Newton's laws and connected particles

3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).

13 A resultant force of \(\left[ \begin{array} { c } - 2 \\ 6 \end{array} \right] \mathrm { N }\) acts on a particle.
The acceleration of the particle is \(\left[ \begin{array} { c } - 6 \\ y \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }\)
Find the value of \(y\)
Circle your answer.
[0pt] [1 mark] 231018

10 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A particle \(R\) of mass 2 kg is moving on a smooth horizontal surface under the action of a single horizontal force \(\mathbf { F }\) N. At time \(t\) seconds, the velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) of \(R\), relative to a fixed origin \(O\), is given by \(\mathbf { v } = \left( p t ^ { 2 } - 3 t \right) \mathbf { i } + ( 8 t + q ) \mathbf { j }\), where \(p\) and \(q\) are constants and \(p < 0\).

1. Given that when \(t = 0.5\) the magnitude of \(\mathbf { F }\) is 20 , find the value of \(p\). When \(t = 0 , R\) is at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m }\).
2. Find, in terms of \(q\), an expression for the displacement vector of \(R\) at time \(t\). When \(t = 1 , R\) is at a point on the line \(L\), where \(L\) passes through \(O\) and the point with position vector \(2 \mathbf { i } - 8 \mathbf { j }\).
3. Find the value of \(q\).
   \includegraphics[max width=\textwidth, alt={}, center]{7d1b7598-8f97-43a0-8366-efa8192d549e-09_544_1297_251_255} The diagram shows a ladder \(A B\), of length \(2 a\) and mass \(m\), resting in equilibrium on a vertical wall of height \(h\). The ladder is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The end \(A\) is in contact with horizontal ground. An object of mass \(2 m\) is placed on the ladder at a point \(C\) where \(A C = d\). The ladder is modelled as uniform, the ground is modelled as being rough, and the vertical wall is modelled as being smooth.
4. Show that the normal contact force between the ladder and the wall is \(\frac { m g ( a + 2 d ) \sqrt { 3 } } { 4 h }\). It is given that the equilibrium is limiting and the coefficient of friction between the ladder and the ground is \(\frac { 1 } { 8 } \sqrt { 3 }\).
5. Show that \(h = k ( a + 2 d )\), where \(k\) is a constant to be determined.
6. Hence find, in terms of \(a\), the greatest possible value of \(d\).
7. State one improvement that could be made to the model.

4 A car of mass 1700 kg is pulling a trailer of mass 300 kg along a straight horizontal road. The car and trailer are connected by a light inextensible cable which is parallel to the road. There are constant resistances to motion of 400 N on the car and 150 N on the trailer. The power of the car's engine is 14000 W . Find the acceleration of the car and the tension in the cable when the speed is \(20 \mathrm {~ms} ^ { - 1 }\).

15 A cyclist is towing a trailer behind her bicycle. She is riding along a straight, horizontal path at a constant speed.
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-22_371_723_447_657} A tension of \(T\) newtons acts on the connecting rod between the bicycle and the trailer.
The cyclist is causing a constant driving force of 40 N to be applied whilst pedalling forwards on her bicycle. The constant resistance force acting on the trailer is 12 N
15

1. State the value of \(T\) giving a clear reason for your answer.
   15
2. State one assumption you have made in reaching your answer to part (a).

2 A van of mass 3600 kg is towing a trailer of mass 1200 kg along a straight horizontal road using a light horizontal rope. There are resistance forces of 700 N on the van and 300 N on the trailer.

1. The driving force exerted by the van is 2500 N . Find the tension in the rope.
   The driving force is now removed and the van driver applies a braking force which acts only on the van. The resistance forces remain unchanged.
2. Find the least possible value of the braking force which will cause the rope to become slack.

