3.04b Equilibrium: zero resultant moment and force

3 A uniform ladder \(P Q\), of length 8 metres and mass 28 kg , rests in equilibrium with its foot, \(P\), on a rough horizontal floor and its top, \(Q\), 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 \(69 ^ { \circ }\). A man, of mass 72 kg , is standing at the point \(C\) on the ladder so that the distance \(P C\) is 6 metres. The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{06c3e260-8167-4616-97d4-0f360a376a0f-3_679_679_685_678}

1. Draw a diagram to show the forces acting on the ladder.
2. With the man standing at the point \(C\), the ladder is on the point of slipping.
   1. Show that the magnitude of the reaction between the ladder and the vertical wall is 256 N , correct to three significant figures.
   2. Find the coefficient of friction between the ladder and the horizontal floor.

9 A smooth hollow hemisphere, of radius \(a\) and centre \(O\), is fixed so that its rim is in a horizontal plane. A smooth uniform \(\operatorname { rod } A B\), of mass \(m\), is in equilibrium, with one end \(A\) resting on the inside of the hemisphere and the point \(C\) on the rod being in contact with the rim of the hemisphere. The rod, of length \(l\), is inclined at an angle \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85514b55-3f13-4746-a3ef-747239b64cca-6_453_828_559_591}

1. Explain why the reaction between the rod and the hemisphere at point \(A\) acts through \(O\).
2. Draw a diagram to show the forces acting on the rod.
3. Show that \(l = \frac { 4 a \cos 2 \theta } { \cos \theta }\).

3 The diagram shows a uniform rod, \(A B\), of mass 10 kg and length 5 metres. The rod is held in equilibrium in a horizontal position, by a support at \(C\) and a light vertical rope attached to \(A\), where \(A C\) is 2 metres. \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-3_237_680_479_648}

1. Draw and label a diagram to show the forces acting on the rod.
2. Show that the tension in the rope is 24.5 N .
3. A package of mass \(m \mathrm {~kg}\) is suspended from \(B\). The tension in the rope has to be doubled to maintain equilibrium.
   1. Find \(m\).
   2. Find the magnitude of the force exerted on the rod by the support.
4. Explain how you have used the fact that the rod is uniform in your solution.

4 A uniform plank is 10 m long and has mass 15 kg . It is placed on horizontal ground at the edge of a vertical river bank, so that 2 m of the plank is projecting over the edge, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{676e753d-1b80-413c-a4b9-21861db8dde5-3_250_1285_1361_388}

1. A woman of mass 50 kg stands on the part of the plank which projects over the river. Find the greatest distance from the river bank at which she can safely stand.
2. The woman wishes to stand safely at the end of the plank which projects over the river. Find the minimum mass which she should place on the other end of the plank so that she can do this.
3. State how you have used the fact that the plank is uniform in your solution.
4. State one other modelling assumption which you have made.

4 Ken is trying to cross a river of width 4 m . He has a uniform plank, \(A B\), of length 8 m and mass 17 kg . The ground on both edges of the river bank is horizontal. The plank rests at two points, \(C\) and \(D\), on fixed supports which are on opposite sides of the river. The plank is at right angles to both river banks and is horizontal. The distance \(A C\) is 1 m , and the point \(C\) is at a horizontal distance of 0.6 m from the river bank. Ken, who has mass 65 kg , stands on the plank directly above the middle of the river, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{31ba38f7-38a8-4e4e-96a3-19e819fabfb0-3_468_1086_1710_479}

1. Draw a diagram to show the forces acting on the plank.
2. Given that the reaction on the plank at the point \(D\) is \(44 g \mathrm {~N}\), find the horizontal distance of the point \(D\) from the nearest river bank.
3. State how you have used the fact that the plank is uniform in your solution.

4 A uniform plank \(A B\), of length 6 m , has mass 25 kg . It is supported in equilibrium in a horizontal position by two vertical inextensible ropes. One of the ropes is attached to the plank at the point \(P\) and the other rope is attached to the plank at the point \(Q\), where \(A P = 1 \mathrm {~m}\) and \(Q B = 0.8 \mathrm {~m}\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5b1c9e8d-459a-474c-bd29-6dadff40de14-2_227_1187_2252_424}

1. 1. Find the tension in each rope.
   2. State how you have used the fact that the plank is uniform in your solution. (1 mark)
2. A particle of mass \(m \mathrm {~kg}\) is attached to the plank at point \(B\), and the tension in each rope is now the same. Find \(m\).

