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

7 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-4_129_798_756_676} A winch drags a \(\log\) of mass 600 kg up a slope inclined at \(10 ^ { \circ }\) to the horizontal by means of an inextensible cable of negligible mass parallel to the slope (see diagram). The coefficient of friction between the \(\log\) and the slope is 0.15 , and the \(\log\) is initially at rest at the foot of the slope. The acceleration of the \(\log\) is \(0.11 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Calculate the tension in the cable. The cable suddenly breaks after dragging the log a distance of 10 m .
2. (a) Show that the deceleration of the log while continuing to move up the slope is \(3.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures.
   (b) Calculate the time taken, after the cable breaks, for the log to return to its original position at the foot of the slope. www.ocr.org.uk after the live examination series.
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4 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_156_1141_258_502} A block \(B\) of mass 0.8 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string inclined at \(10 ^ { \circ }\) to the horizontal. They are pulled across a horizontal surface with acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), by a horizontal force of 2 N applied to \(B\) (see diagram).

1. Given that contact between \(B\) and the surface is smooth, calculate the tension in the string.
2. Calculate the coefficient of friction between \(P\) and the surface.

7 A particle \(P\) of mass 0.6 kg is projected up a line of greatest slope of a plane inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) moves with deceleration \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and comes to rest before reaching the top of the plane.

1. Calculate the frictional force acting on \(P\), and the coefficient of friction between \(P\) and the plane.
2. Find the magnitude of the contact force exerted on \(P\) by the plane and the angle between the contact force and the upward direction of the line of greatest slope,
   1. when \(P\) is in motion,
   2. when \(P\) is at rest.

2 In the sport of curling, a heavy stone is projected across a horizontal ice surface. One player projects a stone of weight 180 N , which moves 36 m in a straight line and comes to rest 24 s after the instant of projection. The only horizontal force acting on the stone after its projection is a constant frictional force between the stone and the ice.

1. Calculate the deceleration of the stone.
2. Find the magnitude of the frictional force acting on the stone, and calculate the coefficient of friction between the stone and the ice.

6 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_328_698_255_657} A particle \(P\) lies on a slope inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is attached to one end of a taut light inextensible string which passes through a small smooth ring \(Q\) of mass \(m \mathrm {~kg}\). The portion \(P Q\) of the string is horizontal and the other portion of the string is inclined at \(40 ^ { \circ }\) to the vertical. A horizontal force of magnitude \(H \mathrm {~N}\), acting away from \(P\), is applied to \(Q\) (see diagram). The tension in the string is 6.4 N , and the string is in the vertical plane containing the line of greatest slope on which \(P\) lies. Both \(P\) and \(Q\) are in equilibrium.

1. Calculate \(m\).
2. Calculate \(H\).
3. Given that the weight of \(P\) is 32 N , and that \(P\) is in limiting equilibrium, show that the coefficient of friction between \(P\) and the slope is 0.879 , correct to 3 significant figures. \(Q\) and the string are now removed.
4. Determine whether \(P\) remains in equilibrium.

7 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-4_310_579_255_721} A block \(B\) is placed on a plane inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.6 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth pulley fixed to the top of the plane. A particle \(Q\) of mass 0.5 kg is attached to the other end of the string. The portion of the string attached to \(P\) is parallel to a line of greatest slope of the plane, the portion of the string attached to \(Q\) is vertical and the string is taut. The particles are released from rest and start to move with acceleration \(1.4 \mathrm {~ms} ^ { - 2 }\) (see diagram). It is given that \(B\) is in equilibrium while \(P\) moves on its upper surface.

1. Find the tension in the string while \(P\) and \(B\) are in contact.
2. Calculate the coefficient of friction between \(P\) and \(B\).
3. Given that the weight of \(B\) is 7 N , calculate the set of possible values of the coefficient of friction between \(B\) and the plane.

5 A particle \(P\) of mass 0.4 kg is at rest on a horizontal surface. The coefficient of friction between \(P\) and the surface is 0.2 . A force of magnitude 1.2 N acting at an angle of \(\theta ^ { \circ }\) above the horizontal is then applied to \(P\). Find the acceleration of \(P\) in each of the following cases:

1. \(\theta = 0\);
2. \(\theta = 20\);
3. \(\theta = 70\);
4. \(\theta = 90\).

7 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-4_392_1192_255_424} \(A B\) and \(B C\) are lines of greatest slope on a fixed triangular prism, and \(M\) is the mid-point of \(B C . A B\) and \(B C\) are inclined at \(30 ^ { \circ }\) to the horizontal. The surface of the prism is smooth between \(A\) and \(B\), and between \(B\) and \(M\). Between \(M\) and \(C\) the surface of the prism is rough. A small smooth pulley is fixed to the prism at \(B\). A light inextensible string passes over the pulley. Particle \(P\) of mass 0.3 kg is fixed to one end of the string, and is placed at \(A\). Particle \(Q\) of mass 0.4 kg is fixed to the other end of the string and is placed next to the pulley on \(B C\). The particles are released from rest with the string taut. \(P\) begins to move towards the pulley, and \(Q\) begins to move towards \(M\) (see diagram).

