3.03l Newton's third law: extend to situations requiring force resolution

7 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-10_551_776_260_689} Two particles \(P\) and \(Q\), of masses 2 kg and 0.25 kg respectively, are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(P\) is on an inclined plane at an angle of \(30 ^ { \circ }\) to the horizontal. Particle \(Q\) hangs below the pulley. Three points \(A , B\) and \(C\) lie on a line of greatest slope of the plane with \(A B = 0.8 \mathrm {~m}\) and \(B C = 1.2 \mathrm {~m}\) (see diagram). Particle \(P\) is released from rest at \(A\) with the string taut and slides down the plane. During the motion of \(P\) from \(A\) to \(C , Q\) does not reach the pulley. The part of the plane from \(A\) to \(B\) is rough, with coefficient of friction 0.3 between the plane and \(P\). The part of the plane from \(B\) to \(C\) is smooth.

1. 1. Find the acceleration of \(P\) between \(A\) and \(B\).
   2. Hence, find the speed of \(P\) at \(C\).
2. Find the time taken for \(P\) to travel from \(A\) to \(C\).
   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 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-12_439_1095_258_525} Two particles \(P\) and \(Q\) of masses 0.5 kg and \(m \mathrm {~kg}\) respectively are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the top of two inclined planes. The particles are initially at rest with \(P\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal and \(Q\) on a plane inclined at \(45 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 0.8 N is applied to \(P\) acting down the plane, causing \(P\) to move down the plane (see diagram).

1. It is given that \(m = 0.3\), and that the plane on which \(Q\) rests is smooth. Find the tension in the string.
2. It is given instead that the plane on which \(Q\) rests is rough, and that after each particle has moved a distance of 1 m , their speed is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The work done against friction in this part of the motion is 0.5 J . Use an energy method to find the value of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8 \includegraphics[max width=\textwidth, alt={}, center]{fcc3d739-5c36-48ad-9c34-f69b28a06dba-14_388_1216_264_461} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to a horizontal plane and to the top of an inclined plane. The particles are initially at rest with \(A\) on the horizontal plane and \(B\) on the inclined plane, which makes an angle of \(30 ^ { \circ }\) with the horizontal. The string is taut and \(B\) can move on a line of greatest slope of the inclined plane. A force of magnitude 3.5 N is applied to \(B\) acting down the plane (see diagram).

1. Given that both planes are smooth, find the tension in the string and the acceleration of \(B\).
2. It is given instead that the two planes are rough. When each particle has moved a distance of 0.6 m from rest, the total amount of work done against friction is 1.1 J . Use an energy method to find the speed of \(B\) when it has moved this distance down the plane. [You should assume that the string is sufficiently long so that \(A\) does not hit the pulley when it moves 0.6 m .]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A car of mass 1600 kg is pulling a caravan of mass 800 kg . The car and the caravan are connected by a light rigid tow-bar. The resistances to the motion of the car and caravan are 400 N and 250 N respectively.

1. The car and caravan are travelling along a straight horizontal road.
   1. Given that the car and caravan have a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the power of the car's engine.
   2. The engine's power is now suddenly increased to 39 kW . Find the instantaneous acceleration of the car and caravan and find the tension in the tow-bar.
2. The car and caravan now travel up a straight hill, inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal, at a constant speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's engine is working at 32.5 kW . Find \(v\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ac4bb5a0-c7c0-4e1d-9e76-64f92ae28066-10_214_1461_255_342} As shown in the diagram, particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the top of two inclined planes. Particle \(A\) is on plane \(P\), which is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. Particle \(B\) is on plane \(Q\), which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The string is taut, and the two parts of the string are parallel to lines of greatest slope of their respective planes.

1. It is given that plane \(P\) is smooth, plane \(Q\) is rough, and the particles are in limiting equilibrium. Find the coefficient of friction between particle \(B\) and plane \(Q\).
2. It is given instead that both planes are smooth and that the particles are released from rest at the same horizontal level. Find the time taken until the difference in the vertical height of the particles is 1 m . [You should assume that this occurs before \(A\) reaches the pulley or \(B\) reaches the bottom of plane \(Q\).] [6]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest 14 s later. At time \(t \mathrm {~s}\) after leaving \(O\), the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is given by $$\begin{array} { l l } v = p t ^ { 2 } - q t & 0 \leqslant t \leqslant 6 \\ v = 63 - 4.5 t & 6 \leqslant t \leqslant 14 \end{array}$$ where \(p\) and \(q\) are positive constants.
The acceleration of \(P\) is zero when \(t = 2\).