9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks] \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}

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 \(\mu\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d1eb99a1-04e5-43bc-87b4-d0f7c962135c-2_599_677_1151_696} A uniform beam \(A B\) of mass 15 kg and length 4 m is freely hinged to a vertical wall at \(A\). The beam is held in equilibrium in a horizontal position by a light rod \(P Q\) of length \(1.5 \mathrm {~m} . P\) is fixed to the wall vertically below \(A\) and \(P Q\) makes an angle of \(30 ^ { \circ }\) with the vertical (see diagram). The force exerted on the beam at \(Q\) by the rod is in the direction \(P Q\). Find

1. the magnitude of the force exerted on the beam at \(Q\),
2. the magnitude and direction of the force exerted on the beam at \(A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-2_542_638_1208_717} A uniform semicircular arc \(A C B\) is freely pivoted at \(A\). The arc has mass 0.3 kg and is held in equilibrium by a force of magnitude \(P\) N applied at \(B\). The line of action of this force lies in the same plane as the arc, and is perpendicular to \(A B\). The diameter \(A B\) has length 4 cm and makes an angle of \(\theta ^ { \circ }\) with the downward vertical (see diagram).

1. Given that \(\theta = 0\), find the magnitude of the force acting on the arc at \(A\).
2. Given instead that \(\theta = 30\), find the value of \(P\).

2 A uniform solid cylinder of height 12 cm and radius \(r \mathrm {~cm}\) is in equilibrium on a rough inclined plane with one of its circular faces in contact with the plane.

1. The cylinder is on the point of toppling when the angle of inclination of the plane to the horizontal is \(21 ^ { \circ }\). Find \(r\). The cylinder is now placed on a different inclined plane with one of its circular faces in contact with the plane. This plane is also inclined at \(21 ^ { \circ }\) to the horizontal. The coefficient of friction between this plane and the cylinder is \(\mu\).
2. The cylinder slides down this plane but does not topple. Find an inequality for \(\mu\).

4 A uniform rod \(P Q\) has weight 18 N and length 20 cm . The end \(P\) rests against a rough vertical wall. A particle of weight 3 N is attached to the rod at a point 6 cm from \(P\). The rod is held in a horizontal position, perpendicular to the wall, by a light inextensible string attached to the rod at \(Q\) and to a point \(R\) on the wall vertically above \(P\), as shown in the diagram. The string is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The system is in limiting equilibrium.

1. Find the tension in the string.
2. Find the magnitude of the force exerted by the wall on the rod.
3. Find the coefficient of friction between the wall and the rod.

6 A particle is projected with speed \(v \mathrm {~ms} ^ { - 1 }\) from a point \(O\) on horizontal ground. The angle of projection is \(\theta ^ { \circ }\) above the horizontal. At time \(t\) seconds after the instant of projection the horizontal displacement of the particle from \(O\) is \(x \mathrm {~m}\) and the upward vertical displacement from \(O\) is \(y \mathrm {~m}\).

1. Show that $$y = x \tan \theta - \frac { 4.9 x ^ { 2 } } { v ^ { 2 } \cos ^ { 2 } \theta } .$$ A stone is thrown from the top of a vertical cliff 100 m high. The initial speed of the stone is \(16 \mathrm {~ms} ^ { - 1 }\) and the angle of projection is \(\theta ^ { \circ }\) to the horizontal. The stone hits the sea 40 m from the foot of the cliff.
2. Find the two possible values of \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{8492ec9b-3327-4d89-aaa4-bf98cdf0ebdc-3_623_995_1475_536} A uniform ladder \(A B\) of weight \(W \mathrm {~N}\) and length 4 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal where \(\tan \theta = \frac { 1 } { 2 }\) (see diagram). A small object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at a point \(C\), which is 1 m from \(A\). The coefficient of friction between the ladder and the ground is \(\mu\) and the system is in limiting equilibrium.

2 A uniform circular cylinder, of radius 6 cm and height 15 cm , is in equilibrium on a fixed inclined plane with one of its ends in contact with the plane.

1. Given that the cylinder is on the point of toppling, find the angle the plane makes with the horizontal. The cylinder is now placed on a horizontal board with one of its ends in contact with the board. The board is then tilted so that the angle it makes with the horizontal gradually increases.
2. Given that the coefficient of friction between the cylinder and the board is \(\frac { 3 } { 4 }\), determine whether or not the cylinder will slide before it topples, justifying your answer.

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.

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 \(\frac { 5 } { 8 }\).

