3.03k Connected particles: pulleys and equilibrium

5. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 4. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 250 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 150 N . The towbar is modelled as a light rod.
At the instant when the speed of the van is \(16 \mathrm {~ms} ^ { - 1 }\), the engine of the van is working at a rate of 10 kW .

1. Find the deceleration of the van at this instant.
2. Find the tension in the towbar at this instant.

2. A trailer of mass 250 kg is towed by a car of mass 1000 kg . The car and the trailer are travelling down a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\) The resistance to motion of the car is modelled as a single force of magnitude 300 N acting parallel to the road. The resistance to motion of the trailer is modelled as a single force of magnitude 100 N acting parallel to the road. The towbar joining the car to the trailer is modelled as a light rod which is parallel to the direction of motion. At a given instant the car and the trailer are moving down the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the power being developed by the car's engine at this instant.
2. Find the tension in the towbar at this instant.

3. Two particles \(P\) and \(Q\), of mass \(2 m\) and \(3 m\) respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed rough plane inclined at \(\theta\) to the horizontal ground, where \(\sin \theta = \frac { 2 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. At time \(t = 0\) the system is released from rest with the string taut. When \(P\) moves the friction between \(P\) and the plane is modelled as a constant force of magnitude \(\frac { 3 } { 5 } m g\). At the instant when each particle has moved a distance \(d\), they are both moving with speed \(v\), particle \(P\) has not reached the pulley and \(Q\) has not reached the ground.

1. Show that the total potential energy lost by the system when each particle has moved a distance \(d\) is \(\frac { 11 } { 5 } m g d\).
2. Use the work-energy principle to find \(v ^ { 2 }\) in terms of \(g\) and \(d\). When \(t = T\) seconds, \(d = 1.5 \mathrm {~m}\).
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

5. \begin{figure}[h] \end{figure} One end \(A\) of a light inextensible string of length \(3 a\) is attached to a fixed point. A particle of mass \(m\) is attached to the other end \(B\) of the string. The particle is held in equilibrium at a distance \(2 a\) below the horizontal through \(A\), with the string taut. The particle is then projected with speed \(\sqrt { } ( 2 a g )\), in the direction perpendicular to \(A B\), in the vertical plane containing \(A\) and \(B\), as shown in Figure 4. In the subsequent motion the string remains taut. When \(A B\) is at an angle \(\theta\) below the horizontal, the speed of the particle is \(v\) and the tension in the string is \(T\).

1. Show that \(v ^ { 2 } = 2 \operatorname { ag } ( 3 \sin \theta - 1 )\).
2. Find the range of values of \(T\).

7 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-4_369_508_246_781} Particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.05 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth pulley. \(Q\) is attached to a particle \(R\) of mass 0.45 kg by a light inextensible string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. \(P\) is in contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the string \(Q R\) is 2.52 N during the descent of \(R\).

1. (a) Find the acceleration of \(R\) during its descent.
   (b) By considering the motion of \(Q\), calculate the tension in the string \(P Q\) during the descent of \(R\).
2. Find the value of \(m\). \(R\) strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, \(P\) does not reach the pulley and \(Q\) does not reach the surface.
3. Calculate the greatest height of \(P\) above the surface.

5 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-3_462_405_258_845} A small smooth pulley is suspended from a fixed point by a light chain. A light inextensible string passes over the pulley. Particles \(P\) and \(Q\), of masses 0.3 kg and \(m \mathrm {~kg}\) respectively, are attached to the opposite ends of the string. The particles are released from rest at a height of 0.2 m above horizontal ground with the string taut; the portions of the string not in contact with the pulley are vertical (see diagram). \(P\) strikes the ground with speed \(1.4 \mathrm {~ms} ^ { - 1 }\). Subsequently \(P\) remains on the ground, and \(Q\) does not reach the pulley.

1. Calculate the acceleration of \(P\) while it is in motion and the corresponding tension in the string.
2. Find the value of \(m\).
3. Calculate the greatest height of \(Q\) above the ground.
4. It is given that the mass of the pulley is 0.5 kg . State the magnitude of the tension in the chain which supports the pulley
   1. when \(P\) is in motion,
   2. when \(P\) is at rest on the ground and \(Q\) is moving upwards.

2 Particles \(P\) and \(Q\), of masses 0.45 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley. The particles are released from rest with the string taut and both particles 0.36 m above a horizontal surface. \(Q\) descends with acceleration \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). When \(Q\) strikes the surface, it remains at rest.