3 The diagram shows three forces which act in the same plane and are in equilibrium.
\includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}

1. Find \(F\).
2. Find \(\alpha\).

3
\includegraphics[max width=\textwidth, alt={}, center]{6853f050-45ed-4a76-b450-648d8ac91468-2_429_826_721_662} A particle is in equilibrium on a smooth horizontal table when acted on by the three horizontal forces shown in the diagram.

1. Find the values of \(F\) and \(\theta\).
2. The force of magnitude 7 N is now removed. State the magnitude and direction of the resultant of the remaining two forces.

3
\includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-04_442_636_264_758} Coplanar forces of magnitudes \(30 \mathrm {~N} , 15 \mathrm {~N} , 33 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram, where \(\tan \alpha = \frac { 4 } { 3 }\). The system is in equilibrium.

1. Show that \(\left( \frac { 14.4 } { 30 - P } \right) ^ { 2 } + \left( \frac { 28.8 } { P + 30 } \right) ^ { 2 } = 1\).
2. Verify that \(P = 6\) satisfies this equation and find the value of \(\theta\).

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

11 A decoration is hanging freely from a fixed point on a ceiling.
The decoration has a mass of 0.2 kilograms.
The decoration is hanging by a light, inextensible wire.
The wire is 0.1 metres long.
Find the tension in the wire. Circle your answer.
0.02 N
0.02 g N
0.2 N
0.2 g N

4
\includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_389_1134_258_468} The diagram shows two particles, \(A\) and \(B\), of masses 0.2 kg and 0.1 kg respectively. The particles are suspended below a horizontal ceiling by two strings, \(A P\) and \(B Q\), attached to fixed points \(P\) and \(Q\) on the ceiling. The particles are connected by a horizontal string, \(A B\). Angle \(A P Q = 45 ^ { \circ }\) and \(B Q P = \theta ^ { \circ }\). Each string is light and inextensible. The particles are in equilibrium.

1. Find the value of the tension in the string \(A B\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_2715_44_110_2006}
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-07_2721_34_101_20}
2. Find the value of \(\theta\) and the tension in the string \(B Q\).

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.

3
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_296_735_1685_705} Three horizontal forces of magnitudes \(150 \mathrm {~N} , 100 \mathrm {~N}\) and \(P \mathrm {~N}\) have directions as shown in the diagram. The resultant of the three forces is shown by the broken line in the diagram. This resultant has magnitude 120 N and makes an angle \(75 ^ { \circ }\) with the 150 N force. Find the values of \(P\) and \(\theta\).

15 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\) In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

11 A moon vehicle has a mass of 212 kg and a length of 3 metres.
On the moon the vehicle has a weight of 345 N
Calculate a value for acceleration due to gravity on the moon.
Circle your answer.
[0pt] [1 mark] $$0.614 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.63 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 1.84 \mathrm {~m} \mathrm {~s} ^ { - 2 } \quad 4.89 \mathrm {~m} \mathrm {~s} ^ { - 2 }$$

16 A block of mass \(m\) kg rests on rough horizontal ground. The coefficient of friction between the block and the ground is \(\mu\). A force of magnitude \(T \mathrm {~N}\) is applied at an angle \(\theta\) radians above the horizontal as shown in the diagram and the block slides without tilting or lifting.
\includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-10_291_707_388_239}

1. Show that the acceleration of the block is given by \(\frac { T } { m } \cos \theta - \mu g + \frac { T } { m } \mu \sin \theta\). For a fixed value of \(T\), the acceleration of the block depends on the value of \(\theta\). The acceleration has its greatest value when \(\theta = \alpha\).
2. Find an expression for \(\alpha\) in terms of \(\mu\).

1 A man of mass 70 kg stands on the floor of a lift which is moving with an upward acceleration of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the force exerted by the floor on the man.

1 A man of mass 70 kg stands on the floor of a lift which is moving with an upward acceleration of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the force exerted by the floor on the man.

4. \begin{figure}[h] \end{figure} A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.

1. Find the acceleration of the lift.
2. Find the magnitude of the force exerted on the lift by the cable.

8
\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_159_896_488_244} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15 ^ { \circ }\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N , find the tension in the rope.

