Moments

A uniform rod, \(AB\), has length 3 metres and mass 24 kg. A particle of mass \(M\) kg is attached to the rod at \(A\). The rod is balanced in equilibrium on a support at \(C\), which is 0.8 metres from \(A\). \includegraphics{figure_11} Find the value of \(M\). [2 marks]

A uniform rod, AB, has length 4 metres. The rod is resting on a support at its midpoint C. A particle of mass 4 kg is placed 0.6 metres to the left of C. Another particle of mass 1.5 kg is placed \(x\) metres to the right of C, as shown. \includegraphics{figure_3} The rod is balanced in equilibrium at C. Find \(x\). Circle your answer. [1 mark] 1.8 m 1.5 m 1.75 m 1.6 m

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.

\includegraphics{figure_5} A uniform rod \(AB\), of mass 3 kg and length 4 m, is in limiting equilibrium with \(A\) on rough horizontal ground. The rod is at an angle of 60° to the horizontal and is supported by a small smooth peg \(P\), such that the distance \(AP\) is 2.5 m (see diagram). Find

1. the force acting on the rod at \(P\), [3]
2. the coefficient of friction between the ground and the rod. [5]

5. A smooth solid hemisphere is fixed with its flat surface in contact with rough horizontal ground. The hemisphere has centre \(O\) and radius \(5 a\).
A uniform rod \(A B\), of length \(16 a\) and weight \(W\), rests in equilibrium on the hemisphere with end \(A\) on the ground. The rod rests on the hemisphere at the point \(C\), where \(A C = 12 a\) and angle \(C A O = \alpha\), as shown in Figure 1. Points \(A , C , B\) and \(O\) all lie in the same vertical plane.

1. Explain why \(A O = 13 a\) The normal reaction on the rod at \(C\) has magnitude \(k W\)
2. Show that \(k = \frac { 8 } { 13 }\) The resultant force acting on the rod at \(A\) has magnitude \(R\) and acts upwards at \(\theta ^ { \circ }\) to the horizontal.
3. Find
   1. an expression for \(R\) in terms of \(W\)
   2. the value of \(\theta\) (8) 5 \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0762451f-b951-4d66-9e01-61ecb7b30d95-16_426_1001_125_475}
      \end{figure} . T a and angle \(C A O = \alpha\), as shown in Figure 1.
      Points \(A , C , B\) and \(O\) all lie in the same vertical plane.
      1. Explain why \(A O = 13 a\)

\includegraphics{figure_2} A uniform smooth disc with centre \(O\) and radius \(a\) is fixed at the point \(D\) on a horizontal surface. A uniform rod of length \(3a\) and weight \(W\) rests on the disc with its end \(A\) in contact with a rough vertical wall. The rod and the disc lie in a vertical plane that is perpendicular to the wall. The wall meets the horizontal surface at the point \(E\) such that \(AE = a\) and \(ED = \frac{5}{4}a\). A particle of weight \(kW\) is hung from the rod at \(B\) (see diagram). The coefficient of friction between the rod and the wall is \(\frac{1}{8}\) and the system is in limiting equilibrium. Find the value of \(k\). [8]

3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).

2 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).

1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
2. Find the magnitude and direction of the resultant of the three forces.
3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.

\includegraphics{figure_7} A uniform solid hemisphere of mass \(M\) and radius \(a\) is placed with its curved surface on rough horizontal ground. A horizontal force \(P\) is applied to the hemisphere at the centre of its flat circular face.

1. Find the minimum value of the coefficient of friction \(\mu\) between the hemisphere and the ground for the hemisphere to slide without toppling.
2. Show that if \(\mu < \frac{3}{8}\), the hemisphere will topple.
3. Find the maximum horizontal distance that the centre of mass of the hemisphere moves before toppling begins, given that \(\mu = \frac{1}{4}\) and the hemisphere starts from rest.
4. Find the angular acceleration of the hemisphere about its point of contact with the ground at the instant when toppling begins.

[16]

A stick of mass \(0.75\) kg is at rest with one end \(X\) on a rough horizontal floor and the other end \(Y\) leaning against a smooth vertical wall. The coefficient of friction between the stick and the floor is \(0.6\). Modelling the stick as a uniform rod, find the smallest angle that the stick can make with the floor before it starts to slip. \includegraphics{figure_2} [6 marks]

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The end \(A\) is in contact with rough horizontal ground and the end \(B\) is in contact with a smooth vertical wall. The rod rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle of \(30 ^ { \circ }\) with the wall, as shown in Figure 2. The coefficient of friction between the rod and the ground is \(\mu\).

