3.04a Calculate moments: about a point

3 Fig. 3.1 shows a thin rectangular frame ABCD , with part of it filled by a triangular lamina ABD . \(\mathrm { AD } = 30 \mathrm {~cm}\) and \(\mathrm { AB } = x \mathrm {~cm}\). Together they form the composite structure S . The centre of mass of \(S\) lies at a point \(M , 16.5 \mathrm {~cm}\) from \(A D\) and 11.7 cm from \(A B\). \begin{figure}[h] \end{figure} The frame and the triangular lamina are both uniform but made of different materials. The mass of the frame is 1.7 kg .

1. Show that the triangular lamina has a mass of 3.3 kg .
2. Determine the value of \(x\), correct to \(\mathbf { 3 }\) significant figures. One end of a light inextensible string is attached to S at D . The other end is attached to a fixed point on a vertical wall. For S to hang in equilibrium with AD vertical, a force of magnitude \(Q N\) is applied to S as shown in Fig. 3.2. The line of action of this force lies in the same plane as S . The string is taut and lies in the same plane as S at an angle \(\phi\) to the downward vertical. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-4_611_994_1756_242} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
3. By taking moments about D , show that \(Q = 50.5\), correct to 3 significant figures.
4. Determine, in degrees, the value of \(\phi\).

5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h] \end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
   \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
   \end{figure} The weight of S is \(W \mathrm {~N}\).
3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .

6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h] \end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.

1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
   \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
   \section*{END OF QUESTION PAPER}

3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
2. 1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
   2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
4. Find the range of possible values for \(\mu\).
5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).

3 Fig. 3 shows a light square lamina ABCD , of side length 0.75 m , suspended vertically by wires attached to A and B so that AB is horizontal. A particle P of mass \(m \mathrm {~kg}\) is attached to the edge DC . The lamina hangs in equilibrium. \begin{figure}[h] \end{figure} The tension in the wire attached to A is 14 N and the tension in the wire attached to B is \(T \mathrm {~N}\). The wire at A makes an angle of \(25 ^ { \circ }\) with the horizontal and the wire at B makes an angle of \(60 ^ { \circ }\) with the horizontal.

1. Determine the value of \(T\).
2. Determine
   1. the value of \(m\),
   2. the distance of P from D . P is moved to the midpoint of CD . A couple is applied to the lamina so that it remains in equilibrium with AB horizontal and the tension in both wires unchanged.
3. Determine

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.

2 The vertices of a triangular lamina, which is in the \(x - y\) plane, are at the origin O and the points \(\mathrm { A } ( 4,0 )\) and \(\mathrm { B } ( 0,3 )\). Forces, of magnitude \(T _ { 1 } \mathrm {~N} , T _ { 2 } \mathrm {~N}\) and 10 N , whose lines of action are in the \(x - y\) plane, are applied to the lamina at \(\mathrm { O } , \mathrm { A }\) and B respectively, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-2_814_922_1135_246}

1. 1. Show that \(\sin \alpha = 0.6\).
   2. Write down the value of \(\cos \alpha\). The lamina is in equilibrium.
2. Determine the values of \(T _ { 1 } , T _ { 2 }\) and \(\theta\).

4 Fig. 4 shows a thin rigid non-uniform rod PQ of length 0.5 m . End P rests on a rough circular peg. A force of \(T \mathrm {~N}\) acts at the end Q at \(60 ^ { \circ }\) to QP . The weight of the rod is 40 N and its centre of mass is 0.3 m from P . \begin{figure}[h] \end{figure} The rod does not slip on the peg and is in equilibrium with PQ horizontal.

1. Show that the vertical component of \(T\) is 24 N .
2. \(F\) is the contact force at P between the rod and the peg. Find

6 A uniform solid cylinder, L, has base radius 5 cm , height 24 cm and mass 5 kg . L is placed on a rough plane inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 6. \begin{figure}[h] \end{figure}

1. On the copy of Fig. 6 in the Printed Answer Booklet mark the forces acting on L . The coefficient of friction between L and the plane is 0.3 . Initially \(\alpha\) is \(15 ^ { \circ }\).
2. Show that L rests in equilibrium on the plane. A couple is applied to L . It is given that L will topple if the couple is applied in an anticlockwise sense, but L will not topple if the couple is applied in a clockwise sense.
3. Find the range of possible values of the magnitude of the couple. The couple is now removed and the plane is slowly tilted so that \(\alpha\) increases.
4. Determine whether L topples first without sliding or slides first without toppling.

