Connected particles via tow-bar on horizontal surface

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 \(R\).

2 A car of mass 1400 kg pulls a trailer of mass 400 kg along a straight horizontal road. The engine of the car produces a driving force of 6000 N . A resistance of 800 N acts on the car. A resistance of 300 N acts on the trailer. The tow-bar between the car and the trailer is light and horizontal.

1. Draw a force diagram showing all the horizontal forces on the car and the trailer.
2. Calculate the acceleration of the car and trailer.

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
   • the magnitude of the new force in the bar,
   • whether the bar is in tension or in compression.

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.

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.

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}

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).

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 \(\beta\).

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 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 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 \(P\).

4 A car, of mass 1200 kg , is connected by a tow rope to a truck, of mass 2800 kg . The truck tows the car in a straight line along a horizontal road. Assume that the tow rope is horizontal. A horizontal driving force of magnitude 3000 N acts on the truck. A horizontal resistance force of magnitude 800 N acts on the car. The car and truck accelerate at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
\includegraphics[max width=\textwidth, alt={}, center]{81f3753c-f148-44be-8b35-0a8e531016dd-3_177_1002_580_513}

1. Find the tension in the tow rope.
2. Show that the magnitude of the horizontal resistance force acting on the truck is 600 N .
3. In fact, the tow rope is not horizontal. Assume that the resistance forces and the driving force are unchanged. Is the tension in the tow rope greater or less than in part (a)? Explain why.

15 A tractor and its driver have a combined mass of \(m\) kilograms.
The tractor is towing a trailer of mass \(4 m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of 11080 N and a total resistance force of 160 N act on the tractor.
A total resistance force of 600 N acts on the trailer.
The tractor and the trailer have an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
15

1. Find \(m\).
   15
2. Find the tension in the tow bar.
   15
3. At the instant the speed of the tractor reaches \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
   \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-22_2488_1719_219_150}
   \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-23_2488_1719_219_150}
   \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-24_2498_1721_213_148}

17
\includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-26_270_1036_274_552} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5 t + 0.01 t ^ { 2 }$$ 17

1. Show that when \(t = 15\) their acceleration is \(0.8 \mathrm {~ms} ^ { - 2 }\)
   17
2. The car has a mass of 1500 kg
   The caravan has a mass of 850 kg
   When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N 17
3. 1. Find the total resistance force experienced by the caravan when \(t = 15\)
      17
4. (ii) Find the driving force being applied by the car when \(t = 15\)
   [0pt] [3 marks]
   17
5. State one assumption you have made about the tow bar.
   \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-28_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-32_2486_1719_221_150}

17 A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e475c96-6bbb-44fb-a411-706dac20e2d6-24_240_1006_402_516} The combined mass of the buggy and driver is 410 kg
A driving force of 300 N and a total resistance force of 140 N act on the buggy.
The mass of the roller-skater is 72 kg
A total resistance force of \(R\) newtons acts on the roller-skater.
The buggy and the roller-skater have an acceleration of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
17

1. 1. Find \(R\).
      

      17	17 (ii) Find the tension in the rope.

      17
   2. The roller-skater releases the rope at a point \(A\), when she reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) She continues to move forward, experiencing the same resistance force.
      The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from \(A\). 17
   3. 1. Determine whether the roller-skater will stop before reaching the stationary buggy.
         Fully justify your answer.
         17
   4. (ii) Explain the change in motion that the driver noticed.

15 A cyclist is towing a trailer behind her bicycle. She is riding along a straight, horizontal path at a constant speed.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-22_371_723_447_657} A tension of \(T\) newtons acts on the connecting rod between the bicycle and the trailer.
The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle. The constant resistance force acting on the trailer is 12 N
15

1. State the value of \(T\) giving a clear reason for your answer.
   15
2. State one assumption you have made in reaching your answer to part (a).