1. Given that there are no instantaneous changes in velocity, find \(p\) and \(q\).
2. Sketch the velocity-time graph.
3. Find the total distance travelled by \(P\) during the 14 s . \includegraphics[max width=\textwidth, alt={}, center]{e1b91e54-a3ae-436c-a4f7-7095891f7034-10_326_1109_255_520} Two particles \(A\) and \(B\) of masses 2 kg and 3 kg respectively are connected by a light inextensible string. Particle \(B\) is on a smooth fixed plane which is at an angle of \(18 ^ { \circ }\) to horizontal ground. The string passes over a fixed smooth pulley at the top of the plane. Particle \(A\) hangs vertically below the pulley and is 0.45 m above the ground (see diagram). The system is released from rest with the string taut. When \(A\) reaches the ground, the string breaks. Find the total distance travelled by \(B\) before coming to instantaneous rest. You may assume that \(B\) does not reach the pulley.
   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 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-4_601_515_699_815} Two particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. Particle \(B\) is held on the horizontal floor and particle \(A\) hangs in equilibrium. Particle \(B\) is released and each particle starts to move vertically with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(a\). Particle \(A\) hits the floor 1.2 s after it starts to move, and does not rebound upwards.
2. Show that \(A\) hits the floor with a speed of \(2.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the gain in gravitational potential energy by \(B\), from leaving the floor until reaching its greatest height.

5 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_250_846_260_648} A small block \(B\) of mass 0.25 kg is attached to the mid-point of a light inextensible string. Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of the string. The string passes over two smooth pulleys fixed at opposite sides of a rough table, with \(B\) resting in limiting equilibrium on the table between the pulleys and particles \(P\) and \(Q\) and block \(B\) are in the same vertical plane (see diagram).

1. Find the coefficient of friction between \(B\) and the table. \(Q\) is now removed so that \(P\) and \(B\) begin to move.
2. Find the acceleration of \(P\) and the tension in the part \(P B\) of the string.

5 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_289_567_1233_788} Particles \(A\) and \(B\), each of mass 0.3 kg , are connected by a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal surface. Particle \(A\) hangs freely and particle \(B\) is held at rest in contact with the surface (see diagram). The coefficient of friction between \(B\) and the surface is 0.7 . Particle \(B\) is released and moves on the surface without reaching the pulley.

1. Find, for the first 0.9 m of \(B\) 's motion,

4 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-2_499_784_1617_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2 . Find the set of values of \(m\) for which the two blocks remain at rest.

5 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.

1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the length of the string.

4 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
2. the greatest height above the floor reached by \(B\).

1 \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_241_823_264_660} Two particles \(P\) and \(Q\), of masses 0.6 kg and 0.4 kg respectively, are connected by a light inextensible string. The string passes over a small smooth light pulley fixed at the edge of a smooth horizontal table. Initially \(P\) is held at rest on the table and \(Q\) hangs vertically (see diagram). \(P\) is then released. Find the tension in the string and the acceleration of \(Q\).

6 A van of mass 3000 kg is pulling a trailer of mass 500 kg along a straight horizontal road at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The system of the van and the trailer is modelled as two particles connected by a light inextensible cable. There is a constant resistance to motion of 300 N on the van and 100 N on the trailer.

1. Find the power of the van's engine.
2. Write down the tension in the cable. The van reaches the bottom of a hill inclined at \(4 ^ { \circ }\) to the horizontal with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The power of the van's engine is increased to 25000 W .
3. Assuming that the resistance forces remain the same, find the new tension in the cable at the instant when the speed of the van up the hill is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_312_1207_1320_468} Particles \(P\) and \(Q\), of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).

1. Find the acceleration of the particles and the speed of \(P\) immediately before it reaches the floor.
2. Determine whether \(Q\) comes to instantaneous rest before it reaches the pulley directly above it.

7 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-10_212_1029_255_557} Two particles \(A\) and \(B\) of masses 0.9 kg and 0.4 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 top of two inclined planes. The particles are initially at rest with \(A\) on a smooth plane inclined at angle \(\theta ^ { \circ }\) to the horizontal and \(B\) on a plane inclined at angle \(25 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 2.5 N is applied to \(B\) acting down the plane (see diagram).

1. For the case where \(\theta = 15\) and the plane on which \(B\) rests is smooth, find the acceleration of \(B\).
2. For a different value of \(\theta\), the plane on which \(B\) rests is rough with coefficient of friction between the plane and \(B\) of 0.8 . The system is in limiting equilibrium with \(B\) on the point of moving in the direction of the 2.5 N force. Find the value of \(\theta\).