4 A uniform beam AB of mass 6 kg and length 5 m rests with its end A on smooth horizontal ground and its end B against a smooth vertical wall. The vertical distance between the ground and B is 4 m , and the angle between the beam and the downward vertical is \(\theta\). To prevent the beam from sliding, one end of a light taut rope of length 2 m is attached to the beam at C and the other end of the rope is attached to a point on the wall 2 m above the ground, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-5_558_556_500_251}

1. By considering the value of \(\cos \theta\), determine the distance BC . An object of mass 75 kg is placed on the beam at a point which is \(x \mathrm {~m}\) from A . It is given that the tension in the rope is \(T \mathrm {~N}\) and the magnitude of the normal contact force between the ground and the beam is \(R \mathrm {~N}\).
2. By taking moments about B for the beam, show that \(25 \mathrm { R } + 3675 \mathrm { x } - 16 \mathrm {~T} = 19110\).
3. Given that the rope can withstand a maximum tension of 720 N , determine the largest possible value of \(x\).

6 Fig. 6.1 shows a light rod ABC , of length 60 cm , where B is the midpoint of AC . Particles of masses \(3.5 \mathrm {~kg} , 1.4 \mathrm {~kg}\) and 2.1 kg are attached to \(\mathrm { A } , \mathrm { B }\) and C respectively. \begin{figure}[h] \end{figure} The centre of mass is located at a point G along the rod.

1. Determine the distance AG . Two light inextensible strings, each of length 40 cm , are attached to the rod, one at A , the other at C. The other ends of these strings are attached to a fixed point D. The rod is allowed to hang in equilibrium.
2. Determine the angle AD makes with the vertical. The two strings are now replaced by a single light inextensible string of length 80 cm . One end of the string is attached to A and the other end of the string is attached to C. The string passes over a smooth peg fixed at D. The rod hangs in equilibrium, but is not vertical, as shown in Fig. 6.2. Fig. 6.2
3. Explain why angle ADG and angle CDG must be equal.
4. Determine the tension in the string.

5 Fig. 5.1 shows a particle P, of mass 5 kg , and a particle Q, of mass 11 kg , which are attached to the ends of a light, inextensible string. The string is taut and passes over a small smooth pulley fixed to the ceiling. \begin{figure}[h] \end{figure} When a force of magnitude \(H \mathrm {~N}\), acting at an angle \(\theta\) to the upward vertical, is applied to Q the particles hang in equilibrium, with the part of the string connecting the pulley to Q making an angle of \(40 ^ { \circ }\) with the upward vertical. It is given that the force acts in the same vertical plane in which the string lies.

1. Determine the values of \(H\) and \(\theta\). Particle Q is now removed. The string is instead attached to one end of a uniform beam B of length 3 m and mass 7 kg . The other end of B is in contact with a rough horizontal floor. The situation is shown in Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_504_978_1557_251}
   \end{figure} With B in equilibrium, at an angle \(\phi\) to the horizontal, the part of the string connecting the pulley to B makes an angle of \(30 ^ { \circ }\) with the upward vertical. It is given that the string and B lie in the same vertical plane.
2. Determine the smallest possible value for the coefficient of friction between B and the floor.
3. Determine the value of \(\phi\).

6 Fig. 6.1 shows three forces of magnitude \(15 \mathrm {~N} , 15 \mathrm {~N}\) and 30 N acting on a rigid beam AB of length 6 m . One of the forces of magnitude 15 N acts at A, and the other force of magnitude 15 N acts at B. The force of magnitude 30 N acts at distance of \(x \mathrm {~m}\) from B. All three forces act in a direction perpendicular to the beam as shown in Fig. 6.1. The beam and the three forces all lie in the same horizontal plane. The three forces form a couple of magnitude 42 Nm in the clockwise direction. \begin{figure}[h] \end{figure}

1. Determine the value of \(x\). Fig. 6.2 shows the same beam, without the three forces from Fig. 6.1, resting in limiting equilibrium against a step. The point of contact, C , between the beam and the edge of the step lies 1.5 m from A. The other end of the beam rests on a horizontal floor. The contacts between the beam and both the step and the floor are rough. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{0a790ad0-7eda-40f1-9894-f156766ae46f-6_348_412_1633_244}
   \end{figure} It is given that the beam is non-uniform, and that its centre of mass lies \(\sqrt { 3 } \mathrm {~m}\) from B .
2. Draw a diagram to show all the forces acting on the beam. The coefficient of friction between the beam and the step and the coefficient of friction between the beam and the floor are the same, and are denoted by \(\mu\).
3. 1. Show that \(\mu ^ { 2 } - 6 \mu + 1 = 0\).
   2. Hence determine the value of \(\mu\).

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 \(\theta\).

7. As part of a design for a new building, an architect wants to support a wooden beam in a horizontal position. The beam is suspended using a vertical steel cable and a smooth fixed support on its underside. The diagram below shows the architect's diagram and the adjacent table shows the categories of steel cable available. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-18_504_1699_559_191} You may use the following modelling assumptions.

• The wooden beam is a rigid uniform rod of mass 100 kg .
• The force exerted on the beam by the support is vertical.
• The steel cable is inextensible.