1. Find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the rod by the wall.
2. Show that \(\mu \geqslant \frac { \sqrt { 3 } } { 6 }\). A particle of mass \(k m\) is now attached to the rod at \(B\). Given that \(\mu = \frac { \sqrt { 3 } } { 5 }\) and that the rod is now in limiting equilibrium,
3. find the value of \(k\).

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

\includegraphics{figure_2} A uniform rod \(AB\) of weight \(W\) rests in equilibrium with \(A\) in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude \(P\) acting at \(B\) in this vertical plane. The rod makes an angle of \(60°\) with the wall, and the force makes an angle of \(30°\) with the rod (see diagram). Find the value of \(P\). [3] Find also the set of possible values of the coefficient of friction between the rod and the wall. [4]

5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).

1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.

\includegraphics{figure_5} Two uniform rods \(AB\) and \(BC\) are smoothly jointed at \(B\) and rest in equilibrium with \(C\) on a rough horizontal floor and with \(A\) against a rough vertical wall. The rod \(AB\) is horizontal and the rods are in a vertical plane perpendicular to the wall. The rod \(AB\) has mass \(3m\) and length \(3a\), the rod \(BC\) has mass \(5m\) and length \(5a\), and \(C\) is at a distance \(6a\) from the wall (see diagram). Show that the normal reaction exerted by the floor on the rod \(BC\) at \(C\) has magnitude \(\frac{1}{2}mg\). [5] The coefficient of friction at both \(A\) and \(C\) is \(\mu\). Find the least possible value of \(\mu\) for which the rods do not slip at either \(A\) or \(C\). [7]

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 uniform ruler AB has mass 28 g and length 30 cm . As shown in Fig. 6, the ruler is placed on a horizontal table so that it overhangs a point C at the edge of the table by 25 cm . A downward force of \(F \mathrm {~N}\) is applied at A . This force just holds the ruler in equilibrium so that the contact force between the table and the ruler acts through C . \begin{figure}[h] \end{figure}

1. Complete the force diagram in the Printed Answer Booklet, labelling the forces and all relevant distances.
2. Calculate the value of \(F\). Answer all the questions.
   Section B (78 marks)

\includegraphics{figure_7} A barrier is modelled as a uniform rectangular plank of wood, \(ABCD\), rigidly joined to a uniform square metal plate, \(DEFG\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m. The metal plate has mass 80 kg and side 0.5 m. The plank and plate are joined in such a way that \(CDE\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(CH\) is 0.25 m.

1. Calculate the contact force at \(H\) when the barrier is in the closed position. [3]

In the open position, the centre of mass of the barrier is vertically above \(D\).

1. Calculate the angle between \(AB\) and the horizontal when the barrier is in the open position. [8]

\includegraphics{figure_4} A rigid lamina of negligible mass is in the form of a rhombus ABCD, where AC = 6 m and BD = 8 m. Forces of magnitude 2 N, 4 N, 3 N and 5 N act along its sides AB, BC, CD and DA, respectively, as shown in the diagram. A further force F N, acting at A, and a couple of magnitude G N m are also applied to the lamina so that it is in equilibrium.

1. Determine the magnitude and direction of F. [4]
2. Determine the value of G. [2]

5 \includegraphics[max width=\textwidth, alt={}, center]{f8961f84-c080-4407-a178-45b76f200111-3_533_698_1343_721} A uniform rectangular lamina \(A B C D\), in which \(A B = 8 a\) and \(B C = 6 a\), has mass \(M\). A uniform circular lamina of radius \(\frac { 5 } { 2 } a\) has mass \(\frac { 1 } { 3 } M\). The two laminas are fixed together in the same plane with their centres coinciding at the point \(O\) (see diagram). A particle \(P\) of mass \(\frac { 1 } { 2 } M\) is attached at \(C\). The system is free to rotate about a fixed smooth horizontal axis through \(A\) and perpendicular to the plane \(A B C D\). Show that the moment of inertia of the system about this axis is \(\frac { 2225 } { 24 } M a ^ { 2 }\). The system is released from rest with \(A C\) horizontal and \(D\) below \(A C\). Find, in the form \(k \sqrt { } \left( \frac { g } { a } \right)\), the greatest angular speed in the subsequent motion, giving the value of \(k\) correct to 3 decimal places.
[0pt] [4]