4 Two particles \(A\) and \(B\) have masses 0.35 kg and 0.45 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small fixed smooth pulley which is 1 m above horizontal ground. Initially particle \(A\) is held at rest on the ground vertically below the pulley, with the string taut. Particle \(B\) hangs vertically below the pulley at a height of 0.64 m above the ground. Particle \(A\) is released.

1. Find the speed of \(A\) at the instant that \(B\) reaches the ground.
2. Assuming that \(B\) does not bounce after it reaches the ground, find the total distance travelled by \(A\) between the instant that \(B\) reaches the ground and the instant when the string becomes taut again.

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.

5. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\) have masses \(2 m\) and \(3 m\) respectively. 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. The hanging parts of the string are vertical and \(A\) and \(B\) are above a horizontal plane, as shown in Figure 2. The system is released from rest.

1. Show that the tension in the string immediately after the particles are released is \(\frac { 12 } { 5 } m g\). After descending \(1.5 \mathrm {~m} , B\) strikes the plane and is immediately brought to rest. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the distance travelled by \(A\) between the instant when \(B\) strikes the plane and the instant when the string next becomes taut. Given that \(m = 0.5 \mathrm {~kg}\),
3. find the magnitude of the impulse on \(B\) due to the impact with the plane.

7. \begin{figure}[h] \end{figure} Three particles \(A , B\) and \(C\) have masses \(3 m , 2 m\) and \(2 m\) respectively. Particle \(C\) is attached to particle \(B\). Particles \(A\) and \(B\) 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, as shown in Figure 5. The system is released from rest and \(A\) moves upwards.

1. 1. Show that the acceleration of \(A\) is \(\frac { g } { 7 }\)
   2. Find the tension in the string as \(A\) ascends. At the instant when \(A\) is 0.7 m above its original position, \(C\) separates from \(B\) and falls away. In the subsequent motion, \(A\) does not reach the pulley.
2. Find the speed of \(A\) at the instant when it is 0.7 m above its original position.
3. Find the acceleration of \(A\) at the instant after \(C\) separates from \(B\).
4. Find the greatest height reached by \(A\) above its original position. \includegraphics[max width=\textwidth, alt={}, center]{ed659098-c1cf-4ee1-a12a-bf8b6c42db95-14_115_161_2455_1784}

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.

1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
2. find, in terms of \(h\), the range of possible values of \(d\).
3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(4 m\) lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The particle \(P\) is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which lies on a smooth horizontal plane. The string lies along the horizontal plane and 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, as shown in Figure 4, and \(P\) moves down the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) For the motion before \(Q\) reaches the pulley

1. write down an equation of motion for \(Q\),
2. find, in terms of \(m\) and \(g\), the tension in the string,
3. find the magnitude of the force exerted on the pulley by the string.
4. State where in your working you have used the information that the string is light.

7. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a small smooth fixed pulley at the top of the plane. The particle \(Q\) hangs freely below the pulley and 0.6 m above the ground, as shown in Figure 3. The part of the string from \(P\) to the pulley is parallel to a line of greatest slope of the plane. The system is released from rest with the string taut. For the motion before \(Q\) hits the ground,

1. 1. show that the acceleration of \(Q\) is \(\frac { 2 g } { 5 }\),
   2. find the tension in the string. On hitting the ground \(Q\) is immediately brought to rest by the impact.
2. Find the speed of \(P\) at the instant when \(Q\) hits the ground. In its subsequent motion \(P\) does not reach the pulley.
3. Find the total distance moved up the plane by \(P\) before it comes to instantaneous rest.
4. Find the length of time between \(Q\) hitting the ground and \(P\) first coming to instantaneous rest.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs at rest vertically below the pulley, at a height \(h\) above a horizontal plane, as shown in Figure 3. The coefficient of friction between \(P\) and the table is 0.5 . Particle \(P\) is released from rest with the string taut and slides along the table.

1. Find, in terms of \(m g\), the tension in the string while both particles are moving. The particle \(P\) does not reach the pulley before \(Q\) hits the plane.
2. Show that the speed of \(Q\) immediately before it hits the plane is \(\sqrt { 1.4 g h }\) When \(Q\) hits the plane, \(Q\) does not rebound and \(P\) continues to slide along the table. Given that \(P\) comes to rest before it reaches the pulley,
3. show that the total length of the string must be greater than 2.4 h

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.