3 A thin rigid non-uniform beam AB of length 6 m has weight 800 N . Its centre of mass, G , is 2 m from B .
Initially the beam is horizontal and in equilibrium when supported by a small round peg at \(\mathrm { C } , 1 \mathrm {~m}\) from A , and a light vertical wire at B . This situation is shown in Fig. 3.1 where the lengths are in metres. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Calculate the tension in the wire and the normal reaction of the peg on the beam. The beam is now held horizontal and in equilibrium with the wire at \(70 ^ { \circ }\) to the horizontal, as shown in Fig. 3.2. The peg at C is rough and still supports the beam 1 m from A. The beam is on the point of slipping.
2. Calculate the new tension in the wire. Calculate also the coefficient of friction between the peg and the beam. The beam is now held in equilibrium at \(30 ^ { \circ }\) to the vertical with the wire at \(\theta ^ { \circ }\) to the beam, as shown in Fig. 3.3. A new small smooth peg now makes contact with the beam at C, still 1 m from A. The tension in the wire is now \(T \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a6297924-579e-4340-8fe6-2b43bd1a8698-4_456_353_1484_861} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. By taking moments about C , resolving in a suitable direction and obtaining two equations in terms of \(\theta\) and \(T\), or otherwise, calculate \(\theta\) and \(T\).

4 A rigid thin uniform rod AB with length 2.4 m and weight 30 N is used in different situations.

1. In the first situation, the rod rests on a small support 0.6 m from B and is held horizontally in equilibrium by a vertical string attached to A, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_196_707_456_680} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Calculate the tension in the string and the force of the support on the rod.
2. In the second situation, the rod rests in equilibrium on the point of slipping with end A on a horizontal floor and the rod resting at P on a fixed block of height 0.9 m , as shown in Fig. 4.2. The rod is perpendicular to the edge of the block on which it rests and is inclined at \(\theta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_208_746_1101_657} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} (A) Suppose that the contact between the block and the rod is rough with coefficient of friction 0.6 and contact between the end A and the floor is smooth. Show that \(\tan \theta = 0.6\).
   (B) Suppose instead that the contact between the block and the rod is smooth and the contact between the end A and the floor is rough. The rod is now in limiting equilibrium at a different angle \(\theta\) such that the distance AP is 1.5 m . Calculate the normal reaction of the block on the rod. Calculate the coefficient of friction between the rod and the floor.

2 Two heavy rods AB and BC are freely jointed together at B and to a wall at A . AB has weight 90 N and centre of mass at \(\mathrm { P } ; \mathrm { BC }\) has weight 75 N and centre of mass at Q . The lengths of the rods and the positions of P and Q are shown in Fig. 2.1, with the lengths in metres. Initially, AB and BC are horizontal. There is a support at R , as shown in Fig. 2.1. The system is held in equilibrium by a vertical force acting at C . \begin{figure}[h] \end{figure}

1. Draw diagrams showing all the forces acting on \(\operatorname { rod } \mathrm { AB }\) and on \(\operatorname { rod } \mathrm { BC }\). Calculate the force exerted on AB by the hinge at B and hence the force required at C . The rods are now set up as shown in Fig. 2.2. AB and BC are each inclined at \(60 ^ { \circ }\) to the vertical and C rests on a rough horizontal table. Fig. 2.3 shows all the forces acting on AB , including the forces \(X \mathrm {~N}\) and \(Y \mathrm {~N}\) due to the hinge at A and the forces \(U \mathrm {~N}\) and \(V \mathrm {~N}\) in the hinge at B . The rods are in equilibrium. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_393_661_1615_429} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{31c38a58-e9d5-4d01-90af-3b41213a9c7d-3_355_438_1530_1178} \captionsetup{labelformat=empty} \caption{Fig. 2.3}
   \end{figure}
2. By considering the equilibrium of \(\operatorname { rod } \mathrm { AB }\), show that \(60 \sqrt { 3 } = U + V \sqrt { 3 }\).
3. Draw a diagram showing all the forces acting on rod BC .
4. Find a further equation connecting \(U\) and \(V\) and hence find their values. Find also the frictional force at C .

3 A uniform plank is 2.8 m long and has weight 200 N . The centre of mass is G.

1. Fig. 3.1 shows the plank horizontal and in equilibrium, resting on supports at A and B . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_229_1125_434_459} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} Calculate the reactions of the supports on the plank at A and at B .
2. Fig. 3.2 shows the plank horizontal and in equilibrium between a support at C and a peg at D . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_236_1141_993_461} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} Calculate the reactions of the support and the peg on the plank at C and at D , showing the directions of these forces on a diagram. Fig. 3.3 shows the plank in equilibrium between a support at P and a peg at Q . The plank is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_424_1099_1692_475} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Calculate the normal reactions at P and at Q .
4. Just one of the contacts is rough. Determine which one it is if the value of the coefficient of friction is as small as possible. Find this value of the coefficient of friction.