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform plane object consists of three identical circular rings, \(X\), \(Y\) and \(Z\), enclosed in a larger circular ring \(W\). Each of the inner rings has mass \(m\) and radius \(r\). The outer ring has mass \(3m\) and radius \(R\). The centres of the inner rings lie at the vertices of an equilateral triangle of side \(2r\). The outer ring touches each of the inner rings and the rings are rigidly joined together. The fixed axis \(AB\) is the diameter of \(W\) that passes through the centre of \(X\) and the point of contact of \(Y\) and \(Z\) (see diagram). It is given that \(R = \left(1 + \frac{2}{3}\sqrt{3}\right)r\).

1. Show that the moment of inertia of the object about \(AB\) is \(\left(7 + 2\sqrt{3}\right)mr^2\). [8]

The line \(CD\) is the diameter of \(W\) that is perpendicular to \(AB\). A particle of mass \(9m\) is attached to \(D\). The object is now held with its plane horizontal. It is released from rest and rotates freely about the fixed horizontal axis \(AB\).

1. Find, in terms of \(g\) and \(r\), the angular speed of the object when it has rotated through \(60°\). [6]

**OR** Fish of a certain species live in two separate lakes, \(A\) and \(B\). A zoologist claims that the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). To test his claim, he catches a random sample of 8 fish from \(A\) and a random sample of 6 fish from \(B\). The lengths of the 8 fish from \(A\), in appropriate units, are as follows. $$15.3 \quad 12.0 \quad 15.1 \quad 11.2 \quad 14.4 \quad 13.8 \quad 12.4 \quad 11.8$$ Assuming a normal distribution, find a 95% confidence interval for the mean length of fish in \(A\). [5] The lengths of the 6 fish from \(B\), in the same units, are as follows. $$15.0 \quad 10.7 \quad 13.6 \quad 12.4 \quad 11.6 \quad 12.6$$ Stating any assumptions that you make, test at the 5% significance level whether the mean length of fish in \(A\) is greater than the mean length of fish in \(B\). [7] Calculate a 95% confidence interval for the difference in the mean lengths of fish from \(A\) and from \(B\). [2]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\includegraphics{figure_1} A seesaw in a playground consists of a beam \(AB\) of length \(4\) m which is supported by a smooth pivot at its centre \(C\). Jill has mass \(25\) kg and sits on the end \(A\). David has mass \(40\) kg and sits at a distance \(x\) metres from \(C\), as shown in Figure 1. The beam is initially modelled as a uniform rod. Using this model,

1. find the value of \(x\) for which the seesaw can rest in equilibrium in a horizontal position. [3]
2. State what is implied by the modelling assumption that the beam is uniform. [1]

David realises that the beam is not uniform as he finds that he must sit at a distance \(1.4\) m from \(C\) for the seesaw to rest horizontally in equilibrium. The beam is now modelled as a non-uniform rod of mass \(15\) kg. Using this model,

1. find the distance of the centre of mass of the beam from \(C\). [4]

When the object is suspended from \(A\), the angle between \(AB\) and the vertical is \(\theta\), where \(\tan \theta = \frac{1}{2}\).

1. Given that \(h = \frac{8}{3}a\), find the possible values of \(k\). [3]

\includegraphics{figure_4} A uniform lamina of weight 15 N is in the form of a trapezium \(ABCD\) with dimensions as shown in the diagram. The lamina is freely hinged at \(A\) to a fixed point. One end of a light inextensible string is attached to the lamina at \(B\). The lamina is in equilibrium with \(AB\) horizontal; the string is taut and in the same vertical plane as the lamina, and makes an angle of \(30°\) upwards from the horizontal (see diagram). Find the tension in the string. [5]

\includegraphics{figure_2} A uniform steel girder \(AB\), of mass 40 kg and length 3 m, is freely hinged at \(A\) to a vertical wall. The girder is supported in a horizontal position by a steel cable attached to the girder at \(B\). The other end of the cable is attached to the point \(C\) vertically above \(A\) on the wall, with \(\angle ABC = \alpha\), where \(\tan \alpha = \frac{4}{3}\). A load of mass 60 kg is suspended by another cable from the girder at the point \(D\), where \(AD = 2\) m, as shown in Fig. 2. The girder remains horizontal and in equilibrium. The girder is modelled as a rod, and the cables as light inextensible strings.

