3.04b Equilibrium: zero resultant moment and force

2 Any non-exact answers to this question should be given correct to four significant figures.
A thin, straight beam AB is 2 m long. It has a weight of 600 N and its centre of mass G is 0.8 m from end A. The beam is freely pivoted about a horizontal axis through A. The beam is held horizontally in equilibrium.
Initially this is done by means of a support at B, as shown in Fig.2.1. \begin{figure}[h] \end{figure}

1. Calculate the force on the beam due to the support at B . The support at B is now removed and replaced by a wire attached to the beam 0.3 m from A and inclined at \(50 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam is still horizontal and in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_275_803_1226_671} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate the tension in the wire. The forces acting on the beam at A due to the hinge are a horizontal force \(X \mathrm {~N}\) in the direction AB , and a downward vertical force \(Y \mathrm {~N}\), which have a resultant of magnitude \(R\) at \(\alpha\) to the horizontal.
3. Calculate \(X , Y , R\) and \(\alpha\). The dotted lines in Fig. 2.3 are the lines of action of the tension in the wire and the weight of the beam. These lines of action intersect at P . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{1dd32b82-020e-45ef-8146-892197fd0985-3_460_791_2074_678} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
4. State with a reason the size of the angle GAP.

2 The rigid object shown in Fig. 2.1 is made of thin non-uniform rods. ABC is a straight line; \(\mathrm { BC } , \mathrm { BE }\) and ED form three sides of a rectangle. The centre of mass of the object is at G. The lengths are in centimetres. The weight of the object is 15 N . \begin{figure}[h] \end{figure} Initially, the object is suspended by light vertical strings attached to B and to C and hangs in equilibrium with AC horizontal.

1. Calculate the tensions in each of the strings. In a new situation the strings are removed. The object can rotate freely in a vertical plane about a fixed horizontal axis through A and perpendicular to ABCDE. The object is held in equilibrium with AC horizontal by a force of magnitude \(T \mathrm {~N}\) in the plane ABCDE acting at C at an angle of \(30 ^ { \circ }\) to CA . This situation is shown in Fig. 2.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ea3c0177-bf3b-4475-9ab1-ae628aeb0bf0-3_356_451_1292_808} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. Calculate \(T\). Calculate also the magnitude of the force exerted on the object by the axis at A . The object is now placed on a rough horizontal table and is in equilibrium with ABCDE in a vertical plane and DE in contact with the table. The coefficient of friction between the edge DE and the table is 0.65 . A force of slowly increasing magnitude (starting at 0 N ) is applied at A in the direction AB . Assume that the object remains in a vertical plane.
3. Determine whether the object slips before it tips.

3 Fig. 3.1 shows a rigid, thin, non-uniform 20 cm by 80 cm rectangular panel ABCD of weight 60 N that is in a vertical plane. Its dimensions and the position of its centre of mass, \(G\), are shown in centimetres. The panel is free to rotate about a fixed horizontal axis through A perpendicular to its plane; the panel rests on a small smooth fixed peg at B positioned so that AB is at \(40 ^ { \circ }\) to the horizontal. A horizontal force in the plane of ABCD of magnitude \(P \mathrm {~N}\) acts at D away from the panel. \begin{figure}[h] \end{figure}

1. Show that the clockwise moment of the weight about A is 9.93 Nm , correct to 3 significant figures.
2. Calculate the value of \(P\) for which the panel is on the point of turning about the axis through A .
3. In the situation where \(P = 0\), calculate the vertical component of the force exerted on the panel by the axis through A . The panel is now placed on a line of greatest slope of a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The panel is at all times in a vertical plane. A horizontal force in the plane ABCD of magnitude 200 N acts at D towards the panel. This situation is shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c8f26b7e-1be1-4abf-8fea-6847185fad81-4_497_842_1653_616} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Given that the panel is moving up the plane with acceleration up the plane of \(1.75 \mathrm {~ms} ^ { - 2 }\), calculate the coefficient of friction between the panel and the plane.