1. Calculate the tension in the string while both particles are in motion.
2. Find the value of \(m\).
3. Calculate the speed at which \(Q\) strikes the surface.
4. Calculate the greatest height of \(P\) above the surface. (You may assume that \(P\) does not reach the pulley.)

2 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h] \end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.

1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
2. Calculate the tension in the rope.

4 As shown in Fig. 4, boxes P and Q are descending vertically supported by a parachute. Box P has mass 75 kg . Box Q has mass 25 kg and hangs from box P by means of a light vertical wire. Air resistance on the boxes should be neglected. At one stage the boxes are slowing in their descent with the parachute exerting an upward vertical force of 1030 N on box P . The acceleration of the boxes is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards and the tension in the wire is \(T \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on box P and another diagram showing all the forces acting on box Q .
2. Write down separate equations of motion for box P and for box Q .
3. Calculate the tension in the wire.

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 \(\beta\).

5 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.

7 \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-5_510_1091_269_479} \(A\) and \(B\) are points at the upper and lower ends, respectively, of a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(M\) is the mid-point of \(A B\). Two particles \(P\) and \(Q\), joined by a taut light inextensible string, are placed on the plane at \(A\) and \(M\) respectively. The particles are simultaneously projected with speed \(0.6 \mathrm {~ms} ^ { - 1 }\) down the line of greatest slope (see diagram). The particles move down the plane with acceleration \(0.9 \mathrm {~ms} ^ { - 2 }\). At the instant 2 s after projection, \(P\) is at \(M\) and \(Q\) is at \(B\). The particle \(Q\) subsequently remains at rest at \(B\).

1. Find the distance \(A B\). The plane is rough between \(A\) and \(M\), but smooth between \(M\) and \(B\).
2. Calculate the speed of \(P\) when it reaches \(B\). \(P\) has mass 0.4 kg and \(Q\) has mass 0.3 kg .
3. By considering the motion of \(Q\), calculate the tension in the string while both particles are moving down the plane.
4. Calculate the coefficient of friction between \(P\) and the plane between \(A\) and \(M\). \section*{END OF QUESTION PAPER}

3. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(3 m\) and \(k m\) respectively, where \(k > 3\). They are connected by a light inextensible string which passes over a smooth fixed pulley. The system is released from rest with the string taut and the hanging parts of the string vertical, as shown in Fig. 3. While the particles are moving freely, \(A\) has an acceleration of magnitude \(\frac { 2 } { 5 } g\).

1. Find, in terms of \(m\) and \(g\), the tension in the string.
   (3 marks)
2. State why \(B\) also has an acceleration of magnitude \(\frac { 2 } { 5 } g\).
3. Find the value of \(k\).
4. State how you have used the fact that the string is light.
   (1 mark)

2 Particles of mass 2 kg and 4 kg are attached to the ends \(X\) and \(Y\) of a light, inextensible string. The string passes round fixed, smooth pulleys at \(\mathrm { P } , \mathrm { Q }\) and R , as shown in Fig. 2. The system is released from rest with the string taut. \begin{figure}[h] \end{figure}

1. State what information in the question tells you that
   (A) the tension is the same throughout the string,
   (B) the magnitudes of the accelerations of the particles at X and Y are the same. The tension in the string is \(T \mathrm {~N}\) and the magnitude of the acceleration of the particles is \(a \mathrm {~ms} ^ { - 2 }\).
2. Draw a diagram showing the forces acting at X and a diagram showing the forces acting at Y .
3. Write down equations of motion for the particles at X and at Y . Hence calculate the values of \(T\) and \(a\).

3 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-2_451_533_1676_808} One end of a light inextensible string of length 1.6 m is attached to a point \(P\). The other end is attached to the point \(Q\), vertically below \(P\), where \(P Q = 0.8 \mathrm {~m}\). A small smooth bead \(B\), of mass 0.01 kg , is threaded on the string and moves in a horizontal circle, with centre \(Q\) and radius \(0.6 \mathrm {~m} . Q B\) rotates with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see diagram).

1. Show that the tension in the string is 0.1225 N .
2. Find \(\omega\).
3. Calculate the kinetic energy of the bead.

2. \begin{figure}[h] \end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
3. State one limitation of the model that will affect the accuracy of your answer to part (a).

2. \begin{figure}[h] \end{figure} One end of a string is attached to a small ball \(P\) of mass \(4 m\).
The other end of the string is attached to another small ball \(Q\) of mass \(3 m\).
The string passes over a fixed pulley.
Ball \(P\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. Ball \(P\) is released.
The string is modelled as being light and inextensible, the balls are modelled as particles, the pulley is modelled as being smooth and air resistance is ignored.