3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
The particle is being pulled by a horizontal force of 10 N .
It remains at rest in equilibrium. \begin{figure}[h] \end{figure}

1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.

7
\includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_414_865_1512_641} Two light strings are attached to a block of mass 20 kg . The block is in equilibrium on a horizontal surface \(A B\) with the strings taut. The strings make angles of \(60 ^ { \circ }\) and \(30 ^ { \circ }\) with the horizontal, on either side of the block, and the tensions in the strings are \(T \mathrm {~N}\) and 75 N respectively (see diagram).

1. Given that the surface is smooth, find the value of \(T\) and the magnitude of the contact force acting on the block.
2. It is given instead that the surface is rough and that the block is on the point of slipping. The frictional force on the block has magnitude 25 N and acts towards \(A\). Find the coefficient of friction between the block and the surface. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

2 An unmanned spacecraft has a weight of 5200 N on Earth. It lands on the surface of the planet Mars where the acceleration due to gravity is \(3.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the weight of the spacecraft on Mars.

2 An unmanned spacecraft has a weight of 5200 N on Earth. It lands on the surface of the planet Mars where the acceleration due to gravity is \(3.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the weight of the spacecraft on Mars.

6
\includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.

1. Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
   The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
2. Find the acceleration of the 4 kg particle and the tensions in the strings.
3. In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a 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.

3
\includegraphics[max width=\textwidth, alt={}, center]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-2_438_621_1676_762} A light inextensible string has its ends attached to two fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A smooth ring \(R\), of mass 0.8 kg , is threaded on the string and is pulled by a horizontal force of magnitude \(X\) newtons. The sections \(A R\) and \(B R\) of the string make angles of \(50 ^ { \circ }\) and \(20 ^ { \circ }\) respectively with the horizontal, as shown in the diagram. The ring rests in equilibrium with the string taut. Find

1. the tension in the string,
2. the value of \(X\).

3
\includegraphics[max width=\textwidth, alt={}, center]{d5acfe31-8614-4508-ac5b-865e15a1f539-2_661_565_1069_790} A small smooth ring \(R\) of weight 8.5 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A horizontal force of magnitude 15.5 N acts on \(R\) so that the ring is in equilibrium with angle \(A R B = 90 ^ { \circ }\). The part \(A R\) of the string makes an angle \(\theta\) with the horizontal and the part \(B R\) makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is \(T \mathrm {~N}\). Show that \(T \sin \theta = 12\) and \(T \cos \theta = 3.5\) and hence find \(\theta\).

3
\includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-04_337_661_262_740} A smooth ring \(R\) of mass 0.2 kg is threaded on a light string \(A R B\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(A B R = 90 ^ { \circ }\). The angle between the part \(A R\) of the string and the vertical is \(60 ^ { \circ }\). The ring is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), acting on the ring in a direction perpendicular to \(A R\) (see diagram). Calculate the tension in the string and the value of \(X\).

19

1. It is given that \(M\) and \(N\) move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
   By forming two equations of motion show that $$a = \frac { 1 } { 11 } g$$ 19
2. The speed of \(N , 0.5\) seconds after its release, is \(\frac { g } { k } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) where \(k\) is a constant. Find the value of \(k\)
   19
3. State one assumption that must be made for the answer in part (b) to be valid.

4
\includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_702_709_269_719} Particles \(P\) and \(Q\), of masses 0.6 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed peg. The particles are held at rest with the string taut. Both particles are at a height of 0.9 m above the ground (see diagram). The system is released and each of the particles moves vertically. Find

1. the acceleration of \(P\) and the tension in the string before \(P\) reaches the ground,
2. the time taken for \(P\) to reach the ground.

6
\includegraphics[max width=\textwidth, alt={}, center]{edf90396-5e17-44ef-bf25-e09cbc5785ba-4_451_729_255_708} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m . The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley.

