3.03m Equilibrium: sum of resolved forces = 0

3 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

3 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).

4 \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

2 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-2_309_520_941_744} A particle rests on a smooth horizontal surface. Three horizontal forces of magnitudes \(2.5 \mathrm {~N} , F \mathrm {~N}\) and 2.4 N act on the particle on bearings \(\theta ^ { \circ } , 180 ^ { \circ }\) and \(270 ^ { \circ }\) respectively (see diagram). The particle is in equilibrium.

1. Find \(\theta\) and \(F\). The 2.4 N force suddenly ceases to act on the particle, which has mass 0.2 kg .
2. Find the magnitude and direction of the acceleration of the particle.

2 \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).

2. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find

1. the magnitude of the reaction on the rod at \(Q\),
2. the distance \(A Q\).
   . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_190_476_964_1905} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
   1. Find the size of the angle \(\alpha\).
      (6 marks)
   2. Find the value of \(T\). You must ensure that your answers to parts of questions are clearly labelled.
      You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.

5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical. 7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)

5 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) 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 .
2. Find the magnitude and direction of the force acting on the rod at \(A\).

10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.

1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
3. Find
   1. the magnitude of the resultant of the two remaining forces,
   2. the direction of the resultant of the two remaining forces.

9 Three forces \(\binom { 7 } { - 6 } \mathrm {~N} , \binom { 2 } { 5 } \mathrm {~N}\) and \(\mathbf { F N }\) act on a particle.
Given that the particle is in equilibrium under the action of these three forces, calculate \(\mathbf { F }\).

8 A particle is in equilibrium under the action of the following three forces: \(( 2 p \mathbf { i } - 4 \mathbf { j } ) N , ( - 3 q \mathbf { i } + 5 p \mathbf { j } ) N\) and \(( - 13 \mathbf { i } - 6 \mathbf { j } ) N\).
Find the values of p and q .

9 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-07_609_894_495_242} A body remains at rest when subjected to the horizontal and vertical forces shown in the diagram.
Determine the value of \(F _ { 1 }\) and the value of \(F _ { 2 }\).

11 A block of mass 3 kg is at rest on a smooth horizontal table. It is attached to a light inextensible string which passes over a smooth pulley. This part of the string is horizontal. A sphere of mass 1.2 kg is attached to the other end of the string. The sphere hangs with this part of the string vertical as shown in the diagram. A horizontal force of magnitude \(F\) N is applied to the block to prevent motion. \includegraphics[max width=\textwidth, alt={}, center]{1d1e41f3-a834-4230-b6e1-4b0be9450d30-7_268_718_493_244}

1. Complete the copy of the diagram in the Printed Answer Booklet to show all the forces acting on the block and the sphere.
2. Find the value of \(F\). The force \(F\) N is removed, and the system begins to move.
3. The equation of motion of the block is \(\mathrm { T } = 3 \mathrm { a }\), where \(T \mathrm {~N}\) is the tension in the string and \(a \mathrm {~ms} ^ { - 2 }\) is the acceleration of the block. Write down the equation of motion of the sphere.
4. Find the value of \(T\).

12 A box hangs from a balloon by means of a light inelastic string. The string is always vertical. The mass of the box is 15 kg . Catherine initially models the situation by assuming that there is no air resistance to the motion of the box. Use Catherine's model to calculate the tension in the string if:

1. the box is held at rest by the tension in the string,
2. the box is instantaneously at rest and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. the box is moving downwards at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Catherine now carries out an experiment to find the magnitude of the air resistance on the box when it is moving.
   At a time when the box is accelerating downwards at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), she finds that the tension in the string is 140 N .
4. Calculate the magnitude of the air resistance at that time. Give, with a reason, the direction of motion of the box. \section*{END OF QUESTION PAPER}

5 A sphere of mass 3 kg hangs on a string. A horizontal force of magnitude \(F \mathrm {~N}\) acts on the sphere so that it hangs in equilibrium with the string making an angle of \(25 ^ { \circ }\) to the vertical. The force diagram for the sphere is shown below. \includegraphics[max width=\textwidth, alt={}, center]{9dd6fc6d-b51e-4a73-ace5-d26a7558032c-05_502_513_408_244}

1. Sketch the triangle of forces for these forces.
2. Hence or otherwise determine each of the following:
   Answer all the questions.
   Section B (76 marks)

5 Two ropes are attached to a load of mass 500 kg . The ropes make angles of \(30 ^ { \circ }\) and \(45 ^ { \circ }\) to the vertical, as shown in the diagram. The tensions in these ropes are \(T _ { 1 }\) and \(T _ { 2 }\) newtons. The load is also supported by a vertical spring. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_533_565_507_744} The system is in equilibrium and \(T _ { 1 } = 200\).

1. Show that \(T _ { 2 } = 141\), correct to three significant figures.
2. Find the force that the spring exerts on the load.

3 The diagram shows three forces which act in the same plane and are in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}

1. Find \(F\).
2. Find \(\alpha\).

6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.

1. Draw a diagram to show the forces acting on the trolley.
2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
3. 1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
   2. Make one criticism of the assumption that the resistance force on the trolley is constant.

3 A particle, of mass 4 kg , is suspended in equilibrium by two light strings, \(A P\) and \(B P\). The string \(A P\) makes an angle of \(30 ^ { \circ }\) to the horizontal and the other string, \(B P\), is horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-2_231_757_1841_639}

1. Draw and label a diagram to show the forces acting on the particle.
2. Show that the tension in the string \(A P\) is 78.4 N .
3. Find the tension in the horizontal string \(B P\).

5 A sledge of mass 8 kg is at rest on a rough horizontal surface. A child tries to move the sledge by pushing it with a pole, as shown in the diagram, but the sledge does not move. The pole is at an angle of \(30 ^ { \circ }\) to the horizontal and exerts a force of 40 newtons on the sledge. \includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-4_221_922_513_552} Model the sledge as a particle.

1. Draw a diagram to show the four forces acting on the sledge.
2. Show that the normal reaction force between the sledge and the surface has magnitude 98.4 N .
3. Find the magnitude of the friction force that acts on the sledge.
4. Find the least possible value of the coefficient of friction between the sledge and the surface.

6 Two particles, \(A\) and \(B\), are connected by a light inextensible string which passes over a smooth peg. Particle \(A\) has mass 2 kg and particle \(B\) has mass 4 kg . Particle \(A\) hangs freely with the string vertical. Particle \(B\) is at rest in equilibrium on a rough horizontal surface with the string at an angle of \(30 ^ { \circ }\) to the vertical. The particles, peg and string are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-14_419_953_571_541}

1. By considering particle \(A\), find the tension in the string.
2. Draw a diagram to show the forces acting on particle \(B\).
3. Show that the magnitude of the normal reaction force acting on particle \(B\) is 22.2 newtons, correct to three significant figures.
4. Find the least possible value of the coefficient of friction between particle \(B\) and the surface.
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-16_2486_1714_221_153}

2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}

1. Show that \(P = 9\).
2. Find the value of \(Q\).

8 The diagram shows a block, of mass 20 kg , being pulled along a rough horizontal surface by a rope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-16_323_1194_411_424} The coefficient of friction between the block and the surface is \(\mu\). Model the block as a particle which slides on the surface.

1. If the tension in the rope is 60 newtons, the block moves at a constant speed.
   1. Show that the magnitude of the normal reaction force acting on the block is 166 N .
   2. Find \(\mu\).
2. If the rope remains at the same angle and the block accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the tension in the rope. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-18_2488_1719_219_150} \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-19_2488_1719_219_150}