3

1. Fig. 3.1 shows a framework in equilibrium in a vertical plane. The framework is made from 3 light rigid rods \(\mathrm { AB } , \mathrm { BC }\) and CA which are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and C . The pin-joint at A is attached to a fixed horizontal beam; the pin-joint at C rests on a smooth horizontal floor. BC is 2 m and angle BAC is \(30 ^ { \circ }\); BC is at right angles to \(\mathrm { AC } . \mathrm { AB }\) is horizontal. Fig. 3.1 also shows the external forces acting on the framework; there is a vertical load of 60 N at B , horizontal and vertical forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) act at A ; the reaction of the floor at C is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-4_323_803_571_580} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure}
   1. Show that \(R = 80\) and find the values of \(X\) and \(Y\).
   2. Using the diagram in your printed answer book, show all the forces acting on the pin-joints, including those internal to the rods.
   3. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CA , stating whether each rod is in tension or thrust (compression). [You may leave your answers in surd form. Your working in this part should correspond to your diagram in part (ii).]
2. Fig 3.2 shows a non-uniform rod of length 6 m and weight 68 N with its centre of mass at G . This rod is free to rotate in a vertical plane about a horizontal axis through B , which is 2 m from A . G is 2 m from B . The rod is held in equilibrium at an angle \(\theta\) to the horizontal by a horizontal force of 102 N acting at C and another force acting at A (not shown in Fig. 3.2). Both of these forces and the force exerted on the rod by the hinge (also not shown in Fig 3.2) act in a vertical plane containing the rod. You are given that \(\sin \theta = \frac { 15 } { 17 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{334b2170-3708-46db-bff7-bcad7d5fab00-4_396_314_1747_852} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
   1. First suppose that the force at A is at right angles to ABC and has magnitude \(P \mathrm {~N}\). Calculate \(P\).
   2. Now instead suppose that the force at A is horizontal and has magnitude \(Q \mathrm {~N}\). Calculate \(Q\).
      Calculate also the magnitude of the force exerted on the rod by the hinge.

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 \(\theta\).

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 \(A P\).

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) of length \(2 a\) and mass 8 kg is smoothly hinged to a vertical wall at \(A\). The rod is held in equilibrium inclined at an angle of \(20 ^ { \circ }\) to the horizontal by a force of magnitude \(F\) newtons acting horizontally at \(B\) which is below the level of \(A\) as shown in Figure 2.

1. Find, correct to 3 significant figures, the value of \(F\).
2. Show that the magnitude of the reaction at the hinge is 133 N , correct to 3 significant figures, and find to the nearest degree the acute angle which the reaction makes with the vertical.

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of mass 2 kg and length \(2 a\). The end \(A\) is attached by a smooth hinge to a fixed point on a vertical wall so that the rod can rotate freely in a vertical plane. A mass of 6 kg is placed at \(B\) and the rod is held in a horizontal position by a light string joining the midpoint of the rod to a point \(C\) on the wall, vertically above \(A\). The string is inclined at an angle of \(60 ^ { \circ }\) to the wall.

1. Show that the tension in the string is \(28 g\).
2. Find the magnitude and direction of the force exerted by the hinge on the rod, giving your answers correct to 3 significant figures.

3. \begin{figure}[h] \end{figure} Figure 1 shows a ladder of mass 20 kg and length 6 m leaning against a rough vertical wall with its lower end on smooth horizontal ground. The ladder is prevented from slipping along the ground by a light rope which is attached to the ladder 2 m from its bottom end and fastened to the wall so that the rope is horizontal and perpendicular to the wall. The ladder is at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 5 } { 2 }\) and the coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\).

1. Draw a diagram showing all the forces acting on the ladder.
2. Show that the magnitude of the tension in the rope is \(5 g\). A man wishes to use the ladder but fears the rope will snap as he climbs the ladder.
3. Suggest, giving a reason for your answer, a more suitable position for the rope.
   (2 marks)

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of length 2 m and mass 6 kg inclined at an angle of \(30 ^ { \circ }\) to the horizontal with \(A\) on smooth horizontal ground and \(B\) supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between \(B\) and the peg is \(\mu\).

1. Find, in terms of \(g\), the magnitude of the reactions at \(A\) and \(B\).
2. Show that \(\mu = \frac { 1 } { \sqrt { 3 } }\).

5 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-3_462_1109_283_569} Two uniform rods \(A B\) and \(B C\) have weights 64 N and 40 N respectively. The rods are freely jointed to each other at \(B\). The rod \(A B\) is freely jointed to a fixed point on horizontal ground at \(A\) and the rod \(B C\) rests against a vertical wall at \(C\). The rod \(B C\) is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod \(B C\) at the point 0.4 m from \(C\). The point \(A\) is 1.2 m below the level of \(B C\) and 3.8 m from the wall (see diagram). The system is in equilibrium.

1. Show that the magnitude of the frictional force at \(C\) is 27 N .
2. Calculate the horizontal and vertical components of the force exerted on \(A B\) at \(B\).
3. Given that friction is limiting at \(C\), find the coefficient of friction between the \(\operatorname { rod } B C\) and the wall.

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 \(C\).

