6.04e Rigid body equilibrium: coplanar forces

3 A uniform lamina \(A B C D E\) consists of a square \(A C D E\) and an equilateral triangle \(A B C\) which are joined along their common edge \(A C\) to form a pentagon whose sides are each 8 cm in length.

1. Calculate the distance of the centre of mass of the lamina from \(A C\).
2. The lamina is freely suspended from \(A\) and hangs in equilibrium. Calculate the angle that \(A C\) makes with the vertical.

7 \includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-4_353_579_248_744} The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium \(A B C D\) with \(A B\) and \(C D\) perpendicular to \(A D\). The lengths of \(A B\) and \(A D\) are each 5 cm and the length of \(C D\) is \(( a + 5 ) \mathrm { cm }\).

1. Show the distance of the centre of mass of the prism from \(A D\) is $$\frac { a ^ { 2 } + 15 a + 75 } { 3 ( a + 10 ) } \mathrm { cm } .$$ The prism is placed with the face containing \(A B\) in contact with a horizontal surface.
2. Find the greatest value of \(a\) for which the prism does not topple. The prism is now placed on an inclined plane which makes an angle \(\theta ^ { \circ }\) with the horizontal. \(A B\) lies along a line of greatest slope with \(B\) higher than \(A\).
3. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the greatest value of \(\theta\) for which the prism does not topple.

3 \includegraphics[max width=\textwidth, alt={}, center]{5bfd0285-71cb-4dcb-8545-a379653f9a3e-2_778_579_1304_744} A uniform lamina \(A B C D E\) consists of a rectangle \(A B D E\) and an isosceles triangle \(B C D\) joined along their common edge. \(A B = D E = 8 \mathrm {~cm} , A E = B D = 10 \mathrm {~cm}\) and \(B C = C D = 13 \mathrm {~cm}\) (see diagram).

1. Find the distance of the centre of mass of the lamina from \(A E\).
2. The lamina is freely suspended from \(B\) and hangs in equilibrium. Calculate the angle that \(B D\) makes with the vertical.

4 \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-2_721_513_1260_762} A uniform solid prism has cross-section \(A B C D E\) in the shape of a rectangle measuring 20 cm by 4 cm joined to a semicircle of radius 8 cm as shown in the diagram. The centre of mass of the solid lies in this cross-section.

1. Find the distance of the centre of mass of the solid from \(A B\). The solid is placed with \(A E\) on rough horizontal ground (so the object does not slide) and is in equilibrium with a horizontal force of magnitude 4 N applied along \(C B\).
2. Find the greatest and least possible values for the weight of the solid.

3 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_389_698_1706_694} A uniform lamina \(A B C D\) has the shape of a square of side \(a\) adjoining a right-angled isosceles triangle whose equal sides are also of length \(a\). The weight of the lamina is \(W\). The lamina rests, in a vertical plane, on smooth supports at \(A\) and \(D\), with \(A D\) horizontal (see diagram).

1. Show that the centre of mass of the lamina is at a horizontal distance of \(\frac { 11 } { 9 } a\) from \(A\).
2. Find, in terms of \(W\), the magnitudes of the forces on the supports at \(A\) and \(D\).

4 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-3_563_707_274_721} A rigid body \(A B C\) consists of two uniform rods \(A B\) and \(B C\), rigidly joined at \(B\). The lengths of \(A B\) and \(B C\) are 13 cm and 20 cm respectively, and their weights are 13 N and 20 N respectively. The distance of \(B\) from \(A C\) is 12 cm . The body hangs in equilibrium, with \(A C\) horizontal, from two vertical strings attached at \(A\) and \(C\). Find the tension in each string.

