3.03k Connected particles: pulleys and equilibrium

2 A box of mass 8 kg slides on a horizontal table against a constant resistance of 11.2 N .

1. What horizontal force is applied to the box if it is sliding with acceleration of magnitude \(2 \mathrm {~ms} ^ { - 2 }\) ? Fig. 7 shows the box of mass 8 kg on a long, rough, horizontal table. A sphere of mass 6 kg is attached to the box by means of a light inextensible string that passes over a smooth pulley. The section of the string between the pulley and the box is parallel to the table. The constant frictional force of 11.2 N opposes the motion of the box. A force of 105 N parallel to the table acts on the box in the direction shown, and the acceleration of the system is in that direction. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{0fbef619-ad15-4e46-be35-e17fed9952c0-2_372_878_870_683} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
2. What information in the question indicates that while the string is taut the box and sphere have the same acceleration?
3. Draw two separate diagrams, one showing all the horizontal forces acting on the box and the other showing all the forces acting on the sphere.
4. Show that the magnitude of the acceleration of the system is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and find the tension in the string. The system is stationary when the sphere is at point P . When the sphere is 1.8 m above P the string breaks, leaving the sphere moving upwards at a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. (A) Write down the value of the acceleration of the sphere after the string breaks.
   (B) The sphere passes through P again at time \(T\) seconds after the string breaks. Show that \(T\) is the positive root of the equation \(4.9 T ^ { 2 } - 3 T - 1.8 = 0\).
   ( \(C\) ) Using part ( \(B\) ), or otherwise, calculate the total time that elapses after the sphere moves from P before the sphere again passes through P .

2 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 \(\beta\).

1 Fig. 2 shows a 6 kg block on a smooth horizontal table. It is connected to blocks of mass 2 kg and 9 kg by two light strings which pass over smooth pulleys at the edges of the table. The parts of the strings attached to the 6 kg block are horizontal. \begin{figure}[h] \end{figure}

1. Draw three separate diagrams showing all the forces acting on each of the blocks.
2. Calculate the acceleration of the system and the tension in each string.

3 Fig. 7 illustrates a train with a locomotive, L, pulling two trucks, A and B. The locomotive has mass 90 tonnes and is subject to a resistance force of 2000 N .
Each of the trucks \(A\) and \(B\) has mass 30 tonnes and is subject to a resistance force of \(500 N\). \begin{figure}[h] \end{figure} Initially the train is travelling along a straight horizontal track. The locomotive is exerting a driving force of 12000 N .

1. Find the acceleration of the train.
2. Find the tension in the coupling between trucks A and B . When the train is travelling at \(10 \mathrm {~ms} ^ { - 1 }\), a fault occurs with truck A and the resistance to its motion changes from 500 N to 5000 N . The driver reduces the driving force to zero and allows the train to slow down under the resistance forces and come to a stop.
3. Find the distance the train travels while slowing down and coming to a stop. Find also the force in the coupling between trucks A and B while the train is slowing down, and state whether it is a tension or a thrust. The fault in truck A is repaired so that the resistance to its motion is again 500 N . The train continues and comes to a place where the track goes up a uniform slope at an angle of \(\alpha ^ { \circ }\) to the horizontal.
4. When the train is on the slope, it travels at uniform speed. The driving force remains at 12000 N . Find the value of \(\alpha\).
5. Show that the force in the coupling between trucks A and B has the same value that it had in part (ii).

4 Fig. 5 shows blocks of mass 4 kg and 6 kg on a smooth horizontal table. They are connected by a light, inextensible string. As shown, a horizontal force \(F \mathrm {~N}\) acts on the 4 kg block and a horizontal force of 30 N acts on the 6 kg block. The magnitude of the acceleration of the system is \(2 \mathrm {~ms} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Find the two possible values of \(F\).
2. Find the tension in the string in each case.

3 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h] \end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.

1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
2. Calculate the tension in the rope.

