3.03o Advanced connected particles: and pulleys

4. \begin{figure}[h] \end{figure} Figure 2 shows a car towing a trailer along a straight horizontal road.
The mass of the car is 800 kg and the mass of the trailer is 600 kg .
The trailer is attached to the car by a towbar which is parallel to the road and parallel to the direction of motion of the car and the trailer. The towbar is modelled as a light rod.
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer is modelled as a constant force of magnitude R newtons. The engine of the car is producing a constant driving force that is horizontal and of magnitude 1740 N. The acceleration of the car is \(0.6 \mathrm {~ms} ^ { - 2 }\) and the tension in the towbar is T newtons.
Using the model,

1. show that \(\mathrm { R } = 500\)
2. find the value of T . At the instant when the speed of the car and the trailer is \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks.
   The trailer moves a further distance d metres before coming to rest.
   The resistance to the motion of the trailer is modelled as a constant force of magnitude 500 N. Using the model,
3. show that, after the towbar breaks, the deceleration of the trailer is \(\frac { 5 } { 6 } \mathrm {~ms} ^ { - 2 }\)
4. find the value of d. In reality, the distance d metres is likely to be different from the answer found in part (d).
5. Give two different reasons why this is the case.

10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}

1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.

11 Two small balls \(P\) and \(Q\) have masses 3 kg and 2 kg respectively. The balls are attached to the ends of a string. \(P\) is held at rest on a rough horizontal surface. The string passes over a pulley which is fixed at the edge of the surface. \(Q\) hangs vertically below the pulley at a height of 2 m above a horizontal floor. \includegraphics[max width=\textwidth, alt={}, center]{d6430776-0b87-4e5e-8f78-c6228ee163d5-7_346_906_445_255} The system is initially at rest with the string taut. A horizontal force of magnitude 40 N acts on \(P\) as shown in the diagram. \(P\) is released and moves directly away from the pulley. A constant frictional force of magnitude 8 N opposes the motion of \(P\). It is given that \(P\) does not leave the horizontal surface and that \(Q\) does not reach the pulley in the subsequent motion. The balls are modelled as particles, the pulley is modelled as being small and smooth, and the string is modelled as being light and inextensible.

1. Show that the magnitude of the acceleration of each particle is \(2.48 \mathrm {~ms} ^ { - 2 }\).
2. Find the tension in the string. When the balls have been in motion for 0.5 seconds, the string breaks.
3. Find the additional time that elapses until \(Q\) hits the floor.
4. Find the speed of \(Q\) as it hits the floor.
5. Write down the magnitude of the normal reaction force acting on \(Q\) when \(Q\) has come to rest on the floor.
6. State one improvement that could be made to the model. \section*{OCR} Oxford Cambridge and RSA

11 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-08_451_1340_251_244} A train consists of an engine \(A\) of mass 50000 kg and a carriage \(B\) of mass 20000 kg . The engine and carriage are connected by a rigid coupling. The coupling is modelled as light and horizontal. The resistances to motion acting on \(A\) and \(B\) are 9000 N and 1250 N respectively (see diagram).
The train passes through station \(P\) with speed \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along a straight horizontal track with constant acceleration \(0.01 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) towards station \(Q\). The distance between \(P\) and \(Q\) is 12.95 km .

1. Determine the time, in minutes, to travel between \(P\) and \(Q\). For the train's motion between \(P\) and \(Q\), determine the following.
2. The driving force of the engine.
3. The tension in the coupling between \(A\) and \(B\).

13 A toy train consists of an engine of mass 0.5 kg pulling a coach of mass 0.4 kg . The coupling between the engine and the coach is light and inextensible. The train is pulled along with a string attached to the front of the engine. At first, the train is pulled from rest along a horizontal carpet where there is a resistance to motion of 0.8 N on each part of the train. The string is horizontal, and the tension in the string is 5 N .

