3.03o Advanced connected particles: and pulleys

6 Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m . The particles are released from rest with both sections of the string taut.

1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
2. Given instead that the surface is rough and the coefficient of friction between \(A\) and the surface is 0.1 , find the speed of \(A\) immediately before it reaches the pulley. \(7 \quad\) A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
3. Find \(k\).
4. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
5. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
6. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).

5 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-3_300_792_274_680} Particles \(A\) and \(B\), of masses 0.4 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a horizontal table with the string passing over a smooth pulley at the edge of the table. Particle \(B\) hangs vertically below the pulley (see diagram). The system is released from rest. In the subsequent motion a constant frictional force of magnitude 0.6 N acts on \(A\). Find

1. the tension in the string,
2. the speed of \(B 1.5 \mathrm {~s}\) after it starts to move.

1 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-2_200_529_269_808} Two particles \(P\) and \(Q\), of masses 1.7 kg and 0.3 kg respectively, are connected by a light inextensible string. \(P\) is held on a smooth horizontal table with the string taut and passing over a small smooth pulley fixed at the edge of the table. \(Q\) is at rest vertically below the pulley. \(P\) is released. Find the acceleration of the particles and the tension in the string.

5 \includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-3_643_481_274_831} Particles \(A\) and \(B\), of masses 0.5 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held at rest on the horizontal floor and particle \(A\) hangs in equilibrium (see diagram). \(B\) is released and each particle starts to move vertically. \(A\) hits the floor 2 s after \(B\) is released. The speed of each particle when \(A\) hits the floor is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion while \(A\) is moving downwards, find
   1. the acceleration of \(A\),
   2. the tension in the string.
   3. Find the value of \(m\).

6 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-4_712_529_264_810} Particles \(P\) and \(Q\), of masses 0.55 kg and 0.45 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The particles are held at rest with the string taut and its straight parts vertical. Both particles are at a height of 5 m above the ground (see diagram). The system is released.

1. Find the acceleration with which \(P\) starts to move. The string breaks after 2 s and in the subsequent motion \(P\) and \(Q\) move vertically under gravity.
2. At the instant that the string breaks, find
   1. the height above the ground of \(P\) and of \(Q\),
   2. the speed of the particles.
   3. Show that \(Q\) reaches the ground 0.8 s later than \(P\). \(7 \quad\) A particle \(P\) starts from rest at the point \(A\) at time \(t = 0\), where \(t\) is in seconds, and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . For \(10 \leqslant t \leqslant 20 , P\) continues to move along the line with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 800 } { t ^ { 2 } } - 2\). Find
      1. the speed of \(P\) when \(t = 10\), and the value of \(a\),
      2. the value of \(t\) for which the acceleration of \(P\) is \(- a \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
      3. the displacement of \(P\) from \(A\) when \(t = 20\).

6 \includegraphics[max width=\textwidth, alt={}, center]{efa7175f-832b-4cd3-82ab-52e402115081-4_686_511_269_817} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(A\) is held on the horizontal floor and particle \(B\) hangs in equilibrium. Particle \(A\) is released and both particles start to move vertically.

1. Find the acceleration of the particles. The speed of the particles immediately before \(B\) hits the floor is \(1.6 \mathrm {~ms} ^ { - 1 }\). Given that \(B\) does not rebound upwards, find
2. the maximum height above the floor reached by \(A\),
3. the time taken by \(A\), from leaving the floor, to reach this maximum height.

3 Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the string taut, with its straight parts vertical, and with both particles at a height of 2 m above horizontal ground. \(P\) moves vertically downwards and does not rebound when it hits the ground. At the instant that \(P\) hits the ground, \(Q\) is at the point \(X\), from where it continues to move vertically upwards without reaching the pulley. Given that \(P\) has mass 0.9 kg and that the tension in the string is 7.2 N while \(P\) is moving, find the total distance travelled by \(Q\) from the instant it first reaches \(X\) until it returns to \(X\).

2 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_212_625_528_761} Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. \(A\) is held at rest on a rough horizontal table with the string passing over a small smooth pulley at the edge of the table. \(B\) hangs vertically below the pulley (see diagram). The system is released and \(B\) starts to move downwards with acceleration \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find

1. the tension in the string after the system is released,
2. the frictional force acting on \(A\).

2 Particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and \(( 1 - m ) \mathrm { kg }\) respectively are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is released from rest with the straight parts of the string vertical. \(A\) moves vertically downwards and 0.3 seconds later it has speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of \(A\),
2. the value of \(m\) and the tension in the string.

7 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.

1. Find the acceleration of \(A\) and the tension in the string. \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
2. \(A\) to reach the floor,
3. \(B\) to reach the pulley.

6 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor. \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).

6 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-3_526_519_902_813} Particles \(A\) of mass 0.4 kg and \(B\) of mass 1.6 kg are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 1.2 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the work done on \(B\) by the tension in the string, as \(B\) moves to the floor. When particle \(B\) reaches the floor it remains at rest. Particle \(A\) continues to move upwards.
2. Find the greatest height above the floor reached by particle \(A\).

2 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).

7 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-12_565_511_260_817} Two particles \(A\) and \(B\) have masses \(m \mathrm {~kg}\) and \(k m \mathrm {~kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle \(B\) reaches the ground 0.9 s later. The particle \(A\) does not reach the pulley in its subsequent motion.