1. Show that the tension in the cable \(BC\) is 980 N. [5]
2. Find the magnitude of the reaction on the girder at \(A\). [6]
3. Explain how you have used the modelling assumption that the cable at \(D\) is light. [1]

\includegraphics{figure_5} Two uniform rods, \(AB\) and \(BC\), each have length \(2a\) and weight \(W\). They are smoothly jointed at \(B\), and \(A\) is attached to a smooth fixed pivot. A light inextensible string of length \((2\sqrt{2})a\) joins \(A\) to \(C\) so that angle \(ABC = 90°\). The system hangs in equilibrium, with \(AB\) making an angle \(\alpha\) with the vertical (see diagram). By taking moments about \(A\) for the system, or otherwise, show that \(\alpha = 18.4°\), correct to the nearest \(0.1°\). [3] Find the tension in the string in the form \(kW\), giving the value of \(k\) correct to 3 significant figures. [3] Find, in terms of \(W\), the magnitude of the force acting on the rod \(BC\) at \(B\). [6]

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. \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 \(D\).

1. Show that \(\mu = \frac { 2 } { 3 }\). A small object of weight \(a W \mathrm {~N}\) is placed on the ladder at its mid-point and the object \(S\) of weight \(2 W \mathrm {~N}\) is placed on the ladder at its lowest point \(A\).
2. Given that the system is in equilibrium, find the set of possible values of \(a\).

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.

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 \(\mu\).

2 A particle of mass 8 kg is suspended in equilibrium by two light inextensible strings which make angles of \(60 ^ { \circ }\) and \(45 ^ { \circ }\) above the horizontal.

1. Draw a diagram showing the forces acting on the particle.
2. Find the tensions in the strings.

1 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).

11 Three light inextensible strings \(A C , C D\) and \(D B\), each of length 10 cm , are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-5_300_670_475_699} The ends \(A\) and \(B\) are fixed to points 20 cm apart on the same horizontal level. Two heavy particles, each of mass 2 kg , are attached at \(C\) and \(D\). The system remains in a vertical plane.

1. Determine the tension in each string.
2. The string \(C D\) is replaced by one of length \(L \mathrm {~cm}\), made of the same material. If the tension in \(A C\) is 50 N , show that \(L = 20 - 4 \sqrt { 21 }\).

4 \(A\) B \(A \quad B\) \(\begin{array} { l l } B & \\ A & B \end{array}\) \(P B\) $$P \theta$$ \(\theta\) P \(\theta\) L \(P\) \(P\)

1. (i)
   P \(\theta\) \(P \quad \underline { \theta }\)
2. - \(\underline { \theta }\) \(5 \theta\) $$\begin{gathered} \\ \theta \end{gathered} \quad P$$
3. \(P\)

4 \(A\) B \(A \quad B\) \(\begin{array} { l l } B & \\ A & B \end{array}\) \(P B\) $$P \theta$$ \(\theta\) P \(\theta\) L \(P\) \(P\)

1. (i)
   P \(\theta\) \(P \quad \underline { \theta }\)
2. - \(\underline { \theta }\) \(5 \theta\) $$\begin{gathered} \\ \theta \end{gathered} \quad P$$
3. \(P\)

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The \(\operatorname { rod } A B\) has length \(2 a\) and mass \(4 m\), and the \(\operatorname { rod } B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium.
   1. A smooth uniform sphere \(S\), of mass \(m\), is moving on a smooth horizontal plane when it collides obliquely with another smooth uniform sphere \(T\), of the same radius as \(S\) but of mass \(2 m\), which is at rest on the plane. Immediately before the collision the velocity of \(S\) makes an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), with the line joining the centres of the spheres. Immediately after the collision the speed of \(T\) is \(V\). The coefficient of restitution between the spheres is \(\frac { 3 } { 4 }\).
   2. Find, in terms of \(V\), the speed of \(S\)
      1. immediately before the collision,
      2. immediately after the collision.
   3. Find the angle through which the direction of motion of \(S\) is deflected as a result of the collision.
      

\includegraphics{figure_1} A non-uniform rod \(AB\), of length 0.6 m and weight 9 N, has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30°\) to the horizontal. The tension in the string is \(T\) N (see diagram).

