3.03k Connected particles: pulleys and equilibrium

9 A tractor of mass 1800 kg uses a towbar to pull a trailer of mass 1000 kg on a level field. The tractor and trailer experience resistances to motion of 1600 N and 800 N respectively. The tractor provides a driving force of 6600 N .

1. Draw a force diagram showing all the horizontal forces acting on the tractor and trailer.
2. Find the tension in the towbar.

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 \(T\).

11 Fig. 11 shows two blocks at rest, connected by a light inextensible string which passes over a smooth pulley. Block A of mass 4.7 kg rests on a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. Block B of mass 4 kg rests on a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. On either side of the pulley, the string is parallel to a line of greatest slope of the plane. Block B is on the point of sliding up the plane. \begin{figure}[h] \end{figure}

1. Show that the tension in the string is 39.9 N correct to 3 significant figures.
2. Find the coefficient of friction between the rough plane and Block B.

11 A block of mass 2 kg is placed on a rough horizontal table. A light inextensible string attached to the block passes over a smooth pulley attached to the edge of the table. The other end of the string is attached to a sphere of mass 0.8 kg which hangs freely. The part of the string between the block and the pulley is horizontal. The coefficient of friction between the table and the block is 0.35 . The system is released from rest.

1. Draw a force diagram showing all the forces on the block and the sphere.
2. Write down the equations of motion for the block and the sphere.
3. Show that the acceleration of the system is \(0.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. Calculate the time for the block to slide the first 0.5 m . Assume the block does not reach the pulley.

9 The diagram shows a toy caterpillar consisting of a head and three body sections each connected by a light inextensible ribbon. The head has a mass of 120 g and the body sections each have a mass of 90 g . The toy is pulled on level ground using a horizontal string attached to the head. The tension in the string is 12 N . There are resistances to motion of 2.5 N for the head and each section of the body. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-08_134_794_536_244}

1. 1. State the equation of motion for the toy caterpillar modelled as a single particle.
   2. Calculate the acceleration of the toy caterpillar.
2. Draw a diagram showing all the forces acting on the head of the toy caterpillar.
3. Calculate the tension in the ribbon that joins the head to the body.

14 Blocks A and B are connected by a light rigid horizontal bar and are sliding on a rough horizontal surface. A light horizontal string exerts a force of 40 N on B .
This situation is shown in Fig. 14, which also shows the direction of motion, the mass of each of the blocks and the resistances to their motion. \begin{figure}[h] \end{figure}

1. Calculate the tension in the bar. The string breaks while the blocks are sliding. The resistances to motion are unchanged.
2. Determine

4 Two particles are connected by a string, which passes over a pulley. Model the string as light and inextensible. The particles have masses of 2 kg and 5 kg . The particles are released from rest. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-3_392_209_1685_909}

1. State one modelling assumption that you should make about the pulley in order to determine the acceleration of the particles.
2. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.

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 \(d\).

3 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal tow bar, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-06_277_901_484_584} Assume that a constant resistance force of magnitude 200 newtons acts on the car and a constant resistance force of magnitude 300 newtons acts on the caravan. A constant driving force of magnitude \(P\) newtons acts on the car in the direction of motion. The car and caravan accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).

1. 1. Find \(P\).
   2. Find the magnitude of the force in the tow bar that connects the car to the caravan.
2. 1. Find the time that it takes for the speed of the car and caravan to increase from \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Find the distance that they travel in this time.
3. Explain why the assumption that the resistance forces are constant is unrealistic.
   (1 mark)

5 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal towbar. A resistance force of magnitude \(R\) newtons acts on the car and a resistance force of magnitude \(2 R\) newtons acts on the caravan. The car and caravan accelerate at a constant \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when a driving force of magnitude 4720 newtons acts on the car.

1. Find \(R\).
2. Find the tension in the towbar.

4 A tractor, of mass 3500 kg , is used to tow a trailer, of mass 2400 kg , across a horizontal field. The trailer is connected to the tractor by a horizontal tow bar. As they move, a constant resistance force of 800 newtons acts on the trailer and a constant resistance force of \(R\) newtons acts on the tractor. A forward driving force of 2500 newtons acts on the tractor. The trailer and tractor accelerate at \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. \(\quad\) Find \(R\).
2. Find the magnitude of the force that the tow bar exerts on the trailer.
3. State the magnitude of the force that the tow bar exerts on the tractor.