1. Using the model, find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(P\) is falling and before \(Q\) hits the pulley.
2. State one limitation of the model, apart from ignoring air resistance, that will affect the accuracy of your answer to part (a).

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.

3. \begin{figure}[h] \end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
The string passes over a fixed pulley.
The system is held at rest with the balls hanging freely and the string taut.
The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit the pulley. The balls are modelled as particles.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Air resistance is modelled as being negligible.
Using this model,

1. 1. write down an equation of motion for \(P\),
   2. write down an equation of motion for \(Q\),
2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
4. State how this would affect the accelerations of the particles.

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2. \begin{figure}[h] \end{figure} A small stone \(A\) of mass \(3 m\) is attached to one end of a string.
A small stone \(B\) of mass \(m\) is attached to the other end of the string.
Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) The string passes over a pulley \(P\) that is fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane.
Stone \(B\) hangs freely below \(P\), as shown in Figure 1.
The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 6 }\) Stone \(A\) is released from rest and begins to move down the plane.
The stones are modelled as particles.
The pulley is modelled as being small and smooth.
The string is modelled as being light and inextensible. Using the model for the motion of the system before \(B\) reaches the pulley,

1. write down an equation of motion for \(A\)
2. show that the acceleration of \(A\) is \(\frac { 1 } { 10 } \mathrm {~g}\)
3. sketch a velocity-time graph for the motion of \(B\), from the instant when \(A\) is released from rest to the instant just before \(B\) reaches the pulley, explaining your answer. In reality, the string is not light.
4. State how this would affect the working in part (b).

10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}

1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.

11 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-08_586_672_1231_242} A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) in the positive \(x\)-direction is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = t ( t - 3 ) ( 8 - t )\). \(P\) attains its maximum velocity at time \(T\) seconds. The diagram shows part of the velocity-time graph for the motion of \(P\).

1. State the acceleration of \(P\) at time \(T\).
2. In this question you must show detailed reasoning. Determine the value of \(T\).
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = T\). \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-09_524_410_251_242} Particles \(P\) and \(Q\), of masses 4 kg and 6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in equilibrium with \(P\) hanging 1.75 m above a horizontal plane and \(Q\) resting on the plane. Both parts of the string below the pulley are vertical (see diagram).
   1. Find the magnitude of the normal reaction force acting on \(Q\). The mass of \(P\) is doubled, and the system is released from rest. You may assume that in the subsequent motion \(Q\) does not reach the pulley.
   2. Determine the magnitude of the force exerted on the pulley by the string before \(P\) strikes the plane.
   3. Determine the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest. When this motion is observed in practice, it is found that the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest is less than the answer calculated in part (c).
   4. State one factor that could account for this difference.

11 Two balls \(P\) and \(Q\) have masses 0.6 kg and 0.4 kg respectively. The balls are attached to the ends of a string. The string passes over a pulley which is fixed at the edge of a rough horizontal surface. Ball \(P\) is held at rest on the surface 2 m from the pulley. Ball \(Q\) hangs vertically below the pulley. Ball \(Q\) is attached to a third ball \(R\) of mass \(m \mathrm {~kg}\) by another string and \(R\) hangs vertically below \(Q\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{8c0b68bd-2257-4994-b444-def0b3f64334-7_419_945_493_246} The system is released from rest with the strings taut. Ball \(P\) moves towards the pulley with acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\) and a constant frictional force of magnitude 4.5 N opposes the motion of \(P\). The balls are modelled as particles, the pulley is modelled as being small and smooth, and the strings are modelled as being light and inextensible.

1. By considering the motion of \(P\), find the tension in the string connecting \(P\) and \(Q\).
2. Hence determine the value of \(m\). Give your answer correct to \(\mathbf { 3 }\) significant figures. When the balls have been in motion for 0.4 seconds the string connecting \(Q\) and \(R\) breaks.
3. Show that, according to the model, \(P\) does not reach the pulley. It is given that in fact ball \(P\) does reach the pulley.
4. Identify one factor in the modelling that could account for this difference.

4 Fig. 4 shows a block of mass \(4 m \mathrm {~kg}\) and a particle of mass \(m \mathrm {~kg}\) connected by a light inextensible string passing over a smooth pulley. The block is on a horizontal table, and the particle hangs freely. The part of the string between the pulley and the block is horizontal. The block slides towards the pulley and the particle descends. In this motion, the friction force between the table and the block is \(\frac { 1 } { 2 } m g \mathrm {~N}\). \begin{figure}[h] \end{figure} Find expressions for

• the acceleration of the system,
• the tension in the string.

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.