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

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

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

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

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

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

2
\includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-03_721_622_296_724} A particle of mass 0.2 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point on a vertical wall. The particle is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), perpendicular to the string, with the string taut and making an angle of \(30 ^ { \circ }\) with the wall (see diagram). Find the tension in the string and the value of \(X\).

1. \begin{figure}[h] \end{figure} Figure 1 shows a light, inextensible string fixed at one end to a point \(P\). The other end is attached to a small object of weight 10 N . The object is subjected to a horizontal force \(H\) so that the string makes an angle of \(30 ^ { \circ }\) with the vertical.

1. Find the magnitude of the tension in the string.
2. Show that the ratio of the magnitude of the tension to the magnitude of \(H\) is \(2 : 1\).

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 5 N is attached to one end of a light string. The other end of the string is attached to a fixed point \(O\). A force of magnitude \(F\) newtons is applied to \(P\). The line of action of the force is inclined to the horizontal at \(30 ^ { \circ }\) and lies in the same vertical plane as the string. The particle \(P\) is in equilibrium with the string making an angle of \(40 ^ { \circ }\) with the downward vertical, as shown in Figure 1. Find

1. the tension in the string,
2. the value of \(F\).

5
\includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_275_663_258_742} A light inextensible string has a particle \(A\) of mass 0.26 kg attached to one end and a particle \(B\) of mass 0.54 kg attached to the other end. The particle \(A\) is held at rest on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 5 } { 13 }\). The string is taut and parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley at the top of the plane. Particle \(B\) hangs at rest vertically below the pulley (see diagram). The coefficient of friction between \(A\) and the plane is 0.2 . Particle \(A\) is released and the particles start to move.

1. Find the magnitude of the acceleration of the particles and the tension in the string. Particle \(A\) reaches the pulley 0.4 s after starting to move.
2. Find the distance moved by each of the particles.

4
\includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-3_499_567_260_788} The diagram shows a vertical cross-section of a triangular prism which is fixed so that two of its faces are inclined at \(60 ^ { \circ }\) to the horizontal. One of these faces is smooth and one is rough. Particles \(A\) and \(B\), of masses 0.36 kg and 0.24 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the highest point of the cross-section. \(B\) is held at rest at a point of the cross-section on the rough face and \(A\) hangs freely in contact with the smooth face (see diagram). \(B\) is released and starts to move up the face with acceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. By considering the motion of \(A\), show that the tension in the string is 3.03 N , correct to 3 significant figures.
2. Find the coefficient of friction between \(B\) and the rough face, correct to 2 significant figures.

6. \begin{figure}[h] \end{figure} Figure 3 shows a block \(A\) with mass \(4 m\) and a block \(B\) with mass \(5 m\).
Block \(A\) is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal.
Block \(B\) is at rest on a rough plane inclined at an angle \(\beta\) to the horizontal.
The blocks are connected by a light inextensible string which passes over a small smooth pulley at the top of each plane. A small smooth ring \(C\), of mass \(8 m\), is threaded on the string between the pulleys so that \(A , B\) and \(C\) all lie in the same vertical plane. The part of the string between \(A\) and its pulley lies along a line of greatest slope of the plane of angle \(\alpha\). The part of the string between \(B\) and its pulley lies along a line of greatest slope of the plane of angle \(\beta\). The angle between the vertical and the string between each pulley and the ring \(C\) is \(\gamma\).
The two blocks, \(A\) and \(B\), are modelled as particles.
Given that

• \(\tan \alpha = \frac { 5 } { 12 }\) and \(\tan \beta = \frac { 7 } { 24 }\) and \(\tan \gamma = \frac { 3 } { 4 }\)
• the coefficient of friction, \(\mu\), is the same between each block and its plane
• one of the blocks is on the point of sliding up its plane
• the tension in the string is \(T\)
  1. determine, in terms of \(m\) and \(g\), an expression for \(T\),
  2. draw a diagram showing the forces on block \(A\), clearly labelling each of the forces acting on the block,
  3. determine the value of \(\mu\), giving a justification for your answer.
     \includegraphics[max width=\textwidth, alt={}, center]{a8e9db6b-dfad-4278-82d8-a8fa5ba61008-20_2266_50_312_1978}

