to part (b) would change (assuming the rest of the system remained unchanged) if
   1. \(\theta < 30 ^ { \circ }\),
   2. the contact between the beam and the block were rough.

5 A uniform rod AB , of mass \(3 m\) and length \(2 a\), rests with the end A on a rough horizontal surface. A small object of mass \(m\) is attached to the rod at B . The rod is maintained in equilibrium at an angle of \(60 ^ { \circ }\) to the horizontal by a force acting at an angle of \(\theta\) to the vertical at a point C , where the distance \(\mathrm { AC } = \frac { 6 } { 5 } a\). The force acting at C is in the same vertical plane as the rod (see Fig. 5). \begin{figure}[h] \end{figure}

1. On the copy of Fig. 5 in the Printed Answer Booklet, mark all the forces acting on the rod. [2]
2. Show that the magnitude of the force acting at C can be expressed as \(\frac { 25 m g } { 6 ( \cos \theta + \sqrt { 3 } \sin \theta ) }\).
3. Given that the rod is in limiting equilibrium and the coefficient of friction between the rod and the surface is \(\frac { 3 } { 4 }\), determine the value of \(\theta\).

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.

\includegraphics{figure_2} A uniform rod \(AB\) of weight \(W\) rests in equilibrium with \(A\) in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude \(P\) acting at \(B\) in this vertical plane. The rod makes an angle of \(60°\) with the wall, and the force makes an angle of \(30°\) with the rod (see diagram). Find the value of \(P\). [3] Find also the set of possible values of the coefficient of friction between the rod and the wall. [4]

\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{1}{2}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]

\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{14}{5}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]

\includegraphics{figure_4} A uniform rod \(AB\), of length \(l\) and mass \(m\), rests in equilibrium with its lower end \(A\) on a rough horizontal floor and the end \(B\) against a smooth vertical wall. The rod is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{4}{3}\), and is in a vertical plane perpendicular to the wall. The rod is supported by a light spring \(CD\) which is in compression in a vertical line with its lower end \(D\) fixed on the floor. The upper end \(C\) is attached to the rod at a distance \(\frac{4l}{5}\) from \(B\) (see diagram). The coefficient of friction at \(A\) between the rod and the floor is \(\frac{1}{2}\) and the system is in limiting equilibrium.

1. Show that the normal reaction of the floor at \(A\) has magnitude \(\frac{1}{2}mg\) and find the force in the spring. [7]
2. Given that the modulus of elasticity of the spring is \(2mg\), find the natural length of the spring. [4]

\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\), [3]
2. the least possible value of the coefficient of friction at \(B\). [4]

\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\), of mass \(m\) and length \(2a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(AC = \frac{2}{3}a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(BD = 2a\).

1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac{1}{2}mg \sin \theta\) [3]
2. Find the magnitude of the normal reaction on the rod at \(B\). [5]

The rod is in limiting equilibrium when \(\tan \theta = \frac{4}{3}\).

1. Find the coefficient of friction between the rod and the ground. [3]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). The cable makes an angle \(β\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is \(\frac{1}{3}\). The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{3}{4}W\). [6]

1. Find the value of \(β\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

END

\includegraphics{figure_2} A uniform rod \(AB\), of mass \(20\) kg and length \(4\) m, rests with one end \(A\) on rough horizontal ground. The rod is held in limiting equilibrium at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\), by a force acting at \(B\), as shown in Figure 2. The line of action of this force lies in the vertical plane which contains the rod. The coefficient of friction between the ground and the rod is \(0.5\). Find the magnitude of the normal reaction of the ground on the rod at \(A\). [7]

\includegraphics{figure_3} A straight log \(AB\) has weight \(W\) and length \(2a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{5}{12}\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6. The log is modelled as a uniform rod and the cable as light.

1. Show that the normal reaction on the log at \(A\) is \(\frac{5}{8}W\). [6]
2. Find the value of \(\beta\). [6]

The tension in the cable is \(kW\).

1. Find the value of \(k\). [2]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(BC\) are of equal length and each has weight \(100\) N. The rods are freely jointed to each other at \(B\), and \(A\) is freely jointed to a fixed point. The rods are in equilibrium in a vertical plane with \(AB\) horizontal and \(C\) resting on a rough horizontal surface. \(C\) is vertically below the mid-point of \(AB\) (see diagram).

1. By taking moments about \(A\) for \(AB\), find the vertical component of the force on \(AB\) at \(B\). Hence find the vertical component of the contact force on \(BC\) at \(C\). [3]
2. Calculate the magnitude of the frictional force on \(BC\) at \(C\) and state its direction. [4]

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), of lengths \(3\) m and \(4\) m respectively, have weights \(300\) N and \(400\) N respectively. The rods are freely jointed at \(A\). The mid-points of the rods are joined by a light inextensible string. The rods are in equilibrium in a vertical plane with the string taut and \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is \(2.4\) m above the surface (see diagram).

1. Show that the force exerted by the surface on \(AB\) is \(374\) N and find the force exerted by the surface on \(AC\). [4]
2. Find the tension in the string. [3]
3. Find the horizontal and vertical components of the force exerted on \(AB\) at \(A\) and state their directions. [3]

\includegraphics{figure_6} Two uniform rods \(AB\) and \(AC\) are freely jointed at \(A\). Rod \(AB\) is of length \(2l\) and weight \(W\); rod \(AC\) is of length \(4l\) and weight \(2W\). The rods rest in equilibrium in a vertical plane on two rough horizontal steps, so that \(AB\) makes an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac{3}{5}\), and \(AC\) makes an angle of \(\varphi\) with the horizontal, where \(\sin \varphi = \frac{1}{5}\) (see diagram). The force of the step acting on \(AB\) at \(B\) has vertical component \(R\) and horizontal component \(F\).

1. By taking moments about \(A\) for the rod \(AB\), find an equation relating \(W\), \(R\) and \(F\). [3]
2. Show that \(R = \frac{75}{68}W\), and find the vertical component of the force acting on \(AC\) at \(C\). [6]
3. The coefficient of friction at \(B\) is equal to that at \(C\). Given that one of the rods is on the point of slipping, explain which rod this must be, and find the coefficient of friction. [4]

\includegraphics{figure_11} A uniform rod \(AB\) of mass 4 kg and length 3 m rests in a vertical plane with \(A\) on rough horizontal ground. A particle of mass 1 kg is attached to the rod at \(B\). The rod makes an angle of \(60°\) with the horizontal and is held in limiting equilibrium by a light inextensible string \(CD\). \(D\) is a fixed point vertically above \(A\) and \(CD\) makes an angle of \(60°\) with the vertical. The distance \(AC\) is \(x\) m (see diagram).

1. Find, in terms of \(g\) and \(x\), the tension in the string. [3]

The coefficient of friction between the rod and the ground is \(\frac{9\sqrt{3}}{35}\).

1. Determine the value of \(x\). [4]

\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).

1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]