1. Determine the velocity of the train after 1.5 s . The train is then pulled up a track inclined at \(20 ^ { \circ }\) to the horizontal. The string is parallel to the track and the tension in the string is \(P \mathrm {~N}\). The resistance on each part of the train along the track is \(R \mathrm {~N}\).
2. Draw a diagram showing all the forces acting on the train modelled as two connected particles.
3. Find the equation of motion for the train modelled as a single particle.
4. The acceleration of the train when \(P = 5.5\) is double the acceleration when \(P = 5\). Calculate the value of \(R\).

4 The diagram shows a block, of mass 13 kg , on a rough horizontal surface. It is attached by a string that passes over a smooth peg to a sphere of mass 7 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_323_974_1256_575} The system is released from rest, and after 4 seconds the block and the sphere both have speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the block has not reached the peg.

1. State two assumptions that you should make about the string in order to model the motion of the sphere and the block.
2. Show that the acceleration of the sphere is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.
4. Find the coefficient of friction between the block and the surface.

6 A tractor, of mass 4000 kg , is used to pull a skip, of mass 1000 kg , over a rough horizontal surface. The tractor is connected to the skip by a rope, which remains taut and horizontal throughout the motion, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-4_243_880_477_571} Assume that only two horizontal forces act on the tractor. One is a driving force, which has magnitude \(P\) newtons and acts in the direction of motion. The other is the tension in the rope. The coefficient of friction between the skip and the ground is 0.4 .
The tractor and the skip accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).

1. Show that the magnitude of the friction force acting on the skip is 3920 N .
2. Show that \(P = 7920\).
3. Find the tension in the rope.
4. Suppose that, during the motion, the rope is not horizontal, but inclined at a small angle to the horizontal, with the higher end of the rope attached to the tractor, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-4_241_880_1665_571} How would the magnitude of the friction force acting on the skip differ from that found in part (a)? Explain why.

4 Two particles, \(A\) and \(B\), are connected by a string that passes over a fixed peg, as shown in the diagram. The mass of \(A\) is 9 kg and the mass of \(B\) is 11 kg .
The particles are released from rest in the position shown, where \(B\) is \(d\) metres higher than \(A\). The motion of the particles is to be modelled using simple assumptions.

1. State one assumption that should be made about the peg.
2. State two assumptions that should be made about the string.
3. By forming an equation of motion for each of the particles \(A\) and \(B\), show that the acceleration of each particle has magnitude \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. When the particles have been moving for 0.5 seconds, they are at the same level.
   1. Find the speed of the particles at this time.
   2. Find \(d\).

5 A block, of mass 14 kg , is held at rest on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.25 . A light inextensible string, which passes over a fixed smooth peg, is attached to the block. The other end of the string is attached to a particle, of mass 6 kg , which is hanging at rest. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-10_264_716_502_708} The block is released and begins to accelerate.

1. Find the magnitude of the friction force acting on the block.
2. By forming two equations of motion, one for the block and one for the particle, show that the magnitude of the acceleration of the block and the particle is \(1.225 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.
4. When the block is released, it is 0.8 metres from the peg. Find the speed of the block when it hits the peg.
5. When the block reaches the peg, the string breaks and the particle falls a further 0.5 metres to the ground. Find the speed of the particle when it hits the ground.
   (3 marks)
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-11_2484_1709_223_153}

6 Two particles, \(A\) and \(B\), have masses 12 kg and 8 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg, as shown in the diagram. $$A ( 12 \mathrm {~kg} )$$ The particles are released from rest and move vertically. Assume that there is no air resistance.

1. By forming two equations of motion, show that the magnitude of the acceleration of each particle is \(1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. After the particles have been moving for 2 seconds, both particles are at a height of 4 metres above a horizontal surface. When the particles are in this position, the string breaks.
   1. Find the speed of particle \(A\) when the string breaks.
   2. Find the speed of particle \(A\) when it hits the surface.
   3. Find the time that it takes for particle \(B\) to reach the surface after the string breaks. Assume that particle \(B\) does not hit the peg.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-13_2484_1709_223_153}

5 Two particles, \(P\) and \(Q\), are connected by a string that passes over a fixed smooth peg, as shown in the diagram. The mass of \(P\) is 5 kg and the mass of \(Q\) is 3 kg . \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-3_209_433_1009_808} The particles are released from rest in the position shown.

