3.03o Advanced connected particles: and pulleys

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?

9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.

1. 1. Show that initially particle \(A\) moves towards the hole in the table.
   2. Show that initially particle \(B\) also moves towards the hole in the table.
2. Calculate the initial elastic potential energy in the string.
3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.

2. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to the motion of the car is a constant force of magnitude 400 N.

The engine of the car is working at a constant rate of 16 kW .

1. Find the value of \(U\). The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N . The resistance to the motion of the trailer is a constant force of magnitude 300 N . The engine of the car is working at a constant rate of 16 kW .
   The tow rope is modelled as being light and inextensible.
   Using the model,
2. find the tension in the tow rope at the instant when the speed of the car is \(\frac { 20 } { 3 } \mathrm {~ms} ^ { - 1 }\)

5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\),
2. the tension in the string,
3. the value of \(\mu\).
4. State how in your calculations you have used the information that the string is inextensible.

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.

13 Particle \(A\), of mass \(m \mathrm {~kg}\), lies on the plane \(\Pi _ { 1 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 3 } { 4 }\) to the horizontal.
Particle \(B\), of \(4 m \mathrm {~kg}\), lies on the plane \(\Pi _ { 2 }\) inclined at an angle of \(\tan ^ { - 1 } \frac { 4 } { 3 }\) to the horizontal.
The particles are attached to the ends of a light inextensible string which passes over a smooth pulley at \(P\).
The coefficient of friction between particle \(A\) and \(\Pi _ { 1 }\) is \(\frac { 1 } { 3 }\) and plane \(\Pi _ { 2 }\) is smooth.
Particle \(A\) is initially held at rest such that the string is taut and lies in a line of greatest slope of each plane. This is shown on the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-10_398_844_868_306}

1. Show that when \(A\) is released it accelerates towards the pulley at \(\frac { 7 g } { 15 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Assuming that \(A\) does not reach the pulley, show that it has moved a distance of \(\frac { 1 } { 4 } \mathrm {~m}\) when its speed is \(\sqrt { \frac { 7 g } { 30 } } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   [0pt] [2]

19 A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod. The train engine has a mass of 1.5 kilograms.
The trailer has a mass 0.7 kilograms.
A string inclined at an angle of \(40 ^ { \circ }\) above the horizontal is attached to the front of the train engine. The tension in the string is 2 newtons.
As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration \(0.06 \mathrm {~ms} ^ { - 2 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-30_373_789_904_756} As it moves the train engine experiences a total resistance force of 0.8 N
19

1. Show that the total resistance force experienced by the trailer is approximately 0.6 N
   19
2. At the instant that the toy reaches a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks. As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of \(h\) metres. It can be assumed that the resistance forces remain unchanged.
   19 (b) (i) Find the tension in the rod after the string has broken.
   

   19 (b) (ii) Find \(h\)

   Do not write outside the box

   \includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-33_2488_1716_219_153}
   Nell and her pet dog Maia are visiting the beach.
   The beach surface can be assumed to be level and horizontal. Nell and Maia are initially standing next to each other.
   Nell throws a ball forward, from a height of 1.8 metres above the surface of the beach, at an angle of \(60 ^ { \circ }\) above the horizontal with a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Exactly 0.2 seconds after the ball is thrown, Maia sets off from Nell and runs across the surface of the beach, in a straight line with a constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Maia catches the ball when it is 0.3 metres above ground level as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-34_778_1287_1027_463}

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.

1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.

\includegraphics{figure_6} Two particles \(P\) and \(Q\), of masses \(0.2\) kg and \(0.1\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley \(B\) which is attached to two inclined planes. Particle \(P\) lies on a smooth plane \(AB\) which is inclined at \(60°\) to the horizontal. Particle \(Q\) lies on a plane \(BC\) which is inclined at an angle of \(\theta°\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes (see diagram).

