3.04c Use moments: beams, ladders, static problems

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{7febbd80-4cbb-4b2e-b022-d6a20e7e13aa-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

3. A plank \(A B\) has length 6 m and mass 30 kg . The point \(C\) is on the plank with \(C B = 2 \mathrm {~m}\). The plank rests in equilibrium in a horizontal position on supports at \(A\) and \(C\). Two people, each of mass 75 kg , stand on the plank. One person stands at the point \(P\) of the plank, where \(A P = x\) metres, and the other person stands at the point \(Q\) of the plank, where \(A Q = 2 x\) metres. The plank remains horizontal and in equilibrium with the magnitude of the reaction at \(C\) five times the magnitude of the reaction at \(A\). The plank is modelled as a uniform rod and each person is modelled as a particle.

1. Find the value of \(x\).
2. State two ways in which you have used the assumptions made in modelling the plank as a uniform rod.

3. \begin{figure}[h] \end{figure} A wooden beam \(A B\), of mass 150 kg and length 9 m , rests in a horizontal position supported by two vertical ropes. The ropes are attached to the beam at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(B D = 3.5 \mathrm {~m}\). A gymnast of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 2. The beam remains horizontal and in equilibrium. By modelling the gymnast as a particle, the beam as a uniform rod and the ropes as light inextensible strings,

1. find the tension in the rope attached to the beam at \(C\). The gymnast at \(P\) remains on the beam at \(P\) and another gymnast, who is also modelled as a particle, stands on the beam at \(B\). The beam remains horizontal and in equilibrium. The mass of the gymnast at \(B\) is the largest possible for which the beam remains horizontal and in equilibrium.
2. Find the tension in the rope attached to the beam at \(D\).

3. \begin{figure}[h] \end{figure} A beam \(A D C B\) has length 5 m . The beam lies on a horizontal step with the end \(A\) on the step and the end \(B\) projecting over the edge of the step. The edge of the step is at the point \(D\) where \(D B = 1.3 \mathrm {~m}\), as shown in Figure 2. When a small boy of mass 30 kg stands on the beam at \(C\), where \(C B = 0.5 \mathrm {~m}\), the beam is on the point of tilting. The boy is modelled as a particle and the beam is modelled as a uniform rod.

1. Find the mass of the beam. A block of mass \(X \mathrm {~kg}\) is now placed on the beam at \(A\).
   The block is modelled as a particle.
2. Find the smallest value of \(X\) that will enable the boy to stand on the beam at \(B\) without the beam tilting.
3. State how you have used the modelling assumption that the block is a particle in your calculations.

4. \begin{figure}[h] \end{figure} A branch \(A B\), of length 1.5 m , rests horizontally in equilibrium on two supports.
The two supports are at the points \(C\) and \(D\), where \(A C = 0.24 \mathrm {~m}\) and \(D B = 0.36 \mathrm {~m}\), as shown in Figure 1. When a force of 150 N is applied vertically upwards at \(B\), the branch is on the point of tilting about \(C\). When a force of 225 N is applied vertically downwards at \(B\), the branch is on the point of tilting about \(D\). The branch is modelled as a non-uniform rod \(A B\) of weight \(W\) newtons.
The distance from the point \(C\) to the centre of mass of the rod is \(x\) metres.
Use the model to find

1. the value of \(W\)
2. the value of \(x\)

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 30 kg and length 3 m .
The beam rests on supports at \(C\) and \(D\) where \(A C = 0.4 \mathrm {~m}\) and \(D B = 0.4 \mathrm {~m}\), as shown in Figure 4. A person of mass 55 kg stands on the beam between \(C\) and \(D\).
The person is modelled as a particle at the point \(P\), where \(C P = x\) metres and \(0 < x < 2.2\) The beam is modelled as a uniform rod resting in equilibrium in a horizontal position.
Using the model,