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. State two modelling assumptions that you have made about the string.
4. Particle \(P\) hits the floor when it has moved 0.196 metres and \(Q\) has not reached the peg.
   1. Find the time that it takes \(P\) to reach the floor.
   2. Find the speed of \(P\) when it hits the floor.

5 A block, of mass \(3 m\), is placed on a horizontal surface at a point \(A\). A light inextensible string is attached to the block and passes over a smooth peg. The string is horizontal between the block and the peg. A particle, of mass \(2 m\), is attached to the other end of the string. The block is released from rest with the string taut and the string between the peg and the particle vertical, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-10_170_726_536_657} Assume that there is no air resistance acting on either the block or the particle, and that the size of the block is negligible. The horizontal surface is smooth between the points \(A\) and \(B\), but rough between the points \(B\) and \(C\). Between \(B\) and \(C\), the coefficient of friction between the block and the surface is 0.8 .

1. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(A\) and \(B\).
2. Given that the distance between the points \(A\) and \(B\) is 1.2 metres, find the speed of the block when it reaches \(B\).
3. By forming equations of motion for both the block and the particle, find the acceleration of the block between \(B\) and \(C\).
4. Given that the distance between the points \(B\) and \(C\) is 0.9 metres, find the speed of the block when it reaches \(C\).
5. Explain why it is important to assume that the size of the block is negligible.
   [0pt] [1 mark]

5 Two particles, of masses 3 kg and 7 kg , are connected by a light inextensible string that passes over a smooth peg. The 3 kg particle is held at ground level with the string above it taut and vertical. The 7 kg particle is at a height of 80 cm above ground level, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5dd17095-18a6-470b-a24a-4456c8e3ed31-10_469_600_486_721} The 3 kg particle is then released from rest.

1. By forming two equations of motion, show that the magnitude of the acceleration of the particles is \(3.92 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the speed of the 7 kg particle just before it hits the ground.
3. When the 7 kg particle hits the ground, the string becomes slack and in the subsequent motion the 3 kg particle does not hit the peg. Find the maximum height of the 3 kg particle above the ground.
   [0pt] [4 marks]

7. A car of mass 1200 kg tows a trailer of mass 800 kg along a straight level road by means of a rigid towbar. The resistances to the motion of the car and the trailer are proportional to their masses. Given that the car experiences a resistance to motion of 300 N ,

1. find the resistance to motion which the trailer experiences. Given that the engine of the car exerts a driving force of 3 kN ,
2. find the acceleration of the system,
3. show that the tension in the towbar is 1200 N . When the system has reached a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it continues at constant speed until an electrical fault causes the engine of the car to switch off. The brakes are used to apply a constant retarding force until the system comes to rest. Given that the retarding force of the brakes has magnitude 1 kN and assuming that the original resistances to motion of the car and the trailer remain constant,
4. calculate the distance that the system travels during the braking period,
5. find the magnitude and direction of the force exerted by the towbar on the car.
6. Comment on the assumption that the original resistances to motion of the car and the trailer remain constant throughout the motion.

7. \begin{figure}[h] \end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is \(\mu\) and when the system is released from rest the 4 kg mass starts to move up the slope.

1. Show that the acceleration of the system is \(\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }\).
2. Hence, find the maximum value of \(\mu\). Given that \(\mu = \frac { 1 } { 2 }\),
3. find the tension in the string in terms of \(g\),
4. show that the magnitude of the force on the pulley is given by \(\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }\). END

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 14 }\). The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

1. show that the acceleration of the system is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
2. find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
3. Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
4. Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.

4. \begin{figure}[h] \end{figure} Figure 1 shows a weight \(A\) of mass 6 kg connected by a light, inextensible string which passes over a smooth, fixed pulley to a box \(B\) of mass 5 kg . There is an object \(C\) of mass 3 kg resting on the horizontal floor of box \(B\). The system is released from rest. Find, giving your answers in terms of \(g\),

1. the acceleration of the system,
2. the force on the pulley.
3. Show that the reaction between \(C\) and the floor of \(B\) is \(\frac { 18 } { 7 } \mathrm {~g}\) newtons.

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.

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.

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.