4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
   (4 marks)

5 A small block \(P\) is attached to another small block \(Q\) by a light inextensible string. The block \(P\) rests on a rough horizontal surface and the string hangs over a smooth peg so that \(Q\) hangs freely, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-4_222_426_507_810} The block \(P\) has mass 0.4 kg and the coefficient of friction between \(P\) and the surface is 0.5 . The block \(Q\) has mass 0.3 kg . The system is released from rest and \(Q\) moves vertically downwards.

1. 1. Draw a diagram to show the forces acting on \(P\).
   2. Show that the frictional force between \(P\) and the surface has magnitude 1.96 newtons.
2. By forming an equation of motion for each block, show that the magnitude of the acceleration of each block is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the speed of the blocks after 3 seconds of motion.
4. After 3 seconds of motion, the string breaks. The blocks continue to move. Comment on how the speed of each block will change in the subsequent motion. For each block, give a reason for your answer.

3 Two particles, \(A\) and \(B\), have masses 4 kg and 6 kg respectively. They are connected by a light inextensible string that passes over a smooth fixed peg. A second light inextensible string is attached to \(A\). The other end of this string is attached to the ground directly below \(A\). The system remains at rest, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-3_457_711_523_845}

1. 1. Write down the tension in the string connecting \(A\) and \(B\).
   2. Find the tension in the string connecting \(A\) to the ground.
2. The string connecting particle \(A\) to the ground is cut. Find the acceleration of \(A\) after the string has been cut.

5 Two particles are connected by a light inextensible string that passes over a smooth peg. The particles have masses of 3 kg and 1 kg . The 1 kg particle is pulled down to ground level, where it is 40 cm below the level of the 3 kg particle, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_490_648_1272_696} The particles are released from rest with the string vertical above each particle. Assume that no resistance forces act on the particles as they move.

1. By forming two equations of motion, one for each particle, find the magnitude of the acceleration of the particles after they have been released but before the 3 kg particle hits the ground.
2. Find the speed of the 1 kg particle when the 3 kg particle hits the ground.
3. After the 3 kg particle has hit the ground, the 1 kg particle continues to move and the string is now slack. Find the maximum height above ground level reached by the 1 kg particle.
4. If a constant air resistance force also acts on the particles as they move, explain how this would change your answer for the acceleration in part (a). Give a reason for your answer.

3 A skip, of mass 800 kg , is at rest on a rough horizontal surface. The coefficient of friction between the skip and the ground is 0.4 . A rope is attached to the skip and then the rope is pulled by a van so that the rope is horizontal while it is taut, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_237_1118_497_463} The mass of the van is 1700 kg . A constant horizontal forward driving force of magnitude \(P\) newtons acts on the van. The skip and the van accelerate at \(0.05 \mathrm {~ms} ^ { - 2 }\). Model both the van and the skip as particles connected by a light inextensible rope. Assume that there is no air resistance acting on the skip or on the van.

1. Find the speed of the van and the skip when they have moved 6 metres.
2. Draw a diagram to show the forces acting on the skip while it is accelerating.
3. Draw a diagram to show the forces acting on the van while it is accelerating. State one advantage of modelling the van as a particle when considering the vertical forces.
4. Find the magnitude of the friction force acting on the skip.
5. Find the tension in the rope.
6. \(\quad\) Find \(P\).
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-06_771_1703_1932_155}

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 particle \(P\), of mass 4 kg , rests on horizontal ground and is attached by a light, inextensible string to another particle \(Q\) of mass 4.5 kg . The string passes over a smooth pulley whose centre is 3 m above the ground. Initially \(Q\) is 1.1 m below the level of the centre of the pulley. The system is released from rest in this position.

