3.03r Friction: concept and vector form

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.

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.

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. }

5 \includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-3_432_710_258_721} Two particles of masses 5 kg and 10 kg are connected by a light inextensible string that passes over a fixed smooth pulley. The 5 kg particle is on a rough fixed slope which is at an angle of \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The 10 kg particle hangs below the pulley (see diagram). The coefficient of friction between the slope and the 5 kg particle is \(\frac { 1 } { 2 }\). The particles are released from rest. Find the acceleration of the particles and the tension in the string.

5 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_394_531_260_806} A block of mass 2.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The block is kept in equilibrium by a light string making an angle of \(20 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\), as shown in the diagram. The coefficient of friction between the block and plane is \(\frac { 1 } { 4 }\). The block is in limiting equilibrium and is about to move up the plane. Find the value of \(T\).

7 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_378_1001_1672_573} A particle \(A\) of mass 1.6 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass 2.4 kg which hangs freely below the pulley. The system is released from rest with the string taut and with \(B\) at a height of 0.5 m above the ground, as shown in the diagram. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground.

1. Given that the table is smooth, find the time taken by \(B\) to reach the ground.
2. Given instead that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac { 3 } { 8 }\), find the total distance travelled by \(A\). You may assume that \(A\) does not reach the pulley.

4 A particle of mass 15 kg is stationary on a rough plane inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.2 . A force of magnitude \(X \mathrm {~N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. Show that the least possible value of \(X\) is 23.1 , correct to 3 significant figures, and find the greatest possible value of \(X\).

4 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal table. \(P\) is released from rest at a point on the table 3.5 m from \(O\). The speed of \(P\) at the instant the string becomes slack is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the work done against friction during the period from the release of \(P\) until the string becomes slack,
2. the coefficient of friction between \(P\) and the table.

2 A uniform solid cone has height 38 cm .

1. Write down the distance of the centre of mass of the cone from its base. \includegraphics[max width=\textwidth, alt={}, center]{ece63d46-5e56-4668-939a-9dbbcfc1a77a-2_497_547_1224_840} The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted, and the cone remains in equilibrium until the angle of inclination of the plane reaches \(31 ^ { \circ }\) (see diagram), when the cone topples.
2. Find the radius of the cone.
3. Show that \(\mu \geqslant 0.601\), correct to 3 significant figures, where \(\mu\) is the coefficient of friction between the cone and the plane.

6 A light elastic string has natural length 2 m and modulus of elasticity 0.8 N . One end of the string is attached to a fixed point \(O\) of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 12 } { 13 }\). A particle \(P\) of mass 0.052 kg is attached to the other end of the string. The coefficient of friction between the particle and the plane is 0.4 . \(P\) is released from rest at \(O\).

1. When \(P\) has moved \(d\) metres down the plane from \(O\), where \(d > 2\), find expressions in terms of \(d\) for
   1. the loss in gravitational potential energy of \(P\),
   2. the gain in elastic potential energy of the string,
   3. the work done by the frictional force acting on \(P\).
   4. Show that \(d ^ { 2 } - 6 d + 4 = 0\) when \(P\) is at its lowest point, and hence find the value of \(d\) in this case.

7 \includegraphics[max width=\textwidth, alt={}, center]{18398d27-15eb-4515-8210-4f0f614d5b28-4_713_933_258_605} A narrow groove is cut along a diameter in the surface of a horizontal disc with centre \(O\). Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, lie in the groove, and the coefficient of friction between each of the particles and the groove is \(\mu\). The particles are attached to opposite ends of a light inextensible string of length 1 m . The disc rotates with angular velocity \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a vertical axis passing through \(O\) and the particles move in horizontal circles (see diagram).

1. Given that \(\mu = 0.36\) and that both \(P\) and \(Q\) move in the same horizontal circle of radius 0.5 m , calculate the greatest possible value of \(\omega\) and the corresponding tension in the string.
2. Given instead that \(\mu = 0\) and that the tension in the string is 0.48 N , calculate
   1. the radius of the circle in which \(P\) moves and the radius of the circle in which \(Q\) moves,
   2. the speeds of the particles.

4 A small object of mass 0.4 kg is released from rest at a point 8 m above the ground. The object descends vertically and when its downwards displacement from its initial position is \(x \mathrm {~m}\) the object has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). While the object is moving, a force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\) opposes the motion.

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - 0.5 v ^ { 2 }\).
2. Express \(v\) in terms of \(x\).
3. Find the increase in the value of \(v\) during the final 4 m of the descent of the object.

2 A horizontal turntable rotates with constant angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle of mass 0.06 kg is placed on the turntable at a point 0.25 m from its centre. The coefficient of friction between the particle and the turntable is \(\mu\). As the turntable rotates, the particle moves with the turntable and no sliding takes place.

1. Find the vertical and horizontal components of the contact force exerted on the particle by the turntable.
2. Show that \(\mu \geqslant 0.225\).

2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\). Find the greatest extension of the string during the subsequent motion.

2 \includegraphics[max width=\textwidth, alt={}, center]{2c6b2e42-09cb-4653-9378-6c6add7771cc-2_463_567_479_790} A uniform rod \(A B\) has weight 6 N and length 0.8 m . The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(A B\), acting at \(A\) at an angle of \(45 ^ { \circ }\) to \(A B\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\),
2. the least possible value of the coefficient of friction at \(B\).

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.

6. A particle \(P\) of mass 3 kg is projected up a line of greatest slope of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The initial speed of P is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the frictional force acting on \(P\) as it moves up the plane,
2. the distance moved by \(P\) up the plane before \(P\) comes to instantaneous rest.

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}

4. \begin{figure}[h] \end{figure} Two identical small rings, \(A\) and \(B\), each of mass \(m\), are threaded onto a rough horizontal wire. The rings are connected by a light inextensible string. A particle \(C\) of mass \(3 m\) is attached to the midpoint of the string. The particle \(C\) hangs in equilibrium below the wire with angle \(B A C = \beta\), as shown in Figure 2. The tension in each of the parts, \(A C\) and \(B C\), of the string is \(T\)

1. By considering particle \(C\), find \(T\) in terms of \(m , g\) and \(\beta\)
2. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction between the wire and \(A\). The coefficient of friction between each ring and the wire is \(\frac { 4 } { 5 }\) The two rings, \(A\) and \(B\), are on the point of sliding along the wire towards each other.
3. Find the value of \(\tan \beta\) \includegraphics[max width=\textwidth, alt={}, center]{916543cb-14f7-486c-ba3c-eda9be134045-11_2255_50_314_34}
   

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