1. It is given that \(\theta = 60\), the plane \(BC\) is rough and the coefficient of friction between \(Q\) and the plane \(BC\) is \(0.7\). The particles are released from rest. Determine whether the particles move. [4]
2. It is given instead that the plane \(BC\) is smooth. The particles are released from rest and in the subsequent motion the tension in the string is \((\sqrt{3} - 1)\) N. Find the magnitude of the acceleration of \(P\) as it moves on the plane, and find the value of \(\theta\). [4]

\includegraphics{figure_7} Two particles \(P\) and \(Q\) of masses 2.5 kg and 0.5 kg respectively are connected by a light inextensible string that passes over a small smooth pulley fixed at the top of a plane inclined at an angle of \(30°\) to the horizontal. Particle \(P\) is on the plane and \(Q\) hangs below the pulley such that the level of \(Q\) is 2 m below the level of \(P\) (see diagram). Particle \(P\) is released from rest with the string taut and slides down the plane. The plane is rough with coefficient of friction 0.2 between the plane and \(P\).

1. Find the acceleration of \(P\). [5]
2. Use an energy method to find the speed of the particles at the instant when they are at the same vertical height. [5]

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 \(F\) 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. [5]
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 \(20\) m s\(^{-1}\). Use an energy method to find the speed of the car at the end of the 50 m. [5]

\includegraphics{figure_5} Two particles of masses 0.8 kg and 0.2 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The system is released from rest with both particles 0.5 m above a horizontal floor (see diagram). In the subsequent motion the 0.2 kg particle does not reach the pulley.

1. Show that the magnitude of the acceleration of the particles is \(6 \text{ m s}^{-2}\) and find the tension in the string. [4]
2. When the 0.8 kg particle reaches the floor it comes to rest. Find the greatest height of the 0.2 kg particle above the floor. [3]

A car of mass 1500 kg is pulling a trailer of mass 750 kg up a straight hill of length 800 m inclined at an angle of \(\sin^{-1} 0.08\) to the horizontal. The resistances to the motion of the car and trailer are 400 N and 200 N respectively. The car and trailer are connected by a light rigid tow-bar. The car and trailer have speed \(30 \text{ m s}^{-1}\) at the bottom of the hill and \(20 \text{ m s}^{-1}\) at the top of the hill.

1. Use an energy method to find the constant driving force as the car and trailer travel up the hill. [5]
2. After reaching the top of the hill the system consisting of the car and trailer travels along a straight level road. The driving force of the car's engine is 2400 N and the resistances to motion are unchanged. Find the acceleration of the system and the tension in the tow-bar. [4]

A constant resistance of magnitude 1400 N acts on a car of mass 1250 kg.

1. The car is moving along a straight level road at a constant speed of 28 m s\(^{-1}\). Find, in kW, the rate at which the engine of the car is working. [2]
2. The car now travels at a constant speed up a hill inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.12\), with the engine working at 43.5 kW. Find this speed. [3]
3. On another occasion, the car pulls a trailer of mass 600 kg up the same hill. The system of the car and the trailer is modelled as particles connected by a light inextensible cable. The car's engine produces a driving force of 5000 N and the resistance to the motion of the trailer is 300 N. The resistance to the motion of the car remains 1400 N. Find the acceleration of the system and the tension in the cable. [4]

\includegraphics{figure_6_1} **Fig. 6.1** Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of 30° to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.

1. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane. [6]
2. \includegraphics{figure_6_2} **Fig. 6.2** It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level. [6]

\includegraphics{figure_5} A block \(A\) of mass 80 kg is connected by a light, inextensible rope to a block \(B\) of mass 40 kg. The rope joining the two blocks is taut and is parallel to a line of greatest slope of a plane which is inclined at an angle of \(20°\) to the horizontal. A force of magnitude 500 N inclined at an angle of \(15°\) above the same line of greatest slope acts on \(A\) (see diagram). The blocks move up the plane and there is a resistance force of 50 N on \(B\), but no resistance force on \(A\).