1. Calculate \(T\). [2]
2. Find the least possible value of the coefficient of friction at \(A\). [3]

1 \includegraphics[max width=\textwidth, alt={}, center]{d1f1f036-1676-443e-b733-ca1fe79972d4-2_334_679_258_731} A non-uniform \(\operatorname { rod } A B\), of length 0.6 m and weight 9 N , has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\) (see diagram).

1. Calculate \(T\).
2. Find the least possible value of the coefficient of friction at \(A\).

\includegraphics{figure_4} A uniform rod \(AB\) of length \(4a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac{5}{2}a\) has one end attached to the point \(C\) on the rod, where \(AC = \frac{3}{2}a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the rod \(AB\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\). [10]

3 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

3 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

A uniform rod \(AB\), of length \(2a\) and mass \(2m\), can rotate freely in a vertical plane about a smooth horizontal axis through \(A\). A small rough ring of mass \(m\) is threaded on the rod. The rod is held in a horizontal position with the ring at rest at the mid-point of the rod. The rod is released from rest. Using energy considerations, show that, until the ring slides, $$a\dot{\theta}^2 = \frac{18}{11}g \sin \theta,$$ where \(\theta\) is the angle turned through by the rod. [3] Show that, until the ring slides, the magnitudes of the friction force and normal contact force acting on the ring are \(\frac{20}{11}mg \sin \theta\) and \(\frac{2}{11}mg \cos \theta\) respectively. [6] The coefficient of friction between the ring and the rod is \(\mu\). Find, in terms of \(\mu\), the value of \(\theta\) when the ring starts to slide. [2]

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of length 2 m and mass 6 kg inclined at an angle of \(30 ^ { \circ }\) to the horizontal with \(A\) on smooth horizontal ground and \(B\) supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between \(B\) and the peg is \(\mu\).

1. Find, in terms of \(g\), the magnitude of the reactions at \(A\) and \(B\).
2. Show that \(\mu = \frac { 1 } { \sqrt { 3 } }\).

\includegraphics{figure_5} A uniform rod \(AB\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).

1. Show that the tension in the string is 4.5 N. [4]
2. Find the magnitude and direction of the force acting on the rod at \(A\). [6]

\includegraphics{figure_1} A uniform rod \(AB\) has length \(2a\) and mass \(M\). The rod is held in equilibrium in a horizontal position by two vertical light strings which are attached to the rod at \(C\) and \(D\), where \(AC = \frac{2}{5}a\) and \(DB = \frac{3}{5}a\), as shown in Figure 1. A particle \(P\) is placed on the rod at \(B\). The rod remains horizontal and in equilibrium.

1. Find, in terms of \(M\), the largest possible mass of the particle \(P\) [3] Given that the mass of \(P\) is \(\frac{1}{2}M\)
2. Find, in terms of \(M\) and \(g\), the tension in the string that is attached to the rod at \(C\). [3]

A non-uniform ladder \(AB\), of length \(3a\), has its centre of mass at \(G\), where \(AG = 2a\). The ladder rests in limiting equilibrium with the end \(B\) against a smooth vertical wall and the end \(A\) resting on rough horizontal ground. The angle between \(AB\) and the horizontal in this position is \(\alpha\), where \(\tan \alpha = \frac{14}{9}\). \includegraphics{figure_3} Calculate the coefficient of friction between the ladder and the ground. [7 marks]

3 \includegraphics[max width=\textwidth, alt={}, center]{e85c2bf4-21a8-4d9a-93c5-d5679b2a8233-2_497_951_1123_598} A uniform beam \(A B\) has weight 70 N and length 2.8 m . The beam is freely hinged to a wall at \(A\) and is supported in a horizontal position by a strut \(C D\) of length 1.3 m . One end of the strut is attached to the beam at \(C , 0.5 \mathrm {~m}\) from \(A\), and the other end is attached to the wall at \(D\), vertically below \(A\). The strut exerts a force on the beam in the direction \(D C\). The beam carries a load of weight 50 N at its end \(B\) (see diagram).

1. Calculate the magnitude of the force exerted by the strut on the beam.
2. Calculate the magnitude of the force acting on the beam at \(A\).

12 With the aid of a suitable diagram, show that the three triangles referred to in line 26 have the areas given in line 27 .