3.03t Coefficient of friction: F <= mu*R model

6 A particle travels in a straight line \(P Q\). The velocity of the particle \(t \mathrm {~s}\) after leaving \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where $$v = 4.5 + 4 t - 0.5 t ^ { 2 }$$

1. Find the velocity of the particle at the instant when its acceleration is zero.
   The particle comes to instantaneous rest at \(Q\).
2. Find the distance \(P Q\). \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-10_625_780_260_744} Two particles \(A\) and \(B\), of masses \(3 m \mathrm {~kg}\) and \(2 m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a plane. The plane is inclined at an angle \(\theta\) to the horizontal. \(A\) lies on the plane and \(B\) hangs vertically, 0.8 m above the floor, which is horizontal. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). Initially \(A\) and \(B\) are at rest.
   1. Given that the plane is smooth, find the value of \(\theta\) for which \(A\) remains at rest.
      It is given instead that the plane is rough, \(\theta = 30 ^ { \circ }\) and the acceleration of \(A\) up the plane is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Show that the coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 10 } \sqrt { 3 }\).
   3. When \(B\) reaches the floor it comes to rest. Find the length of time after \(B\) reaches the floor for which \(A\) is moving up the plane. [You may assume that \(A\) 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.

3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8 . A force of magnitude 8 N acts on the ring. This force acts at an angle of \(10 ^ { \circ }\) above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, 0.6 m along the rod.

4 A particle of mass 12 kg is stationary on a rough plane inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A pulling force of magnitude \(P \mathrm {~N}\) acts at an angle of \(8 ^ { \circ }\) above a line of greatest slope of the plane. This force is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is 0.3 . Find the greatest possible value of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ba29ddb2-3558-4be1-a8a8-134e27a70149-10_220_609_260_769} A particle \(P\) of mass 0.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 7 } { 25 }\). A horizontal force of magnitude 4 N , acting in the vertical plane containing a line of greatest slope of the plane, is applied to \(P\) (see diagram). The particle is on the point of sliding up the plane.

1. Show that the coefficient of friction between the particle and the plane is \(\frac { 3 } { 4 }\).
   The force acting horizontally is replaced by a force of magnitude 4 N acting up the plane parallel to a line of greatest slope.
2. Find the acceleration of \(P\).
3. Starting with \(P\) at rest, the force of 4 N parallel to the plane acts for 3 seconds and is then removed. Find the total distance travelled until \(P\) comes to instantaneous rest.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-07_366_567_258_790} A particle of mass 0.6 kg is placed on a rough plane which is inclined at an angle of \(35 ^ { \circ }\) to the horizontal. The particle is kept in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\) acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and plane is 0.4 . Find the least possible value of \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-10_525_885_264_625} The diagram shows the vertical cross-section \(X Y Z\) of a rough slide. The section \(Y Z\) is a straight line of length 2 m inclined at an angle of \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The section \(Y Z\) is tangential to the curved section \(X Y\) at \(Y\), and \(X\) is 1.8 m above the level of \(Y\). A child of mass 25 kg slides down the slide, starting from rest at \(X\). The work done by the child against the resistance force in moving from \(X\) to \(Y\) is 50 J .

1. Find the speed of the child at \(Y\).
   It is given that the child comes to rest at \(Z\).
2. Use an energy method to find the coefficient of friction between the child and \(Y Z\), giving your answer as a fraction in its simplest form.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 A particle \(P\) of mass 0.4 kg is on a rough horizontal floor. The coefficient of friction between \(P\) and the floor is \(\mu\). A force of magnitude 3 N is applied to \(P\) upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is initially at rest and accelerates at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the time it takes for \(P\) to travel a distance of 1.44 m from its starting point.
2. Find \(\mu\).

5 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-08_286_661_260_742} A block of mass 5 kg is being pulled along a rough horizontal floor by a force of magnitude \(X \mathrm {~N}\) acting at \(30 ^ { \circ }\) above the horizontal (see diagram). The block starts from rest and travels 2 m in the first 5 s of its motion.

1. Find the acceleration of the block.
2. Given that the coefficient of friction between the block and the floor is 0.4 , find \(X\).
   The block is now placed on a part of the floor where the coefficient of friction between the block and the floor has a different value. The value of \(X\) is changed to 25, and the block is now in limiting equilibrium.
3. Find the value of the coefficient of friction between the block and this part of the floor.

4 A particle of mass 20 kg is on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of magnitude 25 N , acting at an angle of \(20 ^ { \circ }\) above a line of greatest slope of the plane, is used to prevent the particle from sliding down the plane. The coefficient of friction between the particle and the plane is \(\mu\).

1. Complete the diagram below to show all the forces acting on the particle. \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-06_495_615_543_726}
2. Find the least possible value of \(\mu\).

2 A basket of mass 5 kg slides down a slope inclined at \(12 ^ { \circ }\) to the horizontal. The coefficient of friction between the basket and the slope is 0.2 .