1. show that the magnitude of the reaction at \(C\) is \(( 686 - 245 x ) \mathrm { N }\). The magnitude of the reaction at \(C\) is four times the magnitude of the reaction at \(D\).
   Using the model,
2. find the value of \(x\) The person steps off the beam and places a package of mass \(M \mathrm {~kg}\) at \(A\).
   The package is modelled as a particle at the point \(A\).
   The beam is now on the point of tilting about \(C\).
   Using the model,
3. find the value of \(M\)

4 Fig. 4 shows a uniform beam of mass 4 kg and length 2.4 m resting on two supports P and Q . P is at one end of the beam and Q is 0.3 m from the other end.
Determine whether a person of mass 50 kg can tip the beam by standing on it. \begin{figure}[h] \end{figure}

6 A shelf consists of a horizontal uniform plank AB of length 0.8 m and mass 5 kg with light inextensible vertical strings attached at each end. A stack of bricks each of mass 2.3 kg is placed on the plank as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-06_397_734_641_242}

1. Explain the meaning of each of the following modelling assumptions.
   Either of the strings will break if the tension exceeds 75 N.
2. Find the greatest number of bricks that can be placed at the centre of the plank without breaking the strings.
3. Find an expression for the moment about A of the weight of a stack of \(n\) bricks when the stack is at a distance of \(x \mathrm {~m}\) from A . State the units for your answer.
4. Calculate the greatest distance from A that the largest stack of bricks can be placed without a string breaking.

4 A ruler PQRS is a uniform rectangular lamina with mass 20 grams. The length of PQ is 30 cm and the length of PS is 4 cm . The ruler is attached at P to a smooth hinge and held with S vertically below P by a horizontal force of magnitude \(F \mathrm {~N}\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{8eeff88d-8b05-43c6-86a5-bd82221c0bea-04_303_1495_1363_239}

1. Calculate the value of \(F\).
2. Explain what would happen to the lamina if the force at S were removed.

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 \(\mu\).

17 A uniform plank \(P Q\), of length 7 metres, lies horizontally at rest, in equilibrium, on two fixed supports at points \(X\) and \(Y\) The distance \(P X\) is 1.4 metres and the distance \(Q Y\) is 2 metres as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_56_689_534_762} \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-26_225_830_607_694} 17

1. The reaction force on the plank at \(X\) is \(4 g\) newtons.
   17
   1. (i) Show that the mass of the plank is 9.6 kilograms.
      17
   2. (ii) Find the reaction force, in terms of \(g\), on the plank at \(Y\) 17
   3. The support at \(Y\) is moved so that the distance \(Q Y = 1.4\) metres. The plank remains horizontally at rest in equilibrium.
      It is claimed that the reaction force at \(Y\) remains unchanged.
      Explain, with a reason, whether this claim is correct.

\includegraphics{figure_1} A plank \(AB\) has mass 40 kg and length 3 m. A load of mass 20 kg is attached to the plank at \(B\). The loaded plank is held in equilibrium, with \(AB\) horizontal, by two vertical ropes attached at \(A\) and \(C\), as shown in Figure 1. The plank is modelled as a uniform rod and the load as a particle. Given that the tension in the rope at \(C\) is three times the tension in the rope at \(A\), calculate

1. the tension in the rope at \(C\), [2]
2. the distance \(CB\). [5]

\includegraphics{figure_3} A uniform beam \(AB\) has mass 12 kg and length 3 m. The beam rests in equilibrium in a horizontal position, resting on two smooth supports. One support is at the end \(A\), the other at a point \(C\) on the beam, where \(BC = 1\) m, as shown in Figure 3. The beam is modelled as a uniform rod.

1. Find the reaction on the beam at \(C\). [3]

A woman of mass 48 kg stands on the beam at the point \(D\). The beam remains in equilibrium. The reactions on the beam at \(A\) and \(C\) are now equal.

1. Find the distance \(AD\). [7]

\includegraphics{figure_2} A beam \(AB\) is supported by two vertical ropes, which are attached to the beam at points \(P\) and \(Q\), where \(AP = 0.3\) m and \(BQ = 0.3\) m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between \(P\) and \(Q\). The gymnast is modelled as a particle attached to the beam at the point \(X\), where \(PX = x\) m, \(0 < x < 1.4\) as shown in Figure 2. The beam rests in equilibrium in a horizontal position.