3 \begin{enumerate}[label=(\alph*)] \item Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and D and to a vertical wall at C and D. There are vertical loads of \(L \mathrm {~N}\) at A and \(3 L \mathrm {~N}\) at B . Angle DAB is \(30 ^ { \circ }\), angle DBC is \(60 ^ { \circ }\) and ABC is a straight, horizontal line. \begin{figure}[h] \end{figure}

1. Draw a diagram showing the loads and the internal forces in the four rods.
2. Find the internal forces in the rods in terms of \(L\), stating whether each rod is in tension or in thrust (compression). [You may leave answers in surd form. Note that you are not required to find the external forces acting at C and at D.]

\item Fig. 3.2 shows uniform beams PQ and QR , each of length 2 lm and of weight \(W \mathrm {~N}\). The beams are freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at \(60 ^ { \circ }\) to the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal. This force, \(U \mathrm {~N}\), is shown in Fig. 3.3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Show that the frictional force between the floor and each beam is \(\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}\).

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-5_641_885_269_671} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing line of length 2 m . The other end of the line, A , is fixed and the line can swing freely. The sphere swings with the line taut from a point where the line is at an angle of \(40 ^ { \circ }\) with the vertical, as shown in Fig. 4.
   1. Explain why no work is done on the sphere by the tension in the line.
   2. Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the lowest point of its swing and calculate the amount of gravitational potential energy lost when it is at this point.
   3. Assuming that there is no air resistance and that the sphere swings from rest from the position shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing.
   4. Now consider the case where
      Calculate the speed of the sphere at the lowest point of its swing.
   5. A block of mass 3 kg slides down a uniform, rough slope that is at an angle of \(30 ^ { \circ }\) to the horizontal. The acceleration of the block is \(\frac { 1 } { 8 } g\). Show that the coefficient of friction between the block and the slope is \(\frac { 1 } { 4 } \sqrt { 3 }\).

3 A non-uniform beam AB has weight 85 N . The length of the beam is 5 m and its centre of mass is 3 m from A . In this question all the forces act in the same vertical plane. Fig. 3.1 shows the beam in horizontal equilibrium, supported at its ends. \begin{figure}[h] \end{figure}

1. Calculate the reactions of the supports on the beam. Using a smooth hinge, the beam is now attached at A to a vertical wall. The beam is held in equilibrium at an angle \(\alpha\) to the horizontal by means of a horizontal force of magnitude 27.2 N acting at B . This situation is shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_725_675_1153_347} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_732_565_1153_1231} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
2. Show that \(\tan \alpha = \frac { 15 } { 8 }\). The hinge and 27.2 N force are removed. The beam now rests with B on a rough horizontal floor and A on a smooth vertical wall, as shown in Fig. 3.3. It is at the same angle \(\alpha\) to the horizontal. There is now a force of 34 N acting at right angles to the beam at its centre in the direction shown. The beam is in equilibrium and on the point of slipping.
3. Draw a diagram showing the forces acting on the beam. Show that the frictional force acting on the beam is 7.4 N .
   Calculate the value of the coefficient of friction between the beam and the floor.

3 Fig. 3 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DA }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D and to a vertical wall at A . ABCD is a parallelogram with AD horizontal and BD vertical; the dimensions of the framework, in metres, are shown. There is a vertical load of 300 N acting at C and a vertical wire attached to D , with tension \(T \mathrm {~N}\), holds the framework in equilibrium. The horizontal and vertical forces, \(X \mathrm {~N}\) and \(Y \mathrm {~N}\), acting on the framework at A due to the wall are also shown. \begin{figure}[h] \end{figure}

1. Show that \(T = 600\) and calculate the values of \(X\) and \(Y\).
2. Draw a diagram showing all the forces acting on the framework, and also the internal forces in the rods.
3. Calculate the internal forces in the five rods, indicating whether each rod is in tension or compression (thrust). (You may leave answers in surd form. Your working in this part should correspond to your diagram in part (ii).) Suppose that the vertical wire is attached at B instead of D and that the framework is still in equilibrium.
4. Without doing any further calculations, state which four of the rods have the same internal forces as in part (iii) and say briefly why this is the case. Determine the new force in the fifth rod.