1. Show that the initial acceleration of the particles is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and find the tension in the string. The particle \(Q\) reaches \(M 1.8 \mathrm {~s}\) after being released from rest.
2. Find the speed of the particles when \(Q\) reaches \(M\). After \(Q\) passes through \(M\), the string remains taut and the particles decelerate uniformly. \(Q\) comes to rest between \(M\) and \(C 1.4 \mathrm {~s}\) after passing through \(M\).
3. Find the deceleration of the particles while \(Q\) is moving from \(M\) towards \(C\).
4. (a) By considering the motion of \(P\), find the tension in the string while \(Q\) is moving from \(M\) towards \(C\).
   (b) Calculate the magnitude of the frictional force which acts on \(Q\) while it is moving from \(M\) towards \(C\). \section*{END OF QUESTION PAPER} \section*{OCR
   Oxford Cambridge and RSA}

7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h] \end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.

1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
4. Calculate the frictional force between the slope and the sledge.

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

3 \includegraphics[max width=\textwidth, alt={}, center]{020ebd88-b920-40ce-84cf-5c26d45e2935-2_355_695_1073_726} A uniform solid hemisphere, of radius \(a\) and mass \(M\), is placed with its curved surface in contact with a rough plane that is inclined at an angle \(\alpha\) to the horizontal. A particle \(P\) of mass \(m\) is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and \(P\) at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is \(\frac { 1 } { 2 }\), show that

1. \(\tan \alpha \leqslant \frac { 1 } { 2 }\),
2. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_419_1102_1859_520} The diagram shows two uniform rods \(B A\) and \(A C\), smoothly hinged at \(A\). The rod \(B A\) has length \(8 a\) and weight \(W\); the rod \(A C\) has length \(6 a\) and weight \(2 W\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) resting on a rough horizontal floor and angle \(C A B\) equal to \(90 ^ { \circ }\). Show that the normal contact force at \(B\) is \(\frac { 26 } { 25 } W\). The coefficient of friction between each rod and the floor is \(\mu\). Find the least possible value of \(\mu\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

4 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-3_567_575_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),

1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).

5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.

1. Find, in terms of \(d , g\) and \(\theta\),
   1. the magnitude of the vertical component of the force exerted on the rod by the ground,
   2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
2. find the value of \(d\).
   

6 A particle \(P\) of weight 8 N rests on a horizontal surface. A horizontal force of magnitude 3 N acts on \(P\), and \(P\) is in limiting equilibrium.

1. Calculate the coefficient of friction between \(P\) and the surface.
2. Find the magnitude and direction of the contact force exerted by the surface on \(P\).
3. \includegraphics[max width=\textwidth, alt={}, center]{66eb8290-3a80-40bf-be40-a936ed7d5a1b-4_190_579_580_598} The initial 3 N force continues to act on \(P\) in its original direction. An additional force of magnitude \(T \mathrm {~N}\), acting in the same vertical plane as the 3 N force, is now applied to \(P\) at an angle of \(\theta ^ { \circ }\) above the horizontal (see diagram). \(P\) is again in limiting equilibrium.
   1. Given that \(\theta = 0\), find \(T\).
   2. Given instead that \(\theta = 30\), calculate \(T\).

3 \includegraphics[max width=\textwidth, alt={}, center]{5addd79d-d502-455c-936f-27005483164e-3_483_787_260_641} A uniform rod \(A B\) of mass 10 kg and length 2.4 m rests with \(A\) on rough horizontal ground. The rod makes an angle of \(60 ^ { \circ }\) with the horizontal and is supported by a fixed smooth peg \(P\). The distance \(A P\) is 1.6 m (see diagram).

1. Calculate the magnitude of the force exerted by the peg on the rod.
2. Find the least value of the coefficient of friction between the rod and the ground needed to maintain equilibrium.

3. \begin{figure}[h] \end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley, \(P\), fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(B\) hangs freely below \(P\), as shown in Figure 1. The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\) The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.

1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
3. Suggest two refinements to the model that would make it more realistic.

2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.

1. \begin{figure}[h] \end{figure} Figure 1 shows a particle \(P\) of mass 0.5 kg at rest on a rough horizontal plane.

1. Find the magnitude of the normal reaction of the plane on \(P\). The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 7 }\) A horizontal force of magnitude \(X\) newtons is applied to \(P\).
   Given that \(P\) is now in limiting equilibrium,
2. find the value of \(X\).

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A brick \(P\) of mass \(m\) is placed on the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\) Brick \(P\) is in equilibrium and on the point of sliding down the plane.
Brick \(P\) is modelled as a particle.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on brick \(P\)
2. show that \(\mu = \frac { 3 } { 4 }\) For parts (c) and (d), you are not required to do any further calculations.
   Brick \(P\) is now removed from the plane and a much heavier brick \(Q\) is placed on the plane. The coefficient of friction between \(Q\) and the plane is also \(\frac { 3 } { 4 }\)
3. Explain briefly why brick \(Q\) will remain at rest on the plane. Brick \(Q\) is now projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of the plane.
   Brick \(Q\) is modelled as a particle.
   Using the model,
4. describe the motion of brick \(Q\), giving a reason for your answer.