1 Three horizontal forces, acting at a single point, have magnitudes \(12 \mathrm {~N} , 14 \mathrm {~N}\) and 5 N and act along bearings \(000 ^ { \circ } , 090 ^ { \circ }\) and \(270 ^ { \circ }\) respectively. Find the magnitude and bearing of their resultant.

1. A ship \(S\) is moving with constant velocity \(( 4 \mathbf { i } - 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.
   Find the speed and direction of \(S\), giving the direction as a three-figure bearing, correct to the nearest degree.
   
2. The diagram below shows a forklift truck being used to raise two boxes, \(P\) and \(Q\), vertically. Box \(Q\) rests on horizontal forks and box \(P\) rests on top of box \(Q\). Box \(P\) has mass 25 kg and box \(Q\) has mass 55 kg .
   \includegraphics[max width=\textwidth, alt={}, center]{d9ef2033-bf8b-4aec-bc88-34dbc8b9c208-17_504_814_504_589}
   1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box \(Q\) is 820 N . Calculate the magnitude of the acceleration.
   2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed.
      
   3. A particle, of mass 4 kg , moves in a straight line under the action of a single force \(F \mathrm {~N}\), whose magnitude at time \(t\) seconds is given by
   $$F = 12 \sqrt { t } - 32 \text { for } t \geqslant 0 .$$
3. Find the acceleration of the particle when \(t = 9\).
4. Given that the particle has velocity \(- 1 \mathrm {~ms} ^ { - 1 }\) when \(t = 4\), find an expression for the velocity of the particle at \(t \mathrm {~s}\).
5. Determine whether the speed of the particle is increasing or decreasing when \(t = 9\). [2]

6. \begin{figure}[h] \end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

• The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
• The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
• The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

1. the value of \(\alpha\)
2. the value of \(P\)

1
\includegraphics[max width=\textwidth, alt={}, center]{cbe25a5a-0ca7-4e1b-b5b1-141a49186944-02_645_609_459_246} Three forces of magnitudes \(4 \mathrm {~N} , 7 \mathrm {~N}\) and P N act at a point in the directions shown in the diagram. The forces are in equilibrium.

1. Draw a closed figure to represent the three forces.
2. Hence, or otherwise, find the following.
   1. The value of \(\theta\).
   2. The value of \(P\).

1 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h] \end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.

7
\includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-4_529_481_255_831} A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end \(A\) of a light inextensible string is attached to the ring. The other end \(C\) of the string is attached to a fixed point of the rod above \(A\). A horizontal force of magnitude 8 N is applied to the point \(B\) of the string, where \(A B = 1.5 \mathrm {~m}\) and \(B C = 2 \mathrm {~m}\). The system is in equilibrium with the string taut and \(A B\) at right angles to \(B C\) (see diagram).

1. Find the tension in the part \(A B\) of the string and the tension in the part \(B C\) of the string. The equilibrium is limiting with the ring on the point of sliding up the rod.
2. Find the coefficient of friction between the ring and the rod.

1. In the case where \(F = 20\), find the tensions in each of the strings.
2. Find the greatest value of \(F\) for which the block remains in equilibrium in the position shown.

2 A minibus of mass 4000 kg is travelling along a straight horizontal road. The resistance to motion is 900 N .

1. Find the driving force when the acceleration of the minibus is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the power required for the minibus to maintain a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. the tension in the cable,
2. the magnitude of the force exerted on the woman by the floor of the lift.
   \item \end{enumerate} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3c8dce6f-367a-42bb-be60-d03d0a23664f-04_616_780_118_584} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), and the plane is at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac { 1 } { 3 }\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope.