Motion on inclined plane with connected system

6 A car of mass 1800 kg is towing a trailer of mass 300 kg up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.05\). The car and trailer are connected by a tow-bar which is light and rigid and is parallel to the road. There is a resistance force of 800 N acting on the car and a resistance force of \(F \mathrm {~N}\) acting on the trailer. The driving force of the car's engine is 3000 N .

1. It is given that \(F = 100\). Find the acceleration of the car and the tension in the tow-bar.
2. It is given instead that the total work done against \(F\) in moving a distance of 50 m up the road is 6000 J . The speed of the car at the start of the 50 m is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Use an energy method to find the speed of the car at the end of the 50 m .
   \includegraphics[max width=\textwidth, alt={}, center]{1ca74dfc-9bef-475c-a7d1-77b95c487f4b-10_680_887_269_596} The diagram shows two particles \(P\) and \(Q\) which lie on a line of greatest slope of a plane \(A B C\). Particles \(P\) and \(Q\) are each of mass \(m \mathrm {~kg}\). The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = 0.6\). The length of \(A B\) is 0.75 m and the length of \(B C\) is 3.25 m . The section \(A B\) of the plane is smooth and the section \(B C\) is rough. The coefficient of friction between each particle and the section \(B C\) is 0.25 . Particle \(P\) is released from rest at \(A\). At the same instant, particle \(Q\) is released from rest at \(B\).
3. Verify that particle \(P\) reaches \(B 0.5 \mathrm {~s}\) after it is released, with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
4. Find the time that it takes from the instant the two particles are released until they collide.
   The two particles coalesce when they collide. The coefficient of friction between the combined particle and the plane is still 0.25 .
5. Find the time that it takes from the instant the particles collide until the combined particle reaches \(C\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 A railway engine of mass 120000 kg is towing a coach of mass 60000 kg up a straight track inclined at an angle of \(\alpha\) to the horizontal where \(\sin \alpha = 0.02\). There is a light rigid coupling, parallel to the track, connecting the engine and coach. The driving force produced by the engine is 125000 N and there are constant resistances to motion of 22000 N on the engine and 13000 N on the coach.

1. Find the acceleration of the engine and find the tension in the coupling.
   At an instant when the engine is travelling at \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a section of track inclined upwards at an angle \(\beta\) to the horizontal. The power produced by the engine is now 4500000 W and, as a result, the engine maintains a constant speed.
2. Assuming that the resistance forces remain unchanged, find the value of \(\beta\).

5 A car 6 mass \(\mathbb { I } \quad\) g is p lig a trailer 6 mass \(\theta \quad\) g ah ll in lin d at an ag e \(6 \sin ^ { - 1 } ( \mathbb { I } )\) ) to th b izo al. Th car and to trailer are co cted b a lig rig d -b r wh ch is \(\boldsymbol { \rho }\) rallel to th ro d Th d iv g fo ce \(\varnothing\) th car's eg A is \(\theta \mathrm { N }\) ad th resistan es to th car ad trailer are \(\theta \mathrm { N }\) ad (1) N resp ctie ly.

1. Fid b acceleratio th sy tem ad b tensio it b tw -b r.
2. Wh it b car ad railer are tra llig tasp e \(\boldsymbol { \Theta } \quad \mathbf { b } \mathrm { ms } ^ { - 1 } , \mathrm { t } \mathbf { b }\) divg \(\mathbf { o }\) ce b cm es zero Fid th time, in sect , \(\mathbf { b }\) fo e the sy tem cm es to rest ad th fo ce in th ro \(\mathbf { r d }\) ig th s time.

5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.

1. Find the acceleration of the system and the tension in the cable.
2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.

6. A breakdown van of mass 2000 kg is towing a car of mass 1200 kg along a straight horizontal road. The two vehicles are joined by a tow bar which remains parallel to the road. The van and the car experience constant resistances to motion of magnitudes 800 N and 240 N respectively. There is a constant driving force acting on the van of 2320 N . Find

1. the magnitude of the acceleration of the van and the car,
2. the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\). The driving force and the resistances to the motion are unchanged.
3. Find the magnitude of the acceleration of the van and the car as they move up the hill and state whether their speed increases or decreases.

6. \begin{figure}[h] \end{figure} A railway engine of mass 1500 kg is attached to a railway truck of mass 500 kg by a straight rigid coupling. The engine pushes the truck up a straight track, which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 7 } { 25 }\). The coupling is parallel to the track and parallel to the direction of motion, as shown in Figure 3. The engine produces a constant driving force of magnitude \(D\) newtons. The engine and the truck experience constant resistances to motion, from non-gravitational forces, of magnitude 1200 N and 500 N respectively. The thrust in the coupling is 2000 N . The coupling is modelled as a light rod.

1. Find the acceleration of the engine and the truck.
2. Find the value of \(D\).

7. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4. The road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\)
The driving force produced by the engine of the car is 3000 N .
The car moves with acceleration \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
The non-gravitational resistance to motion of

• the car is modelled as a constant force of magnitude \(2 R\) newtons
• the trailer is modelled as a constant force of magnitude \(R\) newtons

The car and the trailer are modelled as particles.
The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,

1. show that the value of \(R\) is 60
2. find the tension in the tow bar. When the car and trailer are moving at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow bar breaks.
   Given that the non-gravitational resistance to motion of the trailer remains unchanged,
3. use the model to find the further distance moved by the trailer before it first comes to rest.

7. A truck of mass 1600 kg is towing a car of mass 960 kg along a straight horizontal road. The truck and the car are joined by a light rigid tow bar. The tow bar is horizontal and is parallel to the direction of motion. The truck and the car experience constant resistances to motion of magnitude 640 N and \(R\) newtons respectively. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitude of the acceleration of the truck and the car is \(0.4 \mathrm {~ms} ^ { - 2 }\).

1. Show that \(R = 436\)
2. Find the tension in the tow bar. The two vehicles come to a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 15 }\). The truck and the car move down a line of greatest slope of the hill with the tow bar parallel to the direction of motion. The truck's engine produces a constant driving force of magnitude 2100 N . The magnitudes of the resistances to motion on the truck and the car are 640 N and 436 N respectively.
3. Find the magnitude of the acceleration of the truck and the car as they move down the hill.
   
   \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-27_67_59_2654_1886}

5 A car is towing a trailer along a straight road using a light tow-bar which is parallel to the road. The masses of the car and the trailer are 900 kg and 250 kg respectively. The resistance to motion of the car is 600 N and the resistance to motion of the trailer is 150 N .

1. At one stage of the motion, the road is horizontal and the pulling force exerted on the trailer is zero.
   (a) Show that the acceleration of the trailer is \(- 0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   (b) Find the driving force exerted by the car.
   (c) Calculate the distance required to reduce the speed of the car and trailer from \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. At another stage of the motion, the car and trailer are moving down a slope inclined at \(3 ^ { \circ }\) to the horizontal. The resistances to motion of the car and trailer are unchanged. The driving force exerted by the car is 980 N . Find
   (a) the acceleration of the car and trailer,
   (b) the pulling force exerted on the trailer.

1 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 \(\beta\).