4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards and the tension in the wire is \(T \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
2. Write down separate equations of motion for box P and for box Q .
3. Calculate the tension in the wire.

2 Boxes A and B slide on a smooth, horizontal plane. Box A has a mass of 4 kg and box B a mass of 5 kg . They are connected by a light, inextensible, horizontal wire. Horizontal forces of 9 N and 135 N act on A and B in the directions shown in Fig. 5. \begin{figure}[h] \end{figure} Calculate the tension in the wire joining the boxes.

3 Fig. 3 shows a system in equilibrium. The rod is firmly attached to the floor and also to an object, P. The light string is attached to P and passes over a smooth pulley with an object Q hanging freely from its other end. \begin{figure}[h] \end{figure}

1. Why is the tension the same throughout the string?
2. Calculate the force in the rod, stating whether it is a tension or a thrust.

4 Two trucks, A and B, each of mass 10000 kg , are pulled along a straight, horizontal track by a constant, horizontal force of \(P \mathrm {~N}\). The coupling between the trucks is light and horizontal. This situation and the resistances to motion of the trucks are shown in Fig. 4. \begin{figure}[h] \end{figure} The acceleration of the system is \(0.2 \mathrm {~ms} ^ { 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck A is now subjected to an extra resistive force of 2000 N while \(P\) does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

5 A train consists of an engine of mass 10000 kg pulling one truck of mass 4000 kg . The coupling between the engine and the truck is light and parallel to the track. The train is accelerating at \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\) along a straight, level track.

1. What is the resultant force on the train in the direction of its motion? The driving force of the engine is 4000 N .
2. What is the resistance to the motion of the train?
3. If the tension in the coupling is 1150 N , what is the resistance to the motion of the truck? With the same overall resistance to motion, the train now climbs a uniform slope inclined at \(3 ^ { \circ }\) to the horizontal with the same acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { 2 }\).
4. What extra driving force is being applied?

6 A box of weight 147 N is held by light strings AB and BC . As shown in Fig. 7.1, AB is inclined at \(\alpha\) to the horizontal and is fixed at \(\mathrm { A } ; \mathrm { BC }\) is held at C . The box is in equilibrium with BC horizontal and \(\alpha\) such that \(\sin \alpha = 0.6\) and \(\cos \alpha = 0.8\). \begin{figure}[h] \end{figure}

1. Calculate the tension in string AB .
2. Show that the tension in string BC is 196 N . As shown in Fig. 7.2, a box of weight 90 N is now attached at C and another light string CD is held at D so that the system is in equilibrium with BC still horizontal. CD is inclined at \(\beta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-4_378_695_1282_687} \captionsetup{labelformat=empty} \caption{Fig. 7.2}
   \end{figure}
3. Explain why the tension in the string BC is still 196 N .
4. Draw a diagram showing the forces acting on the box at C . Find the angle \(\beta\) and show that the tension in CD is 216 N , correct to three significant figures. The string section CD is now taken over a smooth pulley and attached to a block of mass \(M \mathrm {~kg}\) on a rough slope inclined at \(40 ^ { \circ }\) to the horizontal. As shown in Fig. 7.3, the part of the string attached to the box is still at \(\beta\) to the horizontal and the part attached to the block is parallel to the slope. The system is in equilibrium with a frictional force of 20 N acting on the block up the slope. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-5_436_1049_524_536} \captionsetup{labelformat=empty} \caption{Fig. 7.3}
   \end{figure}
5. Calculate the value of \(M\).

9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.

1. 1. Show that initially particle \(A\) moves towards the hole in the table.
   2. Show that initially particle \(B\) also moves towards the hole in the table.
2. Calculate the initial elastic potential energy in the string.
3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.