3 Fig. 3.1 shows an object made up as follows. ABCD is a uniform lamina of mass \(16 \mathrm {~kg} . \mathrm { BE } , \mathrm { EF }\), FG, HI, IJ and JD are each uniform rods of mass 2 kg . ABCD, BEFG and HIJD are squares lying in the same plane. The dimensions in metres are shown in the figure. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of the object, referred to the axes shown in Fig.3.1. The rods are now re-positioned so that BEFG and HIJD are perpendicular to the lamina, as shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5dd6ba0d-e516-4b9e-ba19-6e90520b171b-004_442_666_1510_722} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. Find the \(x\)-, \(y\)-and \(z\)-coordinates of the centre of mass of the object, referred to the axes shown in Fig. 3.2. Calculate the distance of the centre of mass from A . The object is now freely suspended from A and hangs in equilibrium with AC at \(\alpha ^ { \circ }\) to the vertical.
3. Calculate \(\alpha\).

2 A uniform beam, AB , is 6 m long and has a weight of 240 N .
Initially, the beam is in equilibrium on two supports at C and D, as shown in Fig. 2.1. The beam is horizontal. \begin{figure}[h] \end{figure}

1. Calculate the forces acting on the beam from the supports at C and D . A workman tries to move the beam by applying a force \(T \mathrm {~N}\) at A at \(40 ^ { \circ }\) to the beam, as shown in Fig. 2.2. The beam remains in horizontal equilibrium but the reaction of support C on the beam is zero. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_318_691_1119_687} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure}
2. (A) Calculate the value of \(T\).
   (B) Explain why the support at D cannot be smooth. The beam is now supported by a light rope attached to the beam at A , with B on rough, horizontal ground. The rope is at \(90 ^ { \circ }\) to the beam and the beam is at \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 2.3. The tension in the rope is \(P \mathrm {~N}\). The beam is in equilibrium on the point of sliding. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-3_438_633_1909_708} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
3. (A) Show that \(P = 60 \sqrt { 3 }\) and hence, or otherwise, find the frictional force between the beam and the ground.
   (B) Calculate the coefficient of friction between the beam and the ground.

3 A fish slice consists of a blade and a handle as shown in Fig. 3.1. The rectangular blade ABCD is of mass 250 g and modelled as a lamina; this is 24 cm by 8 cm and is shown in the \(\mathrm { O } x y\) plane. The handle EF is of mass 125 g and is modelled as a thin rod; this is 30 cm long and E is attached to the mid-point of \(\mathrm { CD } . \mathrm { EF }\) is at right angles to CD and inclined at \(\alpha\) to the plane containing ABCD , where \(\sin \alpha = 0.6\) (and \(\cos \alpha = 0.8\) ). Coordinates refer to the axes shown in Fig. 3.1. Lengths are in centimetres. The \(y\) and \(z\)-coordinates of the centre of mass of the fish slice are \(\bar { y }\) and \(\bar { z }\) respectively. \begin{figure}[h] \end{figure}

1. Show that \(\bar { y } = 9 \frac { 1 } { 3 }\) and \(\bar { z } = 3\).
2. Suppose that the plane \(\mathrm { O } x y\) in Fig. 3.1 is horizontal and represents a table top and that the fish slice is placed on it as shown. Determine whether the fish slice topples. The 'superior' version of the fish slice has an extra mass of 125 g uniformly distributed over the existing handle for 10 cm from F towards E , as shown in Fig. 3.2. This section of the handle may still be modelled as a thin rod. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-4_513_1065_1683_539} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. In this new situation show that \(\bar { y } = 14\) and \(\bar { z } = 6\). A sales feature of the 'superior' version is the ability to suspend it using a very small hole in the blade. This situation is modelled as the fish slice hanging in equilibrium when suspended freely about an axis through O .
4. Indicate the position of the centre of mass on a diagram and calculate the angle of the line OE with the vertical.