2 \includegraphics[max width=\textwidth, alt={}, center]{7e0f600a-18f1-458b-8549-27fca592b19c-2_515_1065_861_541} Two uniform rods \(A B\) and \(B C\), each of length 2 m , are freely jointed at \(B\). The weights of the rods are \(W \mathrm {~N}\) and 50 N respectively. The end \(A\) of \(A B\) is hinged at a fixed point. The rods \(A B\) and \(B C\) make angles \(\tan ^ { - 1 } \left( \frac { 3 } { 4 } \right)\) and \(\beta\) respectively with the downward vertical, and are held in equilibrium in a vertical plane by a horizontal force of magnitude 75 N acting at \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), show that \(\tan \beta = 3\).
2. Write down the horizontal and vertical components of the force acting on \(A B\) at \(B\).
3. Find the value of \(W\).

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Two uniform rods \(A B\) and \(B C\), of weights 70 N and 110 N respectively, are freely jointed at \(B\). The rods are in equilibrium in a vertical plane with \(A\) and \(C\) at the same horizontal level and \(A C = 2 \mathrm {~m}\). The \(\operatorname { rod } A B\) is freely jointed to a fixed point at \(A\) and the rod \(B C\) is freely jointed to a fixed point at \(C\). The horizontal distance between \(B\) and \(A\) is 4 m and \(B\) is 4 m below \(A C\); angle \(B A C\) is obtuse (see Fig. 1). The force exerted on the \(\operatorname { rod } A B\) at \(B\), by the \(\operatorname { rod } B C\), has horizontal and vertical components as shown in Fig. 2.

1. By taking moments about \(A\) for the \(\operatorname { rod } A B\) find the value of \(X - Y\).
2. By taking moments about \(C\) for the rod \(B C\) show that \(2 X - 3 Y + 165 = 0\).
3. Find the magnitude of the force acting between \(A B\) and \(B C\) at \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{08760a55-da6c-41f2-a88a-289ecc227f69-3_812_773_260_685} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\), have weights \(2 W\) and \(W\) respectively. The rods are freely jointed to each other at \(B\), and \(B C\) is freely jointed to a fixed point at \(C\). The rods are held in equilibrium in a vertical plane by a light string attached to \(A\) and perpendicular to \(A B\). The rods \(A B\) and \(B C\) make angles \(45 ^ { \circ }\) and \(\alpha\), respectively, with the horizontal. The tension in the string is \(T\) (see diagram).

1. By taking moments about \(B\) for \(A B\), show that \(W = \sqrt { 2 } T\).
2. Find the value of \(\tan \alpha\).

3 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_586_1435_1537_354} A uniform \(\operatorname { rod } P Q\) has weight 72 N . A non-uniform \(\operatorname { rod } Q R\) has weight 54 N and its centre of mass is at \(C\), where \(Q C = 2 C R\). The rods are freely jointed to each other at \(Q\). The rod \(P Q\) is freely jointed to a fixed point of a vertical wall at \(P\) and the rod \(Q R\) rests on horizontal ground at \(R\). The rod \(P Q\) is 2.8 m long and is horizontal. The point \(R\) is 1.44 m below the level of \(P Q\) and 4 m from the wall (see diagram).

1. Find the vertical component of the force exerted by the wall on \(P Q\).
2. Hence show that the normal component of the force exerted by the ground on \(Q R\) is 90 N .
3. Given that the friction at \(R\) is limiting, find the coefficient of friction between the rod \(Q R\) and the ground.

5 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-3_441_450_213_808} Two uniform rods \(A B\) and \(B C\), each of length \(2 L \mathrm {~m}\) and of weight 84.5 N , are freely jointed at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). The rods are held in equilibrium in a vertical plane by a light string attached at \(C\) and perpendicular to \(B C\). The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) to the horizontal, respectively (see diagram). It is given that \(\cos \beta = \frac { 12 } { 13 }\).

1. Find the tension in the string.
2. Hence show that the force acting on \(B C\) at \(B\) has horizontal component of magnitude 15 N and vertical component of magnitude 48.5 N , and state the direction of the component in each case.
3. Find \(\alpha\).

6 A uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(2 l\) is in equilibrium at \(60 ^ { \circ }\) to the horizontal with \(A\) resting against a smooth vertical plane and \(B\) resting on a rough section of a horizontal plane. Another uniform rod \(C D\), of length \(\sqrt { 3 } l\) and weight \(W\), is freely jointed to the mid-point of \(A B\) at \(C\); its other end \(D\) rests on a smooth section of the horizontal plane. \(C D\) is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-4_508_1075_438_495}

1. Show that the force exerted by the horizontal plane on \(C D\) is \(\frac { 1 } { 2 } W\). Find the normal component of the force exerted by the horizontal plane on \(A B\).
2. Find the magnitude and direction of the force exerted by \(C D\) on \(A B\).
3. Given that \(A B\) is in limiting equilibrium, find the coefficient of friction between \(A B\) and the horizontal plane.

