Connected particles via tow-bar on horizontal surface

2 A car of mass 1800 kg is towing a trailer of mass 400 kg along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There are constant resistance forces of 250 N on the car and 100 N on the trailer.

1. Find the tension in the tow-bar.
2. Find the power of the engine of the car at the instant when the speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A car of mass 1500 kg is towing a trailer of mass \(m \mathrm {~kg}\) along a straight horizontal road. The car and the trailer are connected by a tow-bar which is horizontal, light and rigid. There is a resistance force of \(F \mathrm {~N}\) on the car and a resistance force of 200 N on the trailer. The driving force of the car's engine is 3200 N , the acceleration of the car is \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and the tension in the tow-bar is 300 N . Find the value of \(m\) and the value of \(F\).

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

6 On a straight horizontal test track, driverless vehicles (with no passengers) are being tested. A car of mass 1600 kg is towing a trailer of mass 700 kg along the track. The brakes are applied, resulting in a deceleration of \(12 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The braking force acts on the car only. In addition to the braking force there are constant resistance forces of 600 N on the car and of 200 N on the trailer.

1. Find the magnitude of the force in the tow-bar.
2. Find the braking force.
3. At the instant when the brakes are applied, the car has speed \(22 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At this instant the car is 17.5 m away from a stationary van, which is directly in front of the car. Show that the car hits the van at a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. After the collision, the van starts to move with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car and trailer continue moving in the same direction with speed \(2 \mathrm {~ms} ^ { - 1 }\). Find the mass of the van.

2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.

1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
   The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
2. Find the least possible value of the braking force which will cause the rope to become slack.

2. A car of mass 1000 kg is towing a caravan of mass 750 kg along a straight horizontal road. The caravan is connected to the car by a tow-bar which is parallel to the direction of motion of the car and the caravan. The tow-bar is modelled as a light rod. The engine of the car provides a constant driving force of 3200 N . The resistances to the motion of the car and the caravan are modelled as constant forces of magnitude 800 newtons and \(R\) newtons respectively. Given that the acceleration of the car and the caravan is \(0.88 \mathrm {~ms} ^ { - 2 }\),

1. show that \(R = 860\),
2. find the tension in the tow-bar.

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time \(t = 0\), a constant force \(\mathbf { F }\) of magnitude 4 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 3. The system moves under the action of this force until \(t = 6 \mathrm {~s}\). During the motion, the resistance to the motion of \(P\) has constant magnitude 1 N and the resistance to the motion of \(Q\) has constant magnitude 2 N . Find

1. the acceleration of the particles as the system moves under the action of \(\mathbf { F }\),
2. the speed of the particles at \(t = 6 \mathrm {~s}\),
3. the tension in the rod as the system moves under the action of \(\mathbf { F }\). At \(t = 6 \mathrm {~s} , \mathbf { F }\) is removed and the system decelerates to rest. The resistances to motion are unchanged. Find
4. the distance moved by \(P\) as the system decelerates,
5. the thrust in the rod as the system decelerates.

7. \begin{figure}[h] \end{figure} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as they enter a zone where the speed limit is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is 100 m .

1. Find the deceleration of the truck and the car. The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that \(\cos \theta = 0.9\), find
2. the force in the towbar,
3. the value of \(R\).

6. A car pulls a trailer along a straight horizontal road using a light inextensible towbar. The mass of the car is \(M \mathrm {~kg}\), the mass of the trailer is 600 kg and the towbar is horizontal and parallel to the direction of motion. There is a resistance to motion of magnitude 200 N acting on the car and a resistance to motion of magnitude 100 N acting on the trailer. The driver of the car spots a hazard ahead. Instantly he reduces the force produced by the engine of the car to zero and applies the brakes of the car. The brakes produce a braking force on the car of magnitude 6500 N and the car and the trailer have a constant deceleration of magnitude \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Given that the resistances to motion on the car and trailer are unchanged and that the car comes to rest after travelling 40.5 m from the point where the brakes were applied, find

1. the thrust in the towbar while the car is braking,
2. the value of \(M\),
3. the time it takes for the car to stop after the brakes are applied.

3. A tractor of mass 6 tonnes is dragging a large block of mass 2 tonnes along rough horizontal ground. The cable connecting the tractor to the block is horizontal and parallel to the direction of motion. The cable is modelled as being light and inextensible.
The driving force of the tractor is 7400 N and the resistance to the motion of the tractor is 200 N . The resistance to the motion of the block is \(R\) newtons, where \(R\) is a constant. Given that the tension in the cable is 6000 N and the tractor is accelerating,

1. find the value of \(R\).
2. State how you have used the fact that the cable is modelled as being inextensible.
   