4

1. A uniform, rigid beam, AB , has a weight of 600 N . It is horizontal and in equilibrium resting on two small smooth pegs at P and Q . Fig. 4.1 shows the positions of the pegs; lengths are in metres. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_229_647_404_790} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Calculate the forces exerted by the pegs on the beam. A force of \(L \mathrm {~N}\) is applied vertically downwards at B . The beam is in equilibrium but is now on the point of tipping.
   2. Calculate the value of \(L\).
2. Fig. 4.2 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC }\) and CA . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and C and to a fixed point at A . The pin-joint at C rests on a smooth, horizontal support. The dimensions of the framework are shown in metres. There is a force of 340 N acting at B in the plane of the framework. This force and the \(\operatorname { rod } \mathrm { BC }\) are both inclined to the vertical at an angle \(\alpha\), which is defined in triangle BCX . The force on the framework exerted by the support at C is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3865b4b3-97c7-412b-aabd-2705a954a847-5_675_869_1434_678} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure}
   1. Show that \(R = 600\).
   2. Draw a diagram showing all the forces acting on the framework and also the internal forces in the rods.
      [0pt]
   3. Calculate the internal forces in the three rods, indicating whether each rod is in tension or in compression (thrust). [Your working in this part should correspond to your diagram in part (ii).]

3 A side view of a free-standing kitchen cupboard on a horizontal floor is shown in Fig. 3.1. The cupboard consists of: a base CE; vertical ends BC and DE; an overhanging horizontal top AD. The dimensions, in metres, of the cupboard are shown in the figure. The cupboard and contents have a weight of 340 N and centre of mass at G . \begin{figure}[h] \end{figure}

1. Calculate the magnitude of the vertical force required at A for the cupboard to be on the point of tipping in the cases where the force acts
   (A) downwards,
   (B) upwards. A force of magnitude \(Q \mathrm {~N}\) is now applied at A at an angle of \(\theta\) to AB , as shown in Fig. 3.2, where \(\cos \theta = \frac { 5 } { 13 } \left( \right.\) and \(\left. \sin \theta = \frac { 12 } { 13 } \right)\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-4_303_1134_1619_504} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
2. By considering the vertical and horizontal components of the force at A , show that the clockwise moment of this force about E is \(\frac { 30 Q } { 13 } \mathrm { Nm }\). With the force of magnitude \(Q \mathrm {~N}\) acting as shown in Fig. 3.2, the cupboard is in equilibrium and is on the point of tipping but not on the point of sliding.
3. Show that \(Q = 221\) and that the coefficient of friction between the cupboard base and the floor must be greater than \(\frac { 5 } { 8 }\).

4 In this question, coordinates refer to the axes shown in the figures and the units are centimetres.
Fig. 4.1 shows a lamina KLMNOP shaded. The lamina is made from uniform material and has the dimensions shown. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the \(x\)-coordinate of the centre of mass of this lamina is 26 and calculate the \(y\)-coordinate. A uniform thin heavy wire KLMNOPQ is bent into the shape of part of the perimeter of the lamina KLMNOP with an extension of the side OP to Q, as shown in Fig. 4.2.
2. Show that the \(x\)-coordinate of the centre of mass of this wire is 23.2 and calculate the \(y\)-coordinate. The wire is freely suspended from Q and hangs in equilibrium.
3. Draw a diagram indicating the position of the centre of mass of the hanging wire and calculate the angle of QO with the vertical. A wall-mounted bin with an open top is shown in Fig. 4.3. The centre part has cross-section KLMNOPQ; the two ends are in the shape of the lamina KLMNOP. The ends are made from the same uniform, thin material and each has a mass of 1.5 kg . The centre part is made from different uniform, thin material and has a total mass of 7 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2aaae62-a5f3-47da-afa5-1dd4b37ea2d6-5_499_540_2017_804} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
4. Calculate the \(x\) - and \(y\)-coordinates of the centre of mass of the bin.

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.

4 Fig. 4.1 shows a hollow circular cylinder open at one end and closed at the other. The radius of the cylinder is 0.1 m and its height is \(h \mathrm {~m} . \mathrm { O }\) and C are points on the axis of symmetry at the centres of the open and closed ends, respectively. The thin material used for the closed end has four times the density of the thin material used for the curved surface. \begin{figure}[h] \end{figure} Cylinders of this type are made with different values of \(h\).

