3.03m Equilibrium: sum of resolved forces = 0

3 Fixed points A and B are 4.8 m apart on the same horizontal level. The midpoint of AB is M . A light elastic string, with natural length 3.9 m and modulus of elasticity 573.3 N , has one end attached to A and the other end attached to \(\mathbf { B }\).

1. Find the elastic energy stored in the string. A particle P is attached to the midpoint of the string, and is released from rest at M . It comes instantaneously to rest when P is 1.8 m vertically below M .
2. Show that the mass of P is 15 kg .
3. Verify that P can rest in equilibrium when it is 1.0 m vertically below M . In general, a light elastic string, with natural length \(a\) and modulus of elasticity \(\lambda\), has its ends attached to fixed points which are a distance \(d\) apart on the same horizontal level. A particle of mass \(m\) is attached to the midpoint of the string, and in the equilibrium position each half of the string has length \(h\), as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5ecb198d-7863-4fc2-81b6-c8b6c37b1859-4_280_755_1064_696} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure} When the particle makes small oscillations in a vertical line, the period of oscillation is given by the formula $$\sqrt { \frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } } } m ^ { \alpha } a ^ { \beta } \lambda ^ { \gamma }$$
4. Show that \(\frac { 8 \pi ^ { 2 } h ^ { 3 } } { 8 h ^ { 3 } - a d ^ { 2 } }\) is dimensionless.
5. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
6. Hence find the period when the particle P makes small oscillations in a vertical line centred on the position of equilibrium given in part (iii).

2. A particle \(P\) is attached to one end of a light elastic string of modulus of elasticity 80 N . The other end of the string is attached to a fixed point \(A\). \begin{figure}[h] \end{figure} When a horizontal force of magnitude 20 N is applied to \(P\), it rests in equilibrium with the string making an angle of \(30 ^ { \circ }\) with the vertical and \(A P = 1.2 \mathrm {~m}\) as shown in Figure 1.

1. Find the tension in the string.
2. Find the elastic potential energy stored in the string.

3. \begin{figure}[h] \end{figure} A popular racket game involves a tennis ball of mass 0.1 kg which is attached to one end of a light inextensible string. The other end of the string is attached to the top of a fixed rigid pole. A boy strikes the ball such that it moves in a horizontal circle with angular speed \(4 \mathrm { rad } \mathrm { s } ^ { - 1 }\) and the string makes an angle of \(60 ^ { \circ }\) with the downward vertical as shown in Figure 1.

1. Find the tension in the string.
2. Find the length of the string.

5. A physics student is set the task of finding the mass of an object without using a set of scales. She decides to use a light elastic string of natural length 2 m and modulus of elasticity 280 N attached to two points \(A\) and \(B\) which are on the same horizontal level and 2.4 m apart. \begin{figure}[h] \end{figure} She attaches the object to the midpoint of the string so that it hangs in equilibrium 0.35 m below \(A B\) as shown in Figure 2.

1. Explain why it is reasonable to assume that the tensions in each half of the string are equal.
2. Find the mass of the object.
3. Find the elastic potential energy of the string when the object is suspended from it.

1 Forces of magnitude \(4 \mathrm {~N} , 3 \mathrm {~N} , 5 \mathrm {~N}\) and \(R \mathrm {~N}\) act on a particle in the directions shown in Fig. 1. \begin{figure}[h] \end{figure} The particle is in equilibrium. Find each of the following.

• The value of \(R\).
• The value of \(\theta\).

6 A uniform rod AB has length \(2 a\) and weight \(W\). The rod is in equilibrium in a horizontal position. The end A rests on a smooth plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The force exerted on AB by the plane is \(R\). The end B is attached to a light inextensible string inclined at an angle of \(\theta\) to AB as shown in Fig. 6. The rod and string are in the same vertical plane, which also contains the line of greatest slope of the plane on which A lies. The tension in the string is \(T\). \begin{figure}[h] \end{figure}

1. Add the forces \(R\) and \(T\) to the copy of Fig. 6 in the Printed Answer Booklet.
2. By taking moments about B , find an expression for \(R\) in terms of \(W\).
3. By resolving horizontally, show that \(6 T \cos \theta = W \sqrt { 3 }\).
4. By finding a second equation connecting \(T\) and \(\theta\), determine

1

1. Fig. 1.1 and Fig. 1.2 show rigid rods with forces acting as marked. The diagrams are to scale, and in each figure the side length of a grid square is 1 metre. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_428_552_443_319} \captionsetup{labelformat=empty} \caption{Fig. 1.1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_431_553_440_1005} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
   • On the copy of Fig. 1.1 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents an anti-clockwise couple of magnitude 24 Nm .
   • On the copy of Fig. 1.2 in the Printed Answer Booklet, add, to scale, a force so that the overall system represents a clockwise couple of magnitude 1 Nm .
   • Fig. 1.3 shows a rectangular lamina with two coplanar forces acting as marked. Each grid square has side length 1 m .
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-2_561_761_1452_315} \captionsetup{labelformat=empty} \caption{Fig. 1.3}
   \end{figure} A third coplanar force, of magnitude \(T \mathrm {~N}\), acts at A so that the resultant force on the lamina is zero.
   1. Calculate the value of \(T\).
   2. Determine the magnitude and direction of the couple represented by this system of three forces.