1. Find the acceleration of the two particles.
2. Find the speed with which \(Q\) hits the ground. Assuming that \(Q\) does not rebound from the ground while the string is slack,
3. show that \(P\) does not reach the pulley before \(Q\) starts to move again.
4. Find the speed with which \(Q\) leaves the ground when the string again becomes taut.
   (3 marks)

4. \(X\) and \(Y\) are two points 1 m apart on a line of greatest slope of a smooth plane inclined at \(60 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 1 kg is released from rest at \(X\).

1. Find the speed with which \(P\) reaches \(Y\). \(P\) is now connected to another particle \(Q\), of mass \(M \mathrm {~kg}\), by a light inextensible string. The system is placed with \(P\) at \(Y\) on the plane and \(Q\) hanging vertically at the other end of the string, which passes over a fixed pulley at the top of the plane.
   The system is released from rest and \(P\) moves up the plane with acceleration \(\frac { g } { 5 }\). \includegraphics[max width=\textwidth, alt={}, center]{cc75a4a5-1c3a-4e36-acfd-21f6246f2a38-1_358_321_2024_1597}
2. Show that \(M = \frac { 5 \sqrt { } 3 + 2 } { 8 }\).
3. State a modelling assumption that you have made about the pulley. Briefly state what would be implied if this assumption were not made. \section*{MECHANICS 1 (A) TEST PAPER 8 Page 2}

6. Corinne and her brother Dermot are lifted by their parents onto the two ends of a rope which is slung over a large, horizontal branch. When their parents let go of them Dermot, whose mass is 54 kg , begins to descend with an acceleration of \(1 \mathrm {~ms} ^ { - 2 }\). By modelling the children as a pair of particles connected by a light inextensible string, and the branch as a smooth pulley,

1. show that Corinne's mass is 44 kg ,
2. calculate the tension in the rope,
3. find the force on the branch. In a more sophisticated model, the branch is assumed to be rough.
4. Explain what effect this would have on the initial acceleration of the children.
   (1 mark)

8. \begin{figure}[h] \end{figure} Figure 3 shows two particles \(A\) and \(B\), of mass \(5 M\) and \(3 M\) respectively, attached to the ends of a light inextensible string of length 4 m . The string passes over a smooth pulley which is fixed to the edge of a rough horizontal table 2 m high. Particle \(A\) lies on the table at a distance of 3 m from the pulley, whilst particle \(B\) hangs freely over the edge of the table 1 m above the ground. The coefficient of friction between \(A\) and the table is \(\frac { 3 } { 20 }\). The system is released from rest with the string taut.

1. Show that the initial acceleration of the system is \(\frac { 9 } { 32 } \mathrm {~g} \mathrm {~ms} ^ { - 2 }\).
2. Find, in terms of \(g\), the speed of \(A\) immediately before \(B\) hits the ground. When \(B\) hits the ground, it comes to rest and the string becomes slack.
3. Calculate how far particle \(A\) is from the pulley when it comes to rest. END

1 A car of mass 1000 kg is travelling along a straight, level road. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the car when a resultant force of 2000 N acts on it in the direction of its motion. How long does it take the car to increase its speed from \(5 \mathrm {~ms} ^ { - 1 }\) to \(12.5 \mathrm {~ms} ^ { - 1 }\) ? The car has an acceleration of \(1.4 \mathrm {~ms} ^ { - 2 }\) when there is a driving force of 2000 N .
2. Show that the resistance to motion of the car is 600 N . A trailer is now atached to the car, as shown in Fig. 6.2. The car still has a driving force of 2000 N and resistance to motion of 600 N . The trailer has a mass of 800 kg . The tow-bar connecting the car and the trailer is light and horizontal. The car and trailer are accelerating at \(0.7 \mathrm {~ms} ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d5a09ed4-a32f-4ff7-aa08-6e54c2ab26a0-1_165_883_1279_554} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Show that the resistance to the motion of the trailer is 140 N .
4. Calculate the force in the tow bar. The driving force is now removed and a braking force of 610 N is applied to the car. All the resistances to motion remain as before. The trailer has no brakes.
5. Calculate the new acceleration. Calculate also the force in the tow-bar, stating whether it is a tension or a thrust (compression).