3.04b Equilibrium: zero resultant moment and force

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}

4. The diagram below shows a uniform rod \(A B\), of weight 10 N , hinged to a vertical wall at \(A\). The rod is held in a horizontal position by means of a light inextensible string. One end of the string is attached to a point \(C\) on the rod and the other end is attached to a point \(D\) on the wall. The point \(D\) is 0.6 m vertically above \(A\) and the length of \(A C\) is 0.8 m . A particle of weight 25 N is attached to the rod at \(B\) and the tension in the string is 75 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-4_572_808_612_625}

1. Find the length of the rod \(A B\).
2. Calculate the magnitude and direction of the reaction at the hinge at \(A\).

1. The diagram shows a uniform rod \(A B\), of length 8 m and mass 23 kg , in limiting equilibrium with its end \(A\) on rough horizontal ground and point \(C\) resting against a smooth fixed cylinder. The rod is inclined at an angle of \(30 ^ { \circ }\) to the ground. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is \(\frac { 2 } { 3 }\).

1. Calculate the magnitude of the normal reaction at \(C\) and the magnitude of the normal reaction to the ground at \(A\).
2. Find the length \(A C\).
3. Suppose instead that the rod is non-uniform with its centre of mass closer to \(A\) than to \(B\). Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.
   

1. \begin{figure}[h] \end{figure} A uniform plane lamina is in the shape of an isosceles trapezium \(A B C D E F\), as shown shaded in Figure 1.

• \(B C E F\) is a square
• \(A B = C D = a\)
• \(B C = 3 a\)
  1. Show that the distance of the centre of mass of the lamina from \(A D\) is \(\frac { 11 a } { 8 }\)

The mass of the lamina is \(M\) The lamina is suspended by two light vertical strings, one attached to the lamina at \(A\) and the other attached to the lamina at \(F\) The lamina hangs freely in equilibrium, with \(B F\) horizontal.

Find, in terms of \(M\) and \(g\), the tension in the string attached at \(A\)

2. \begin{figure}[h] \end{figure} Uniform wire is used to form the framework shown in Figure 2.
In the framework

• \(A B C D\) is a rectangle with \(A D = 2 a\) and \(D C = a\)
• \(B E C\) is a semicircular arc of radius \(a\) and centre \(O\), where \(O\) lies on \(B C\)

The diameter of the semicircle is \(B C\) and the point \(E\) is such that \(O E\) is perpendicular to \(B C\). The points \(A , B , C , D\) and \(E\) all lie in the same plane.

1. Show that the distance of the centre of mass of the framework from \(B C\) is $$\frac { a } { 6 + \pi }$$ The framework is freely suspended from \(A\) and hangs in equilibrium with \(A E\) at an angle \(\theta ^ { \circ }\) to the downward vertical.
2. Find the value of \(\theta\). The mass of the framework is \(M\).
   A particle of mass \(k M\) is attached to the framework at \(B\).
   The centre of mass of the loaded framework lies on \(O A\).
3. Find the value of \(k\).

4. \begin{figure}[h] \end{figure} A uniform triangular lamina \(A B C\) is isosceles, with \(A C = B C\). The midpoint of \(A B\) is \(M\). The length of \(A B\) is \(18 a\) and the length of \(C M\) is \(18 a\). The triangular lamina \(C D E\), with \(D E = 6 a\) and \(C D = 12 a\), has \(E D\) parallel to \(A B\) and \(M D C\) is a straight line. Triangle \(C D E\) is removed from triangle \(A B C\) to form the lamina \(L\), shown shaded in Figure 1. The distance of the centre of mass of \(L\) from \(M C\) is \(d\).

1. Show that \(d = \frac { 4 } { 7 } a\) The lamina \(L\) is suspended by two light inextensible strings. One string is attached to \(L\) at \(A\) and the other string is attached to \(L\) at \(B\).
   The lamina hangs in equilibrium in a vertical plane with the strings vertical and \(A B\) horizontal.
   The weight of \(L\) is \(W\)
2. Find, in terms of \(W\), the tension in the string attached to \(L\) at \(B\) The string attached to \(L\) at \(B\) breaks, so that \(L\) is now suspended from \(A\). When \(L\) is hanging in equilibrium in a vertical plane, the angle between \(A B\) and the downward vertical through \(A\) is \(\theta ^ { \circ }\)
3. Find the value of \(\theta\)

4. \begin{figure}[h] \end{figure} The uniform triangular lamina \(A B C\) has \(A B\) perpendicular to \(A C\), \(A B = 9 a\) and \(A C = 6 a\). The point \(D\) on \(A B\) is such that \(A D = a\). The rectangle \(D E F G\), with \(D E = 2 a\) and \(E F = 3 a\), is removed from the lamina to form the template shown shaded in Figure 3. The distance of the centre of mass of the template from \(A C\) is \(d\).