3 \includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-2_661_711_918_717} Two uniform rods \(A B\) and \(A C\), of equal lengths, and of weights 200 N and 360 N respectively, are freely jointed at \(A\). The mid-points of the rods are joined by a taut light inextensible string. The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is 2.1 m above the surface and \(B C = 1.4 \mathrm {~m}\) (see diagram).

1. Show that the force exerted on \(A B\) at \(B\) has magnitude 240 N and find the tension in the string.
2. Find the horizontal and vertical components of the force exerted on \(A B\) at \(A\) and state their directions.

5 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods \(A B\) and \(B C\), each of length 1.4 m and weight 80 N , are freely jointed to each other at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). They are held in equilibrium with \(A B\) at an angle \(\alpha\) to the horizontal, and \(B C\) at an angle of \(60 ^ { \circ }\) to the horizontal, by a light string, perpendicular to \(B C\), attached to \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on \(B C\) at \(B\).
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer \(P\) of mass 80 kg is suspended from a fixed point \(O\) by an elastic rope of natural length 5.25 m and modulus of elasticity \(2058 \mathrm {~N} . P\) is in equilibrium at a point 5 m above a safety net. A second performer \(Q\), also of mass 80 kg , falls freely under gravity from a point above \(P\). \(P\) catches \(Q\) and together they begin to descend vertically with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) (see diagram). The performers are modelled as particles.

5 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods \(X A\) and \(X B\) are freely jointed at \(X\). The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with \(A\) and \(B\) in contact with a rough horizontal surface. \(A\) and \(B\) are at distances horizontally from \(X\) of 0.9 m and 0.5 m respectively, and \(X\) is 1.2 m above the surface (see diagram).

1. The normal components of the contact forces acting on the rods at \(A\) and \(B\) are \(R _ { A } \mathrm {~N}\) and \(R _ { B } \mathrm {~N}\) respectively. Show that \(R _ { A } = 125\) and find \(R _ { B }\).
2. Find the frictional components of the contact forces acting on the rods at \(A\) and \(B\).
3. Find the horizontal and vertical components of the force exerted on \(X A\) at \(X\), stating their directions.

6 Two uniform rods \(A B\) and \(B C\), each of length \(2 l\), are freely jointed at \(B\). The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are in a vertical plane with \(A\) freely pivoted at a fixed point and \(C\) resting in equilibrium on a rough horizontal plane. The normal and frictional components of the force acting on \(B C\) at \(C\) are \(R\) and \(F\) respectively. The rod \(A B\) makes an angle \(30 ^ { \circ }\) to the horizontal and the rod \(B C\) makes an angle \(60 ^ { \circ }\) to the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-4_682_901_479_587}

1. By considering the equilibrium of \(\operatorname { rod } B C\), show that \(W + \sqrt { 3 } F = R\).
2. By taking moments about \(A\) for the equilibrium of the whole system, find another equation involving \(W , F\) and \(R\).
3. Given that the friction at \(C\) is limiting, calculate the value of the coefficient of friction at \(C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-3_510_716_662_676} Two uniform rods \(A B\) and \(B C\), each of length \(4 L\), are freely jointed at \(B\), and rest in a vertical plane with \(A\) and \(C\) on a smooth horizontal surface. The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are joined by a horizontal light inextensible string fixed to each rod at a point distance \(L\) from \(B\), so that each rod is inclined at an angle of \(60 ^ { \circ }\) to the horizontal (see diagram).

1. By considering the equilibrium of the whole body, show that the force acting on \(B C\) at \(C\) is \(1.75 W\) and find the force acting on \(A B\) at \(A\).
2. Find the tension in the string in terms of \(W\).
3. Find the horizontal and vertical components of the force acting on \(A B\) at \(B\), and state the direction of the component in each case.

5 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack. \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).

1. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
2. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.

1. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
2. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
3. Show that the motion of \(P\) is simple harmonic.
4. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

5. \begin{figure}[h] \end{figure} The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and mass \(4 m\), is smoothly hinged to a fixed point. The end \(B\) is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as \(A\). The distance from \(A\) to the pulley is \(4 a\). The other end of the string carries a particle of mass \(m\) which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.

1. Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
2. Hence, or otherwise, show that any value of \(\theta\) which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0 .$$
3. Given that \(\theta = \frac { \pi } { 6 }\) corresponds to a position of equilibrium, determine its stability. \section*{L \(\_\_\_\_\)}