1. Show that the tension in the rope attached to the beam at \(P\) is \((588 - 350x)\) N. [3]
2. Find, in terms of \(x\), the tension in the rope attached to the beam at \(Q\). [3]
3. Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]

Given that the tension in the rope attached at \(Q\) is three times the tension in the rope attached at \(P\),

1. find the value of \(x\). [3]

In a theatre, three lights \(A\), \(B\) and \(C\) are suspended from a horizontal beam \(XY\) of length 4.5 m. \(A\) and \(C\) are each of mass 8 kg and \(B\) is of mass 6 kg. The beam \(XY\) is held in place by vertical ropes \(PX\) and \(QY\), as shown. \includegraphics{figure_8} In a simple mathematical model of this situation, \(XY\) is modelled as a light rod.

1. Calculate the tension in each of \(PX\) and \(QY\). [6 marks]

In a refined model, \(XY\) is modelled as a uniform rod of mass \(m\) kg.

1. If the tension in \(PX\) is 1.5 times that in \(QY\), calculate the value of \(m\). [6 marks]

A plank of wood \(AB\), of mass 8 kg and length 6 m, rests on a support at \(P\), where \(AP = 4\) m. When particles of mass 1 kg and \(k\) kg are suspended from \(A\) and \(B\) respectively, the plank rests horizontally in equilibrium. Modelling the plank as a uniform rod, find

1. the value of \(k\), [3 marks]
2. the magnitude of the force exerted by the support on the plank at \(P\). [2 marks]

An iron bar \(AB\), of length \(4\) m, is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(PB = 0.85\) m. The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(AG = 1.4\) m, and the wire is modelled as a light inextensible string. Given that the tension in the wire is \(12\) N, calculate

1. the weight of the bar, \hfill [4 marks]
2. the magnitude of the reaction on the bar at \(A\). \hfill [2 marks]
3. State briefly how you have used the given modelling assumption about the bar. \hfill [1 mark]

\includegraphics{figure_1} Figure 1 shows two window cleaners, Alan and Baber, of mass 60 kg and 100 kg respectively standing on a platform \(PQ\) of length 3 metres and mass 20 kg. The platform is suspended by two vertical cables attached to the ends \(P\) and \(Q\). Alan is standing at the point \(A\), 1.25 metres from \(P\), Baber is standing at the point \(B\) and the tension in the cable at \(P\) is twice the tension in the cable at \(Q\). Modelling the platform as a uniform rod and Alan and Baber as particles,

1. find the tension in the cable at \(P\), [2 marks]
2. find the distance \(BP\). [5 marks]
3. State how you have used the modelling assumptions that
   1. the platform is uniform,
   2. the platform is a rod.
   [2 marks]

A non-uniform plank \(AB\) of mass 20 kg and length 6 m is supported at both ends so that it is horizontal. When a woman of mass 60 kg stands on the plank at a distance of 2 m from \(B\), the magnitude of the reaction at \(A\) is 35g N.

1. Suggest a suitable model for
   1. the plank, [2 marks]
   2. the woman.
2. Calculate the magnitude of the reaction at \(B\), giving your answer in terms of \(g\). [2 marks]
3. Explain briefly, in the context of the problem, the term 'non-uniform'. [2 marks]
4. Find the distance of the centre of mass of the plank from \(A\). [4 marks]

A uniform ladder \(AB\), of length 6 metres and mass 22 kg, rests with its foot, \(A\), on rough horizontal ground. The ladder rests against the top of a smooth vertical wall at the point \(C\), where the length \(AC\) is 5 metres. The vertical plane containing the ladder is perpendicular to the wall, and the angle between the ladder and the ground is \(60°\). A man, of mass 88 kg, is standing on the ladder. The man may be modelled as a particle at the point \(D\), where the length of \(AD\) is 4 metres. The ladder is on the point of slipping. \includegraphics{figure_7}

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find the coefficient of friction between the ladder and the horizontal ground. [6 marks]

A uniform ladder, of weight \(W\), rests with its top against a rough vertical wall and its base on rough horizontal ground. The coefficient of friction between the wall and the ladder is \(\mu\) and the coefficient of friction between the ground and the ladder is \(2\mu\). When the ladder is on the point of slipping, the ladder is inclined at an angle of \(\theta\) to the horizontal.