1. Show that \(d = \frac { 23 } { 7 } a\) The template is freely suspended from \(A\) and hangs in equilibrium with \(A B\) at an angle \(\theta ^ { \circ }\) to the downward vertical through \(A\).
2. Find the value of \(\theta\) A new piece, of exactly the same size and shape as the template, is cut from a lamina of a different uniform material. The template and the new piece are joined together to form the model shown in Figure 4. Both parts of the model lie in the same plane. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-12_369_1185_1667_440} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The weight of \(C P Q R S T A\) is \(W\) The weight of \(A D G F E B C\) is \(4 W\) The model is freely suspended from \(A\).
   A horizontal force of magnitude \(X\), acting in the same vertical plane as the model, is now applied to the model at \(T\) so that \(A C\) is vertical, as shown in Figure 4.
3. Find \(X\) in terms of \(W\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the shape and dimensions of a template \(O P Q R S T U V\) made from thin uniform metal. \(O P = 5 \mathrm {~m} , P Q = 2 \mathrm {~m} , Q R = 1 \mathrm {~m} , R S = 1 \mathrm {~m} , T U = 2 \mathrm {~m} , U V = 1 \mathrm {~m} , V O = 3 \mathrm {~m}\).
Figure 1 also shows a coordinate system with \(O\) as origin and the \(x\)-axis and \(y\)-axis along \(O P\) and \(O V\) respectively. The unit of length on both axes is the metre. The centre of mass of the template has coordinates \(( \bar { x } , \bar { y } )\).

1. 1. Show that \(\bar { y } = 1\)
   2. Find the value of \(\bar { x }\). A new design requires the template to have its centre of mass at the point (2.5,1). In order to achieve this, two circular discs, each of radius \(r\) metres, are removed from the template which is shown in Figure 1, to form a new template \(L\). The centre of the first disc is ( \(0.5,0.5\) ) and the centre of the second disc is ( \(0.5 , a\) ) where \(a\) is a constant.
2. Find the value of \(r\).
3. 1. Explain how symmetry can be used to find the value of \(a\).
   2. Find the value of \(a\). The template \(L\) is now freely suspended from the point \(U\) and hangs in equilibrium.
4. Find the size of the angle between the line \(T U\) and the horizontal.

A flagpole, \(A B\), is 4 m long. The flagpole is modelled as a non-uniform rod so that, at a distance \(x\) metres from \(A\), the mass per unit length of the flagpole, \(m \mathrm {~kg} \mathrm {~m} ^ { - 1 }\), is given by \(m = 18 - 3 x\).

1. Show that the mass of the flagpole is 48 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9b995178-a4be-4d5a-95f8-6c2978ff01b3-12_515_439_502_806} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The end \(A\) of the flagpole is fixed to a point on a vertical wall. A cable has one end attached to the midpoint of the flagpole and the other end attached to a point on the wall that is vertically above \(A\). The cable is perpendicular to the flagpole. The flagpole and the cable lie in the same vertical plane that is perpendicular to the wall. A small ball of mass 4 kg is attached to the flagpole at \(B\). The cable holds the flagpole and ball in equilibrium, with the flagpole at \(45 ^ { \circ }\) to the wall, as shown in Figure 3. The tension in the cable is \(T\) newtons.
   The cable is modelled as a light inextensible string and the ball is modelled as a particle.
2. Using the model, find the value of \(T\).
3. Give a reason why the answer to part (b) is not likely to be the true value of \(T\).

9 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-08_302_992_260_539} The diagram shows a plank of wood \(A B\), of mass 10 kg and length 6 m , resting with its end \(A\) on rough horizontal ground and its end \(B\) in contact with a fixed cylindrical oil drum. The plank is in a vertical plane perpendicular to the axis of the drum, and the line \(A B\) is a tangent to the circular cross-section of the drum, with the point of contact at \(B\). The plank is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The plank is modelled as a uniform rod and the oil drum is modelled as being smooth.

1. Find, in terms of \(g\), the normal contact force between the drum and the plank.
2. Given that the plank is in limiting equilibrium, find the coefficient of friction between the plank and the ground.

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).

3 A uniform plank, of length 8 metres, has mass 30 kg . The plank is supported in equilibrium in a horizontal position by two smooth supports at the points \(A\) and \(B\), as shown in the diagram. A block, of mass 20 kg , is placed on the plank at point \(A\). \includegraphics[max width=\textwidth, alt={}, center]{06b431ca-d3a8-46d6-b9f8-bac08d3fd51e-3_193_1216_477_404}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the magnitude of the force exerted on the plank by the support at \(B\) is \(19.2 g\) newtons.
3. Find the magnitude of the force exerted on the plank by the support at \(A\).
4. Explain how you have used the fact that the plank is uniform in your solution.

