Range of equilibrium positions

4. \begin{figure}[h] \end{figure} A plank \(A B\) of mass 20 kg and length 8 m is resting in a horizontal position on two supports at \(C\) and \(D\), where \(A C = 1.5 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\). A package of mass 8 kg is placed on the plank at \(C\), as shown in Figure 2. The plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the package is modelled as a particle.

1. Find the magnitude of the normal reaction
   1. between the plank and the support at \(C\),
   2. between the plank and the support at \(D\).
      (6) The package is now moved along the plank to the point \(E\). When the package is at \(E\), the magnitude of the normal reaction between the plank and the support at \(C\) is \(R\) newtons and the magnitude of the normal reaction between the plank and the support at \(D\) is \(2 R\) newtons.
2. Find the distance \(A E\).
3. State how you have used the fact that the package is modelled as a particle.

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 \(M\)

6. \begin{figure}[h] \end{figure} A plank \(A B\) has length 4 m and mass 6 kg . The plank rests in a horizontal position on two supports, one at \(B\) and one at \(C\), where \(A C = 1.5 \mathrm {~m}\). A load of mass 15 kg is placed on the plank at the point \(X\), as shown in Figure 2, and the plank remains horizontal and in equilibrium. The plank is modelled as a uniform rod and the load is modelled as a particle. The magnitude of the reaction on the plank at \(C\) is twice the magnitude of the reaction on the plank at \(B\).

1. Find the magnitude of the reaction on the plank at \(C\).
2. Find the distance \(A X\). The load is now moved along the plank to a point \(Y\), between \(A\) and \(C\). Given that the plank is on the point of tipping about \(C\),
3. find the distance \(A Y\).

2. \begin{figure}[h] \end{figure} A uniform wooden beam \(A B\), of mass 20 kg and length 4 m , rests in equilibrium in a horizontal position on two supports. One support is at \(C\), where \(A C = 1.6 \mathrm {~m}\), and the other support is at \(D\), where \(D B = 0.4 \mathrm {~m}\). A boy of mass 60 kg stands on the beam at the point \(P\), where \(A P = 3 \mathrm {~m}\), as shown in Figure 1. The beam remains in equilibrium in a horizontal position. By modelling the boy as a particle and the beam as a uniform rod,

1. 1. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(C\),
   2. find, in terms of \(g\), the magnitude of the force exerted on the beam by the support at \(D\). The boy now starts to walk slowly along the beam towards the end \(A\).
2. Find the greatest distance he can walk from \(P\) without the beam tilting.

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has length 5 m and mass 5 kg . The rod rests in equilibrium in a horizontal position on two supports \(C\) and \(D\), where \(A C = 1 \mathrm {~m}\) and \(D B = 2 \mathrm {~m}\), as shown in Figure 2 . A particle of mass 10 kg is placed on the rod at \(A\) and a particle of mass \(M \mathrm {~kg}\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(C\).
2. Find, in terms of \(M\), the magnitude of the reaction on the rod at \(D\).
3. Hence, or otherwise, find the range of possible values of \(M\). \includegraphics[max width=\textwidth, alt={}, center]{61cb5bce-2fad-48f0-b6a4-e9899aa0acec-14_2256_51_310_1983}

5. \begin{figure}[h] \end{figure} A beam \(A B\) has mass 12 kg and length 5 m . It is held in equilibrium in a horizontal position by two vertical ropes attached to the beam. One rope is attached to \(A\), the other to the point \(C\) on the beam, where \(B C = 1 \mathrm {~m}\), as shown in Figure 2. The beam is modelled as a uniform rod, and the ropes as light strings.

1. Find
   1. the tension in the rope at \(C\),
   2. the tension in the rope at \(A\). A small load of mass 16 kg is attached to the beam at a point which is \(y\) metres from \(A\). The load is modelled as a particle. Given that the beam remains in equilibrium in a horizontal position,
2. find, in terms of \(y\), an expression for the tension in the rope at \(C\). The rope at \(C\) will break if its tension exceeds 98 N. The rope at \(A\) cannot break.
3. Find the range of possible positions on the beam where the load can be attached without the rope at \(C\) breaking.

4. \begin{figure}[h] \end{figure} A bench consists of a plank which is resting in a horizontal position on two thin vertical legs. The plank is modelled as a uniform rod \(P S\) of length 2.4 m and mass 20 kg . The legs at \(Q\) and \(R\) are 0.4 m from each end of the plank, as shown in Figure 1. Two pupils, Arthur and Beatrice, sit on the plank. Arthur has mass 60 kg and sits at the middle of the plank and Beatrice has mass 40 kg and sits at the end \(P\). The plank remains horizontal and in equilibrium. By modelling the pupils as particles, find

1. the magnitude of the normal reaction between the plank and the leg at \(Q\) and the magnitude of the normal reaction between the plank and the leg at \(R\). Beatrice stays sitting at \(P\) but Arthur now moves and sits on the plank at the point \(X\). Given that the plank remains horizontal and in equilibrium, and that the magnitude of the normal reaction between the plank and the leg at \(Q\) is now twice the magnitude of the normal reaction between the plank and the leg at \(R\),
2. find the distance \(Q X\).

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.

15 Fig. 15 shows a uniform shelf AB of weight \(W \mathrm {~N}\).
The shelf is 180 cm long and rests on supports at points C and D . Point C is 30 cm from A and point D is 60 cm from B .
side view \begin{figure}[h] \end{figure} Determine the range of positions a point load of \(3 W\) could be placed on the shelf without the shelf tipping. \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, The Triangle Building, Shaftesbury Road, Cambridge CB2 8EA.
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6. \begin{figure}[h] \end{figure} Figure 2 shows a uniform plank \(A B\) of length 8 m and mass 50 kg suspended horizontally by two light vertical inextensible strings attached at either end of the plank. The maximum tension that either string can support is 40 gN . A rock of mass \(M \mathrm {~kg}\) is placed on the plank at \(A\) and rolled along the plank to \(B\) without either string breaking.

1. Explain, with the aid of a sketch-graph, how the tension in the string at \(A\) varies with \(x\), the distance of the rock from \(A\).
2. Show that \(M \leq 15\). The first rock is removed and a second rock of mass 20 kg is placed on the plank.
3. Find the fraction of the plank on which the rock can be placed without one of the strings breaking.

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 \(\mu\).

A uniform rod \(AC\), of weight \(W\) and length \(3l\), rests horizontally on two supports, one at \(A\) and one at \(B\), where \(AB = 2l\). A particle of weight \(2W\) is placed on the rod at a distance \(x\) from \(A\). The rod remains horizontal and in equilibrium.

1. Find the greatest possible value of \(x\). [5]

The magnitude of the reaction of the support at \(A\) is \(R\). Due to a weakness in the support at \(A\), the greatest possible value of \(R\) is \(4W\).

1. find the least possible value of \(x\). [5]

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]