1. Find the frictional force acting on the basket.
2. Determine whether the speed of the basket is increasing or decreasing.

4 A box of mass 4.5 kg is pulled at a constant speed of \(2 \mathrm {~ms} ^ { - 1 }\) along a rough horizontal floor by a horizontal force of magnitude 15 N .

1. Find the coefficient of friction between the box and the floor. The horizontal pulling force is now removed. Find
2. the deceleration of the box in the subsequent motion,
3. the distance travelled by the box from the instant the horizontal force is removed until the box comes to rest.
4. A cyclist travels in a straight line from \(A\) to \(B\) with constant acceleration \(0.06 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). His speed at \(A\) is \(3 \mathrm {~ms} ^ { - 1 }\) and his speed at \(B\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find
   1. the time taken by the cyclist to travel from \(A\) to \(B\),
   2. the distance \(A B\).
   3. A car leaves \(A\) at the same instant as the cyclist. The car starts from rest and travels in a straight line to \(B\). The car reaches \(B\) at the same instant as the cyclist. At time \(t \mathrm {~s}\) after leaving \(A\) the speed of the car is \(k t ^ { 2 } \mathrm {~ms} ^ { - 1 }\), where \(k\) is a constant. Find
      (a) the value of \(k\),
      (b) the speed of the car at \(B\).
      1. A lorry \(P\) of mass 15000 kg climbs a straight hill of length 800 m at a steady speed. The hill is inclined at \(2 ^ { \circ }\) to the horizontal. For \(P\) 's journey from the bottom of the hill to the top, find
         (a) the gain in gravitational potential energy,
         (b) the work done by the driving force, which has magnitude 7000 N ,
      2. the work done against the force resisting the motion.
      3. A second lorry, \(Q\), also has mass 15000 kg and climbs the same hill as \(P\). The motion of \(Q\) is subject to a constant resisting force of magnitude 900 N , and \(Q\) s speed falls from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill to \(10 \mathrm {~ms} ^ { - 1 }\) at the top. Find the work done by the driving force as \(Q\) climbs from the bottom of the hill to the top. \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-3_483_231_1537_973} Particles \(A\) and \(B\), of masses 0.15 kg and 0.25 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley. The system is held at rest with the string taut and with \(A\) and \(B\) at the same horizontal level, as shown in the diagram. The system is then released.
         1. Find the downward acceleration of \(B\). After \(2 \mathrm {~s} B\) hits the floor and comes to rest without rebounding. The string becomes slack and \(A\) moves freely under gravity.
         2. Find the time that elapses until the string becomes taut again.
         3. Sketch on a single diagram the velocity-time graphs for both particles, for the period from their release until the instant that \(B\) starts to move upwards.

6 A small block of mass 0.15 kg moves on a horizontal surface. The coefficient of friction between the block and the surface is 0.025 .

1. Find the frictional force acting on the block.
2. Show that the deceleration of the block is \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The block is struck from a point \(A\) on the surface and, 4 s later, it hits a boundary board at a point \(B\). The initial speed of the block is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance \(A B\). The block rebounds from the board with a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves along the line \(B A\). Find
4. the speed with which the block passes through \(A\),
5. the total distance moved by the block, from the instant when it was struck at \(A\) until the instant when it comes to rest.

1 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_200_588_267_781} A ring of mass 1.1 kg is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of magnitude 13 N at an angle \(\alpha\) below the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) (see diagram). The ring is in equilibrium.

1. Find the frictional component of the contact force on the ring.
2. Find the normal component of the contact force on the ring.
3. Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.

5 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-3_305_599_1717_774} Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string. \(P\) is at rest on a rough horizontal table. The string passes over a small smooth pulley which is fixed at the edge of the table. \(Q\) hangs vertically below the pulley (see diagram). The force exerted on the string by the pulley has magnitude \(4 \sqrt { } 2 \mathrm {~N}\). The coefficient of friction between \(P\) and the table is 0.8 .

1. Show that the tension in the string is 4 N and state the mass of \(Q\).
2. Given that \(P\) is on the point of slipping, find its mass. A particle of mass 0.1 kg is now attached to \(Q\) and the system starts to move.
3. Find the tension in the string while the particles are in motion.

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-3_314_867_1457_639} A block \(B\) of mass 0.6 kg and a particle \(A\) of mass 0.4 kg are attached to opposite ends of a light inextensible string. The block is held at rest on a rough horizontal table, and the coefficient of friction between the block and the table is 0.5 . The string passes over a small smooth pulley \(C\) at the edge of the table and \(A\) hangs in equilibrium vertically below \(C\). The part of the string between \(B\) and \(C\) is horizontal and the distance \(B C\) is 3 m (see diagram). \(B\) is released and the system starts to move.

1. Find the acceleration of \(B\) and the tension in the string.
2. Find the time taken for \(B\) to reach the pulley.

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-4_246_665_253_739} Two rectangular boxes \(A\) and \(B\) are of identical size. The boxes are at rest on a rough horizontal floor with \(A\) on top of \(B\). Box \(A\) has mass 200 kg and box \(B\) has mass 250 kg . A horizontal force of magnitude \(P\) N is applied to \(B\) (see diagram). The boxes remain at rest if \(P \leqslant 3150\) and start to move if \(P > 3150\).

1. Find the coefficient of friction between \(B\) and the floor. The coefficient of friction between the two boxes is 0.2 . Given that \(P > 3150\) and that no sliding takes place between the boxes,
2. show that the acceleration of the boxes is not greater than \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. find the maximum possible value of \(P\).

4 A block of mass 11 kg is at rest on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. A force acts on the block in a direction up the plane parallel to a line of greatest slope. When the magnitude of the force is \(2 X \mathrm {~N}\) the block is on the point of sliding down the plane, and when the magnitude of the force is \(9 X \mathrm {~N}\) the block is on the point of sliding up the plane. Find

1. the value of \(X\),
2. the coefficient of friction between the block and the plane.

6 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-3_387_1095_1724_525} A small smooth ring \(R\), of mass 0.6 kg , is threaded on a light inextensible string of length 100 cm . One end of the string is attached to a fixed point \(A\). A small bead \(B\) of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through \(A\). The system is in equilibrium with \(B\) at a distance of 80 cm from \(A\) (see diagram).

1. Find the tension in the string.
2. Find the frictional and normal components of the contact force acting on \(B\).
3. Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

5 \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-4_620_623_255_760} A small block of mass 1.25 kg is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle \(\theta\) is such that \(\cos \theta = 0.28\) and \(\sin \theta = 0.96\). A horizontal frictional force also acts on the block, and the block is in equilibrium.

1. Show that the magnitude of the frictional force is 7.5 N and state the direction of this force.
2. Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. The force of magnitude 6.1 N is now replaced by a force of magnitude 8.6 N acting in the same direction, and the block begins to move.
3. Find the magnitude and direction of the acceleration of the 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.

4 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_168_711_1612_717} A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of \(25 ^ { \circ }\) below the horizontal (see diagram). With a tension in the string of \(T \mathrm {~N}\) the ring is in equilibrium.

1. Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod. The coefficient of friction between the ring and the rod is 0.4 .
2. Given that the equilibrium is limiting, find the value of \(T\).

7 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-4_333_1001_264_573} A light inextensible string of length 5.28 m has particles \(A\) and \(B\), of masses 0.25 kg and 0.75 kg respectively, attached to its ends. Another particle \(P\), of mass 0.5 kg , is attached to the mid-point of the string. Two small smooth pulleys \(P _ { 1 }\) and \(P _ { 2 }\) are fixed at opposite ends of a rough horizontal table of length 4 m and height 1 m . The string passes over \(P _ { 1 }\) and \(P _ { 2 }\) with particle \(A\) held at rest vertically below \(P _ { 1 }\), the string taut and \(B\) hanging freely below \(P _ { 2 }\). Particle \(P\) is in contact with the table halfway between \(P _ { 1 }\) and \(P _ { 2 }\) (see diagram). The coefficient of friction between \(P\) and the table is 0.4 . Particle \(A\) is released and the system starts to move with constant acceleration of magnitude \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the part \(A P\) of the string is \(T _ { A } \mathrm {~N}\) and the tension in the part \(P B\) of the string is \(T _ { B } \mathrm {~N}\).

1. Find \(T _ { A }\) and \(T _ { B }\) in terms of \(a\).
2. Show by considering the motion of \(P\) that \(a = 2\).
3. Find the speed of the particles immediately before \(B\) reaches the floor.
4. Find the deceleration of \(P\) immediately after \(B\) reaches the floor. \end{document}

7 \includegraphics[max width=\textwidth, alt={}, center]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-4_657_618_255_760} A small ring \(R\) is attached to one end of a light inextensible string of length 70 cm . A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point \(A\) on the wire, vertically above \(R\). A horizontal force of magnitude 5.6 N is applied to the point \(J\) of the string 30 cm from \(A\) and 40 cm from \(R\). The system is in equilibrium with each of the parts \(A J\) and \(J R\) of the string taut and angle \(A J R\) equal to \(90 ^ { \circ }\) (see diagram).

1. Find the tension in the part \(A J\) of the string, and find the tension in the part \(J R\) of the string. The ring \(R\) has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
2. Show that the coefficient of friction between \(R\) and the wire is 0.341 , correct to 3 significant figures. A particle of mass \(m \mathrm {~kg}\) is attached to \(R\) and \(R\) is now in limiting equilibrium, on the point of moving down the wire.
3. Given that the coefficient of friction is unchanged, find the value of \(m\). {www.cie.org.uk} after the live examination series. }