2 Three forces, of magnitudes \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. The system is in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-3_501_703_342_239}

1. Draw a triangle of forces for the system shown above. Your diagram should include the magnitudes of the forces ( \(33 \mathrm {~N} , 45 \mathrm {~N}\) and \(P \mathrm {~N}\) ) and angle \(\theta\).
2. If \(P = 38\), find, in degrees, the value of \(\theta\).
3. If \(\theta = 40 ^ { \circ }\), determine the possible values for \(P\).

1 Two horizontal forces of magnitudes 7 N and 15 N act at a point O .
The 15 N force acts an angle of \(\theta ^ { \circ }\) above the positive \(x\)-axis.
The 7 N force acts at an angle of \(70 ^ { \circ }\) below the negative \(x\)-axis (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-2_606_773_402_239} The resultant of the two forces acts only in the positive \(x\)-direction.

1. Calculate the value of \(\theta\).
2. Calculate the magnitude of the resultant of the two forces.

3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
2. 1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
   2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
4. Find the range of possible values for \(\mu\).
5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).

5 Jack and Jemima are pulling a boat along a straight level canal.
The resistance to the motion of the boat is modelled as constant and equal to 1200 N .
Jack and Jemima walk in the same direction on paths on opposite sides of the canal. They each walk forwards at the same steady speed, keeping level with each other so that the distance between them is always 6 m . Jack and Jemima each pull a long light inextensible rope attached to the boat; initially they hold their ropes so the distance from each of them to the boat is 5 m , as shown in Fig. 5.1. \begin{figure}[h] \end{figure}

1. Explain why the tension will be the same in each rope.
2. Find the tension in each rope. Jemima then gradually releases more rope, so that the distance between her and the boat is 7 m . Jack and Jemima continue to walk at the same steady speed along the paths, but the position of the boat changes so that Jemima's rope makes an angle of \(\theta\) with the path and Jack's rope makes an angle of \(\phi\) with the path, as shown in Fig. 5.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-4_513_1109_1610_246} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
3. - Show that \(\sin \phi = \frac { 1 } { 5 }\).

2 The diagram below shows the cross-section through the centre of mass of a uniform block of weight \(W \mathrm {~N}\), resting on a slope inclined at an angle \(\alpha\) to the horizontal. The cross-section is a rectangle ABCD . The slope exerts a frictional force of magnitude \(F \mathrm {~N}\) and a normal contact force of magnitude \(R \mathrm {~N}\). \includegraphics[max width=\textwidth, alt={}, center]{9b624694-edb6-4000-838f-3557e078952d-3_546_940_450_242}

1. Explain why a triangle of forces may be used to model the scenario.
2. In the space provided in the Printed Answer Booklet, draw such a triangle, fully annotated, including the angle \(\alpha\) in the correct position. The coefficient of friction between the block and the slope is \(\mu\).
3. Given that the block is in limiting equilibrium, use your diagram in part (b) to show that \(\mu = \tan \alpha\). It is given that \(\mathrm { AB } = 8.9 \mathrm {~cm}\) and \(\mathrm { AD } = 11.6 \mathrm {~cm}\). The coefficient of friction between the slope and the block is 1.35 . The slope is slowly tilted so that \(\alpha\) increases.
4. Determine whether the block topples first without sliding or slides first without toppling.

5 Fig. 5.1 shows a particle P, of mass 5 kg , and a particle Q, of mass 11 kg , which are attached to the ends of a light, inextensible string. The string is taut and passes over a small smooth pulley fixed to the ceiling. \begin{figure}[h] \end{figure} When a force of magnitude \(H \mathrm {~N}\), acting at an angle \(\theta\) to the upward vertical, is applied to Q the particles hang in equilibrium, with the part of the string connecting the pulley to Q making an angle of \(40 ^ { \circ }\) with the upward vertical. It is given that the force acts in the same vertical plane in which the string lies.

1. Determine the values of \(H\) and \(\theta\). Particle Q is now removed. The string is instead attached to one end of a uniform beam B of length 3 m and mass 7 kg . The other end of B is in contact with a rough horizontal floor. The situation is shown in Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-5_504_978_1557_251}
   \end{figure} With B in equilibrium, at an angle \(\phi\) to the horizontal, the part of the string connecting the pulley to B makes an angle of \(30 ^ { \circ }\) with the upward vertical. It is given that the string and B lie in the same vertical plane.
2. Determine the smallest possible value for the coefficient of friction between B and the floor.
3. Determine the value of \(\phi\).

8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h] \end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.

1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).

1. The diagram below shows four coplanar horizontal forces of magnitude \(F \mathrm {~N} , 12 \mathrm {~N} , 16 \mathrm {~N}\) and 20 N acting at a point \(P\) in the directions shown. \includegraphics[max width=\textwidth, alt={}, center]{8f47b2ff-f954-42ec-8ecc-fc64313a7b89-14_792_862_593_607}

Given that the forces are in equilibrium, calculate the value of \(F\) and the size of the angle \(\alpha\). [7]

6. The diagram shows a particle \(P\), of mass 4 kg , lying on a smooth horizontal surface. It is attached by two light springs to fixed points \(A\) and \(B\), where \(A B = 2.8 \mathrm {~m}\).
Spring \(A P\) has natural length 0.8 m and modulus of elasticity 60 N .
Spring \(P B\) has natural length 1.2 m and modulus of elasticity 30 N . \includegraphics[max width=\textwidth, alt={}, center]{b9c63cb4-d446-4548-be42-e30b10cb4b99-5_231_1253_612_404} When \(P\) is in equilibrium, it is at the point \(C\).

1. Show that \(A C = 1 \mathrm {~m}\).
2. The particle \(P\) is pulled horizontally and is initially held at rest at the midpoint of \(A B\). The system is then released.
   1. Show that \(P\) performs Simple Harmonic Motion about centre \(C\) and find the period of its motion.
   2. Determine the shortest time taken for \(P\) to reach a position where there is no tension in the spring \(A P\). \section*{END OF PAPER}

6. The diagram shows a playground ride consisting of a seat \(P\), of mass 12 kg , attached to a vertical spring, which is fixed to a horizontal board. When the ride is at rest with nobody on it, the compression of the spring is 0.05 m . \includegraphics[max width=\textwidth, alt={}, center]{3efc4ef6-8a80-4267-8e95-733200e875c5-4_305_654_1032_667} The spring is of natural length 0.75 m and modulus of elasticity \(\lambda\).

1. Find the value of \(\lambda\). The seat \(P\) is now pushed vertically downwards a further 0.05 m and is then released from rest.
2. Show that \(P\) makes Simple Harmonic oscillations of period \(\frac { \pi } { 7 }\) and write down the amplitude of the motion.
3. Find the maximum speed of \(P\).
4. Calculate the speed of \(P\) when it is at a distance 0.03 m from the equilibrium position.
5. Find the distance of \(P\) from the equilibrium position 1.6 s after it is released.[3]
6. State one modelling assumption you have made about the seat and one modelling assumption you have made about the spring.

5 In this question use \(\boldsymbol { g } = 9.8 \mathbf { m ~ s } ^ { \mathbf { - 2 } }\).
A conical pendulum consists of a string of length 60 cm and a particle of mass 400 g . The string is at an angle of \(30 ^ { \circ }\) to the vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-08_501_606_644_854} 5

1. Show that the tension in the string is 4.5 N . 5
2. Find the angular speed of the particle.
   [0pt] [3 marks]
   5
3. State two assumptions that you have made about the string.

1 A train consists of a locomotive pulling 17 identical trucks. The mass of the locomotive is 120 tonnes and the mass of each truck is 40 tonnes. The locomotive gives a driving force of 121000 N . The resistance to motion on each truck is \(R \mathrm {~N}\) and the resistance on the locomotive is \(5 R \mathrm {~N}\).
Initially the train is travelling on a straight horizontal track and its acceleration is \(0.11 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 1500\).
2. Find the tensions in the couplings between
   (A) the last two trucks,
   (B) the locomotive and the first truck. The train now comes to a place where the track goes up a straight, uniform slope at an angle \(\alpha\) with the horizontal, where \(\sin \alpha = \frac { 1 } { 80 }\). The driving force and the resistance forces remain the same as before.
3. Find the magnitude and direction of the acceleration of the train. The train then comes to a straight uniform downward slope at an angle \(\beta\) to the horizontal.
   The driver of the train reduces the driving force to zero and the resistance forces remain the same as before. The train then travels at a constant speed down the slope.
4. Find the value of \(\beta\).

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.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

1. the normal reaction of the plane on \(P\),
2. the value of \(X\). The force of magnitude \(X\) newtons is now removed.
3. Show that \(P\) remains in equilibrium on the plane.