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1 A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N , find the resistance to motion of the trailer.
2. Given also that the driving force of the car is 2100 N , find the resistance to motion of the car.

2 A trailer of mass 500 kg is attached to a car of mass 1250 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road. The resistance to motion of the trailer is 400 N and the resistance to motion of the car is 900 N . Find both the tension in the tow-bar and the driving force of the car in each of the following cases.

1. The car and trailer are travelling at constant speed.
2. The car and trailer have acceleration \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-4_314_997_267_577} A train of total mass 80000 kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E , A\) and \(B\) are \(10500 \mathrm {~N} , 3000 \mathrm {~N}\) and 1500 N respectively (see diagram).

1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than 15000 N .
2. Show that, when the magnitude of the driving force is 35000 N , the acceleration of the train is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is 8500 N when the magnitude of the driving force is 35000 N . The driving force is replaced by a braking force of magnitude 15000 N acting on the engine. The force exerted by the coupling \(Y\) is zero.
4. Find the mass of \(B\).
5. Show that the coupling \(X\) exerts a forward force of magnitude 1500 N on the engine.

4 A car and its trailer travel along a straight, horizontal road. The coupling between them is light and horizontal. The car has mass 900 kg and resistance to motion 100 N , the trailer has mass 700 kg and resistance to motion 300 N , as shown in Fig. 4. The car and trailer have an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h] \end{figure}

1. Calculate the driving force of the car.
2. Calculate the force in the coupling.

5 Fig. 5 shows two boxes, A of mass 12 kg and B of mass 6 kg , sliding in a straight line on a rough horizontal plane. The boxes are connected by a light rigid rod which is parallel to the line of motion. The only forces acting on the boxes in the line of motion are those due to the rod and a constant force of \(F \mathrm {~N}\) on each box. \begin{figure}[h] \end{figure} The boxes have an initial speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and come to rest after sliding a distance of 0.375 m .

1. Calculate the deceleration of the boxes and the value of \(F\).
2. Calculate the magnitude of the force in the rod and state, with a reason, whether it is a tension or a thrust (compression).

3 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?

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 \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) 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.

7 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 \(\beta\).

17

1. 1. Draw a diagram to show the forces acting on the crate while it is being lifted. 17
2. (ii) Draw a diagram to show the forces acting on the van while the crate is being lifted. [1 mark] 17
3. Show that the acceleration of the van is \(0.80 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) [0pt] [4 marks]
   17
4. Find the tension in the rope.
   17
5. Suggest how the assumption of a constant resistance force could be refined to produce a better model.

4. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.
The rope is horizontal and parallel to the direction of motion of the car.

• The resistance to motion of the car is modelled as a constant force of magnitude \(2 R\) newtons
• The resistance to motion of the trailer is modelled as a constant force of magnitude \(R\) newtons
• The rope is modelled as being light and inextensible
• The acceleration of the car is modelled as \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N . Using the model,

1. find the value of \(a\) In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be \(a _ { 1 } \mathrm {~ms} ^ { - 2 }\)
2. State how the value of \(a _ { 1 }\) compares with the value of \(a\)
3. State one limitation of the model used for the resistance to motion of the car.

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.

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 \(B\).

6 Fig. 6 shows a train consisting of an engine of mass 80 tonnes pulling two trucks each of mass 25 tonnes. \begin{figure}[h] \end{figure} The engine exerts a driving force of \(D \mathrm {~N}\) and experiences a resistance to motion of 2000 N . Each truck experiences a resistance of 600 N . The train travels in a straight line on a level track with an acceleration of \(0.1 \mathrm {~ms} ^ { - 2 }\).

1. Complete the force diagram in the Printed Answer Booklet to show all the forces acting on the engine and each of the trucks.
2. Calculate the value of \(D\).
3. The tension in the coupling between the engine and truck A is larger than that in the coupling between the trucks. Determine how much larger.

10 A boat pulls a water skier of mass 65 kg with a light inextensible horizontal towrope. The mass of the boat is 985 kg . There is a driving force of 2400 N acting on the boat. There are horizontal resistances to motion of 400 N and 1200 N acting on the skier and the boat respectively.

1. Draw a diagram showing all the horizontal forces acting on the skier and the boat.
2. 1. Write down the equation of motion of the skier.
   2. Find the equation of motion of the boat.
3. Find the acceleration of the skier and the boat. The driving force of the boat is increased. The skier can only hold on to the towrope when the tension is no greater than her weight.
4. Determine her greatest acceleration, assuming that the resistances to motion stay the same.