2 A uniform plank, of length 6 metres, has mass 40 kg . The plank is held in equilibrium in a horizontal position by two vertical ropes attached to the plank at \(A\) and \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{03994596-21ad-4201-8d64-ba2d7b7e0a77-2_323_1162_1464_440}

1. Draw a diagram to show the forces acting on the plank.
2. Show that the tension in the rope attached to the plank at \(B\) is \(21 g \mathrm {~N}\).
3. Find the tension in the rope that is attached to the plank at \(A\).
4. State where in your solution you have used the fact that the plank is uniform.

3 A uniform ladder, of length 6 metres and mass 22 kg , rests with its foot, \(A\), on a rough horizontal floor and its top, \(B\), 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 \(\theta\). A man, of mass 90 kg , is standing at point \(C\) on the ladder so that the distance \(A C\) is 5 metres. With the man in this position, the ladder is on the point of slipping. The coefficient of friction between the ladder and the horizontal floor is 0.6 . The man may be modelled as a particle at \(C\). \includegraphics[max width=\textwidth, alt={}, center]{9cfa110c-ee11-447a-b21a-3f436432e27d-3_707_702_742_646}

1. Show that the magnitude of the frictional force between the ladder and the horizontal floor is 659 N , correct to three significant figures.
2. Find the angle \(\theta\).

4. \begin{figure}[h] \end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.

1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
3. Determine the stability of the position of equilibrium.

14 A uniform ladder \(A B\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall.
The ladder is inclined at an angle of \(45 ^ { \circ }\) to the horizontal.
A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\).
The coefficient of friction between the ladder and the wall is \(\frac { 1 } { 3 }\) and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 2 }\).
The system is in limiting equilibrium. Find \(x\). 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.
OCR is part of the \section*{...day June 20XX - Morning/Afternoon} A Level Mathematics A
H240/03 Pure Mathematics and Mechanics \section*{SAMPLE MARK SCHEME} MAXIMUM MARK 100 \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-13_259_1320_1242_826} \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for A Level Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
If you are in any doubt whatsoever you should contact your Team Leader.
The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored. Sometimes this is reinforced in the mark scheme by the abbreviation isw. However, this would not apply to a case where a candidate passes through the correct answer as part of a wrong argument.
d When a part of a question has two or more 'method' steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. (The notation 'dep*' is used to indicate that a particular mark is dependent on an earlier, asterisked, mark in the scheme.) Of course, in practice it may happen that when a candidate has once gone wrong in a part of a question, the work from there on is worthless so that no more marks can sensibly be given. On the other hand, when two or more steps are successfully run together by the candidate, the earlier marks are implied and full credit must be given.
e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
Sometimes the answer to one part of a question is used in a later part of the same question. In this case, A marks will often be 'follow through'. In such cases you must ensure that you refer back to the answer of the previous part question even if this is not shown within the image zone. You may find it easier to mark follow through questions candidate-by-candidate rather than question-by-question.
f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question. (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question.
g Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
h For a genuine misreading (of numbers or symbols) which is such that the object and the difficulty of the question remain unaltered, mark according to the scheme but following through from the candidate's data. A penalty is then applied; 1 mark is generally appropriate, though this may differ for some units. This is achieved by withholding one A mark in the question. Marks designated as cao may be awarded as long as there are no other errors. E marks are lost unless, by chance, the given results are established by equivalent working. 'Fresh starts' will not affect an earlier decision about a misread. Note that a miscopy of the candidate's own working is not a misread but an accuracy error.
i If a calculator is used, some answers may be obtained with little or no working visible. Allow full marks for correct answers (provided, of course, that there is nothing in the wording of the question specifying that analytical methods are required). Where an answer is wrong but there is some evidence of method, allow appropriate method marks. Wrong answers with no supporting method score zero. If in doubt, consult your Team Leader.
j If in any case the scheme operates with considerable unfairness consult your Team Leader. PS = Problem Solving
M = Modelling

16 A straight uniform rod, \(A B\), has length 6 m and mass 0.2 kg A particle of weight \(w\) newtons is fixed at \(A\).
A second particle of weight \(3 w\) newtons is fixed at \(B\).
The rod is suspended by a string from a point \(x\) metres from \(B\).
The rod rests in equilibrium with \(A B\) horizontal and the string hanging vertically as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-24_410_1148_767_445} Show that $$x = \frac { 3 w + 0.3 g } { 2 w + 0.1 g }$$ \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-25_2488_1716_219_153}

14 A \(\pounds 2\) coin has a diameter of 28 mm and a mass of 12 grams. A uniform rod \(A B\) of length 160 mm and a fixed load of mass \(m\) grams are used to check that a \(\pounds 2\) coin has the correct mass. The rod rests with its midpoint on a support.
A \(\pounds 2\) coin is placed face down on the rod with part of its curved edge directly above \(A\). The fixed load is hung by a light inextensible string from a point directly below the other end of the rod at \(B\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-22_195_766_854_639} 14

1. Given that the rod is horizontal and rests in equilibrium, find \(m\).
   14
2. State an assumption you have made about the \(\pounds 2\) coin to answer part (a).

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.