1. Find the value of \(k\) and show that the tension in the string before \(B\) reaches the ground is equal to \(12 m \mathrm {~N}\).
   At the instant when \(B\) reaches the ground, the string breaks.
2. Show that the speed of \(A\) when it reaches the ground is \(5.97 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures, and find the time taken, after the string breaks, for \(A\) to reach the ground.
3. Sketch a velocity-time graph for the motion of particle \(A\) from the instant when the system is released until \(A\) reaches the ground. If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.

1. Find the magnitude of the acceleration of \(B\) immediately after release.
2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.

6. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 0.5 kg and \(m \mathrm {~kg}\) respectively, where \(m < 0.5\). The particles are connected by a light inextensible string which passes over a smooth, fixed pulley. Initially \(P\) is 3.15 m above horizontal ground. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 4. After \(P\) has been descending for 1.5 s , it strikes the ground. Particle \(P\) reaches the ground before \(Q\) has reached the pulley.

1. Show that the acceleration of \(P\) as it descends is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string as \(P\) descends.
3. Show that \(m = \frac { 5 } { 18 }\).
4. State how you have used the information that the string is inextensible. When \(P\) strikes the ground, \(P\) does not rebound and the string becomes slack. Particle \(Q\) then moves freely under gravity, without reaching the pulley, until the string becomes taut again.
5. Find the time between the instant when \(P\) strikes the ground and the instant when the string becomes taut again.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find

1. the acceleration of \(Q\),
2. the value of \(\mu\),
3. the tension in the string.
4. State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.

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.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 4 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough plane, which is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 4 } { 3 }\). The coefficient of friction between \(P\) and the plane is 0.5 . The string lies along the plane and passes over a small smooth light pulley which is fixed at the top of the plane. Particle \(Q\) hangs freely at rest vertically below the pulley. The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 4. Particle \(P\) is released from rest with the string taut and slides down the plane. Given that \(Q\) has not hit the pulley, find

1. the tension in the string during the motion,
2. the magnitude of the resultant force exerted by the string on the pulley.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 5 }\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table. Assuming that \(P\) has not reached the pulley, find

1. the tension in the string during the motion,
2. the magnitude and direction of the resultant force exerted on the pulley by the string.

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
4. find \(d\) in terms of \(h\).
5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)

   (Total 15 marks)	Leave blank Q8

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(4 m\) is held at rest at the point \(X\) on the surface of a rough inclined plane which is fixed to horizontal ground. The point \(X\) is a distance \(h\) from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle \(P\) is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which hangs freely at a distance \(d\), where \(d > h\), below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(P\) moves down the plane. For the motion of the particles before \(P\) hits the ground,

1. state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,
2. write down an equation of motion for each particle,
3. find the acceleration of each particle. When \(P\) hits the ground, it immediately comes to rest. Given that \(Q\) comes to instantaneous rest before reaching the pulley,
4. show that \(d > \frac { 28 h } { 25 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-27_56_20_109_1950}
   

   END

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 3 kg and \(m \mathrm {~kg}\) respectively ( \(m > 3\) ). The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut and the hanging parts of the string vertical. The particle \(Q\) is at a height of 10.5 m above the horizontal ground, as shown in Figure 5. The system is released from rest and \(Q\) moves downwards. In the subsequent motion \(P\) does not reach the pulley. After the system is released, the tension in the string is 33.6 N .

1. Show that the magnitude of the acceleration of \(P\) is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(m\). The system is released from rest at time \(t = 0\). At time \(T _ { 1 }\) seconds after release, \(Q\) strikes the ground and does not rebound. The string goes slack and \(P\) continues to move upwards.
3. Find the value of \(T _ { 1 }\) At time \(T _ { 2 }\) seconds after release, \(P\) comes to instantaneous rest.
4. Find the value of \(T _ { 2 }\) At time \(T _ { 3 }\) seconds after release ( \(T _ { 3 } > T _ { 1 }\) ) the string becomes taut again.
5. Sketch a velocity-time graph for the motion of \(P\) in the interval \(0 \leqslant t \leqslant T _ { 3 }\)

8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a block \(A\) of mass 3 kg . Block \(A\) is held at rest on a smooth fixed plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal ground. The string lies along a line of greatest slope of the plane and passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a block \(B\) of mass 5 kg . Block \(B\) hangs freely at rest below the pulley, as shown in Figure 4. The system is released from rest with the string taut. By modelling the two blocks as particles,

1. find the tension in the string as \(B\) descends. After falling for 1.5 s , block \(B\) hits the ground and is immediately brought to rest. In its subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(B\) at the instant it hits the ground.
3. Find the total distance moved up the plane by \(A\) before it comes to instantaneous rest. \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_97_141_2519_1804} \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-28_125_161_2624_1779}

7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(m\) and \(3 m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a rough horizontal table. The coefficient of friction between particle \(A\) and the table is \(\frac { 1 } { 5 }\). The string lies along the table and passes over a small smooth light pulley that is fixed at the edge of the table. Particle \(B\) is at rest on a rough plane that is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 4. The coefficient of friction between particle \(B\) and the inclined plane is \(\frac { 1 } { 3 }\). The string lies in the vertical plane that contains the pulley and a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(B\) slides down the inclined plane. Given that \(A\) does not reach the pulley,

1. find the tension in the string,
2. state where in your working you have used the fact that the string is modelled as being light,
3. find the magnitude of the force exerted on the pulley by the string.
   
   \includegraphics[max width=\textwidth, alt={}, center]{0d5a56ba-6a33-4dc8-b612-d2957211124f-24_172_1824_2581_123} \includegraphics[max width=\textwidth, alt={}, center]{0d5a56ba-6a33-4dc8-b612-d2957211124f-24_157_85_2595_1966}