2 Fig. 2 shows a wedge of angle \(30 ^ { \circ }\) fixed to a horizontal floor. Small objects P , of mass 8 kg , and Q , of mass 10 kg , are connected by a light inextensible string that passes over a smooth pulley at the top of the wedge. The part of the string between P and the pulley is parallel to a line of greatest slope of the wedge. Q hangs freely and at no time does either P or Q reach the pulley or P reach the floor. \begin{figure}[h] \end{figure}

1. Assuming the string remains taut, find the change in the gravitational potential energy of the system when Q descends \(h \mathrm {~m}\), stating whether it is a loss or a gain. Object P makes smooth contact with the wedge. The system is set in motion with the string taut.
2. Find the speed at which Q hits the floor if
   (A) the system is released from rest with Q a distance of 1.2 m above the floor,
   (B) instead, the system is set in motion with Q a distance of 0.3 m above the floor and P travelling down the slope at \(1.05 \mathrm {~ms} ^ { - 1 }\). The sloping face is roughened so that the coefficient of friction between object P and the wedge is 0.9 . The system is set in motion with the string taut and P travelling down the slope at \(2 \mathrm {~ms} ^ { - 1 }\).
3. How far does P move before it reaches its lowest point?
4. Determine what happens to the system after P reaches its lowest point.
5. Calculate the power of the frictional force acting on P in part (iii) at the moment the system is set in motion. \section*{Question 3 begins on page 4.}

4 The diagram shows two particles P and Q , of masses 10 kg and 5 kg respectively, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. The pulley is fixed at the highest point A on a smooth curved surface, the vertical cross-section of which is a quadrant of a circle with centre O and radius 2 m . Particle Q hangs vertically below the pulley and P is in contact with the surface, where the angle AOP is equal to \(\theta ^ { \circ }\). The pulley, P and Q all lie in the same vertical plane. \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-4_499_492_559_251} Throughout this question you may assume that there are no resistances to the motion of either P or Q and the force acting on P due to the tension in the string is tangential to the curved surface at P .

1. Given that P is in equilibrium at the point where \(\theta = \alpha\), determine the value of \(\alpha\). Particle P is now released from rest at the point on the surface where \(\theta = 35\), and starts to move downwards on the surface. In the subsequent motion it is given that P does not leave the surface.
2. By considering energy, determine the speed of P at the instant when \(\theta = 45\).
3. State one modelling assumption you have made in determining the answer to part (b).

2 A car of mass 800 kg is driven with its engine generating a power of 15 kW .

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds.
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude 150 N . Given that \(\sin \theta = \frac { 1 } { 20 }\), find the speed of the car.
3. The car is now driven at a constant speed of \(30 \mathrm {~ms} ^ { - 1 }\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming that the resistance to motion of the car is three times the resistance to motion of the trailer, find

5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\),
2. the tension in the string,
3. the value of \(\mu\).
4. State how in your calculations you have used the information that the string is inextensible.

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

7 A builder ties two identical buckets, \(P\) and \(Q\), to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .

1. 1. State the magnitude of the tension in the rope.
   2. State the magnitude and direction of the force exerted on the beam by the rope.
2. The bucket \(Q\) is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket \(Q\) moves vertically downwards with the stone inside.
   1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
   2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

6 A small train at an amusement park consists of an engine and two carriages connected to each other by light horizontal rods, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-4_190_1038_420_493} The engine has mass 2000 kg and each carriage has mass 500 kg . The train moves along a straight horizontal track. A resistance force of magnitude 400 newtons acts on the engine, and resistance forces of magnitude 300 newtons act on each carriage. The train is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Draw a diagram to show the horizontal forces acting on Carriage 2.
2. Show that the magnitude of the force that the rod exerts on Carriage 2 is 550 newtons.
3. Find the magnitude of the force that the rod attached to the engine exerts on Carriage 1.
4. A forward driving force of magnitude \(P\) newtons acts on the engine. Find \(P\).

9. \begin{figure}[h] \end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

1. find, in terms of \(m\) and \(g\), the tension in the string,
2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
3. find the value of \(k\).
4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).