1. Show that the centres of mass of these cylinders are on the line OC at a distance \(\frac { 5 h ^ { 2 } + 2 h } { 2 + 10 h } \mathrm {~m}\) from O . Fig. 4.2 shows one of the cylinders placed with its open end on a slope inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 2 } { 3 }\). The cylinder does not slip but is on the point of tipping.
2. Show that \(50 h ^ { 2 } + 5 h - 3 = 0\) and hence that \(h = 0.2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_383_497_1178_1402} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} Fig. 4.3 shows another of the cylinders that has weight 42 N and \(h = 0.5\). This cylinder has its open end on a rough horizontal plane. A force of magnitude \(T \mathrm {~N}\) is applied to a point P on the circumference of the closed end. This force is at an angle \(\beta\) with the horizontal such that \(\tan \beta = \frac { 3 } { 4 }\) and the force is in the vertical plane containing \(\mathrm { O } , \mathrm { C }\) and P . The cylinder does not slip but is on the point of tipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-5_451_679_1955_685} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure}
3. Calculate \(T\).

5. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plane lamina \(A B C D E G\) in the shape of a letter ' \(L\) ' consisting of a rectangle \(A B F G\) joined to another rectangle \(C D E F\). The sides \(A B\) and \(D E\) are both 8 cm long and the sides \(E G\) and \(G A\) are of length 24 cm and 32 cm respectively.

1. Show that the centre of mass of the lamina lies on the line \(B F\).
2. Find the distance of the centre of mass from the line \(A B\). The uniform lamina in Figure 2 is a model of the letter ' \(L\) ' in a sign above a shop. The letter is normally suspended from a wall at \(A\) and \(B\) so that \(A B\) is horizontal but the fixing at \(B\) has broken and the letter hangs in equilibrium from the point \(A\).
3. Find, in degrees to one decimal place, the acute angle \(A G\) makes with the vertical.

3. \begin{figure}[h] \end{figure} Figure 1 shows a uniform triangular lamina \(A B C\) placed with edge \(B C\) along the line of greatest slope of a plane inclined at an angle \(\theta\) to the horizontal. The lengths \(A C\) and \(B C\) are 15 cm and 9 cm respectively and \(\angle A B C\) is a right angle.

1. Find the distance of the centre of mass of the lamina from
   1. \(\quad A B\),
   2. \(B C\). Assuming that the plane is rough enough to prevent the lamina from slipping,
2. find in degrees, correct to 1 decimal place, the maximum value of \(\theta\) for which the lamina remains in equilibrium.
   (4 marks)

7. \begin{figure}[h] \end{figure} Figure 2 shows a uniform lamina \(A B C D\) formed by removing an isosceles triangle \(B C D\) from an equilateral triangle \(A B D\) of side \(2 d\). The point \(C\) is the centroid of triangle \(A B D\).

1. Find the area of triangle \(B C D\) in terms of \(d\).
2. Show that the distance of the centre of mass of the lamina from \(B D\) is \(\frac { 4 } { 9 } \sqrt { 3 } d\).
   (8 marks)
   The lamina is freely suspended from the point \(B\) and hangs at rest.
3. Find in degrees, correct to 1 decimal place, the acute angle that the side \(A B\) makes with the vertical.

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)

5 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_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.

4 \includegraphics[max width=\textwidth, alt={}, center]{5bb3bd29-a2eb-4124-802c-fb17b68c50e4-2_283_711_1754_722} Two uniform smooth spheres \(A\) and \(B\), of equal radius, have masses 5 kg and 2 kg respectively. They are moving on a horizontal surface when they collide. Immediately before the collision, \(A\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is moving perpendicular to the line of centres, and \(B\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along the line of centres (see diagram). The coefficient of restitution is 0.75 . Find the speed and direction of motion of each sphere immediately after the collision.

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.