1. Draw a diagram to show the forces acting on the ladder. [2 marks]
2. Find \(\tan \theta\) in terms of \(\mu\). [7 marks]

A metal rod, of mass \(m\) kilograms and length 20 cm, lies at rest on a horizontal shelf. The end of the rod, \(B\), extends 6 cm beyond the edge of the shelf, \(A\), as shown in the diagram below. \includegraphics{figure_14}

1. The rod is in equilibrium when an object of mass 0.28 kilograms hangs from the midpoint of \(AB\). Show that \(m = 0.21\) [3 marks]
2. The object of mass 0.28 kilograms is removed. A number, \(n\), of identical objects, each of mass 0.048 kg, are hung from the rod all at a distance of 1 cm from \(B\). Find the maximum value of \(n\) such that the rod remains horizontal. [4 marks]
3. State one assumption you have made about the rod. [1 mark]

\includegraphics{figure_9} Fig. 9 shows a uniform rod AB of length \(2a\) and weight \(8W\) which is smoothly hinged at the end A to a point on a fixed horizontal rough bar. A small ring of weight \(W\) is threaded on the bar and is connected to the rod at B by a light inextensible string of length \(2a\). The system is in equilibrium with the rod inclined at an angle \(\theta\) to the horizontal.

1. Determine, in terms of \(W\) and \(\theta\), the tension in the string. [4] It is given that, for equilibrium to be possible, the greatest distance the ring can be from A is \(2.4a\).
2. Determine the coefficient of friction between the bar and the ring. [6]

A uniform ladder of length 8 m and weight 180 N stands on a rough horizontal surface and rests against a smooth vertical wall. The ladder makes an angle of 20° with the wall. A woman of weight 720 N stands on the ladder. Fig. 7 shows this situation modelled with the woman's weight acting at a distance \(x\) m from the lower end of the ladder. The system is in equilibrium. \includegraphics{figure_7}

1. Show that the frictional force between the ladder and the horizontal surface is \(F\) N, where \(F = 90(1 + x)\tan 20°\). [4]
2. 1. State with a reason whether \(F\) increases, stays constant or decreases as \(x\) increases. [1]
   2. Hence determine the set of values of the coefficient of friction between the ladder and the surface for which the woman can stand anywhere on the ladder without it slipping. [4]

The diagram below shows a spotlight system that consists of a symmetrical track \(XY\) that is suspended horizontally from the ceiling by means of two vertical wires. \includegraphics{figure_9} Each of the three spotlights \(A\), \(B\), \(C\) may be moved horizontally along its corresponding shaded section of the track. The system remains in equilibrium. The track may be modelled as a light uniform rod of length \(1.8\) m and the wires are fixed at a distance of \(0.4\) m from each end. Each of the spotlights may be modelled as a particle of mass \(m\) kg, positioned at the points where they are in contact with the track. The distances of the spotlights relative to the wires are given in the diagram and are such that $$0 \leqslant d_A \leqslant 0.3, \quad 0.1 \leqslant d_B \leqslant 0.9, \quad 0 \leqslant d_C \leqslant 0.3.$$

1. Given that \(T_1\) and \(T_2\) represent the tension in wires 1 and 2 respectively, show that $$T_1 = mg(2 + d_A - d_B - d_C),$$ and find a similar expression for \(T_2\). [6]
2. 1. Find the maximum possible value of \(T_1\).
   2. Without carrying out any further calculations, write down the maximum possible value of \(T_2\). Give a reason for your answer. [3]