\section*{SAFETY REQUIREMENT} Both the steel cable and the support must be capable of withstanding forces of at least four times those present in the architect's diagram above. The wooden beam is held in horizontal equilibrium.
[0pt]

1. 1. Given that the support is capable of withstanding loads of up to 2000 N , show that the force exerted on the beam by the support satisfies the safety requirement. [3]
   2. Determine which categories of steel cable in the table opposite could meet the safety requirements.
2. State how you have used the modelling assumption that the beam is a uniform rod. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

1 A uniform beam, \(A B\), has mass 20 kg and length 7 metres. A rope is attached to the beam at \(A\). A second rope is attached to the beam at the point \(C\), which is 2 metres from \(B\). Both of the ropes are vertical. The beam is in equilibrium in a horizontal position, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{88aec6ab-af83-4d5e-84b6-5fd84c16a6c9-003_298_906_756_552} Find the tensions in the two ropes.

2 A hotel sign consists of a uniform rectangular lamina of weight \(W\). The sign is suspended in equilibrium in a vertical plane by two vertical light chains attached to the sign at the points \(A\) and \(B\), as shown in the diagram. The edge containing \(A\) and \(B\) is horizontal. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_289_529_1859_726} The tensions in the chains attached at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively.

1. Draw a diagram to show the forces acting on the sign.
2. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(W\).
3. Explain how you have used the fact that the lamina is uniform in answering part (b).

9. \begin{figure}[h] \end{figure} A plank, \(A B\), of mass \(M\) and length \(2 a\), rests with its end \(A\) against a rough vertical wall. The plank is held in a horizontal position by a rope. One end of the rope is attached to the plank at \(B\) and the other end is attached to the wall at the point \(C\), which is vertically above \(A\). A small block of mass \(3 M\) is placed on the plank at the point \(P\), where \(A P = x\). The plank is in equilibrium in a vertical plane which is perpendicular to the wall. The angle between the rope and the plank is \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 3 .
The plank is modelled as a uniform rod, the block is modelled as a particle and the rope is modelled as a light inextensible string.

1. Using the model, show that the tension in the rope is \(\frac { 5 M g ( 3 x + a ) } { 6 a }\) The magnitude of the horizontal component of the force exerted on the plank at \(A\) by the wall is \(2 M g\).
2. Find \(x\) in terms of \(a\). The force exerted on the plank at \(A\) by the wall acts in a direction which makes an angle \(\beta\) with the horizontal.
3. Find the value of \(\tan \beta\) The rope will break if the tension in it exceeds \(5 M g\).
4. Explain how this will restrict the possible positions of \(P\). You must justify your answer carefully.

9. Figure 1 A uniform ladder \(A B\), of length \(2 a\) and weight \(W\), has its end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
The end \(B\) of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
A builder of weight \(7 W\) stands at the top of the ladder.
To stop the ladder from slipping, the builder's assistant applies a horizontal force of magnitude \(P\) to the ladder at \(A\), towards the wall.
The force acts in a direction which is perpendicular to the wall.
The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 5 } { 2 }\).
The builder is modelled as a particle and the ladder is modelled as a uniform rod.

1. Show that the reaction of the wall on the ladder at \(B\) has magnitude \(3 W\).
2. Find, in terms of \(W\), the range of possible values of \(P\) for which the ladder remains in equilibrium. Often in practice, the builder's assistant will simply stand on the bottom of the ladder.
3. Explain briefly how this helps to stop the ladder from slipping.

2 The force \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) acts at the point with coordinates \(( 0,2 )\) The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed along the \(x\)-axis and the \(y\)-axis respectively.
Calculate the magnitude of the moment of this force about the origin.
Circle your answer.
[0pt] [1 mark]
6 Nm
8 Nm
10 Nm
14 Nm

2. \begin{figure}[h] \end{figure} A non-uniform beam \(A B\) has length 6 m and mass 50 kg . The beam rests horizontally on two supports at \(C\) and \(D\), where \(A C = 0.9 \mathrm {~m}\) and \(D B = 1.8 \mathrm {~m}\). A child of mass 25 kg stands on the beam at \(E\), where \(A E = E B = 3 \mathrm {~m}\), as shown in Figure 1. The beam is in equilibrium.
The magnitude of the normal reaction between the beam and the support at \(C\) is \(R _ { C }\) newtons. The magnitude of the normal reaction between the beam and the support at \(D\) is \(R _ { D }\) newtons. The beam is modelled as a rod and the child is modelled as a particle.
The centre of mass of the beam is between \(C\) and \(D\) and is a distance \(x\) metres from \(D\).
Given that \(2 R _ { D } = 3 R _ { C }\)

1. show that \(x = 1.38\) The child remains at \(E\) and a block of mass \(M \mathrm {~kg}\) is placed on the beam at \(B\).
   The block is modelled as a particle.
   Given that the beam is on the point of tilting,
2. find the value of \(M\).

8 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-4_657_655_1128_705}

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