1. Find the acceleration of the blocks and the tension in the rope. [5]

1. Find the time that it takes for the blocks to reach a speed of \(1.2 \text{ m s}^{-1}\) from rest. [2]

\includegraphics{figure_6} The diagram shows a fixed block with a horizontal top surface and a surface which is inclined at an angle of \(\theta°\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). A particle \(A\) of mass \(0.3\) kg rests on the horizontal surface and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the block. The other end of the string is attached to a particle \(B\) of mass \(1.5\) kg which rests on the sloping surface of the block. The system is released from rest with the string taut.

1. Given that the block is smooth, find the acceleration of particle \(A\) and the tension in the string. [5]
2. It is given instead that the block is rough. The coefficient of friction between \(A\) and the block is \(\mu\) and the coefficient of friction between \(B\) and the block is also \(\mu\). In the first \(3\) seconds of the motion, \(A\) does not reach \(P\) and \(B\) does not reach the bottom of the sloping surface. The speed of the particles after \(3\) s is \(5\) m s\(^{-1}\). Find the acceleration of particle \(A\) and the value of \(\mu\). [9]

\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).

1. Find the acceleration of the blocks and the tension in the string. [5]

1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]

\includegraphics{figure_7} Two particles \(P\) and \(Q\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a smooth plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). \(P\) lies on the plane and \(Q\) hangs vertically below the pulley at a height of 0.8 m above the floor (see diagram). The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. \(P\) is released from rest and \(Q\) moves vertically downwards.

1. Find the tension in the string and the magnitude of the acceleration of the particles. [5]

\(Q\) hits the floor and does not bounce. It is given that \(P\) does not reach the pulley in the subsequent motion.

1. Find the time, from the instant at which \(P\) is released, for \(Q\) to reach the floor. [2]
2. When \(Q\) hits the floor the string becomes slack. Find the time, from the instant at which \(P\) is released, for the string to become taut again. [4]

\includegraphics{figure_3} A particle \(A\) of mass 4 kg moves on the inclined face of a smooth wedge. This face is inclined at 30° to the horizontal. The wedge is fixed on horizontal ground. Particle \(A\) is connected to a particle \(B\), of mass 3 kg, by a light inextensible string. The string passes over a small light smooth pulley which is fixed at the top of the plane. The section of the string from \(A\) to the pulley lies in a line of greatest slope of the wedge. The particle \(B\) hangs freely below the pulley, as shown in Fig. 3. The system is released from rest with the string taut. For the motion before \(A\) reaches the pulley and before \(B\) hits the ground, find

1. the tension in the string, [6]
2. the magnitude of the resultant force exerted by the string on the pulley. [3]

1. The string in this question is described as being 'light'.
   1. Write down what you understand by this description.
   2. State how you have used the fact that the string is light in your answer to part (a). [2]

\includegraphics{figure_4} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s. Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\), [3]
2. the tension in the string, [4]
3. the value of \(\mu\). [5]
4. State how in your calculations you have used the information that the string is inextensible. [1]

\includegraphics{figure_3} A fixed wedge has two plane faces, each inclined at \(30°\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3. The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\frac{1}{10}g\).

1. By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. [3]
2. By considering the motion of \(B\), find the value of \(\mu\). [8]
3. Find the resultant force exerted by the string on the pulley, giving its magnitude and direction. [3]

\includegraphics{figure_4} Figure 4 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small smooth light pulley \(A\) fixed at the top of the plane. The part of the string from \(P\) to \(A\) is parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below \(A\). The system is released from rest with the string taut.

1. Write down an equation of motion for \(P\) and an equation of motion for \(Q\). [4]
2. Hence show that the acceleration of \(Q\) is 0.98 m s\(^{-2}\). [2]
3. Find the tension in the string. [2]
4. State where in your calculations you have used the information that the string is inextensible. [1]

On release, \(Q\) is at a height of 0.8 m above the ground. When \(Q\) reaches the ground, it is brought to rest immediately by the impact with the ground and does not rebound. The initial distance of \(P\) from \(A\) is such that in the subsequent motion \(P\) does not reach \(A\). Find

1. the speed of \(Q\) as it reaches the ground, [2]
2. the time between the instant when \(Q\) reaches the ground and the instant when the string becomes taut again. [5]