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

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find

1. the magnitude of the normal reaction of the plane on \(P\),
2. the coefficient of friction between \(P\) and the plane. The force of magnitude 15 N is removed.
3. Determine whether \(P\) moves, justifying your answer.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have mass 4 kg and 0.5 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough plane, which is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 4 } { 3 }\). The coefficient of friction between \(P\) and the plane is 0.5 . The string lies along the plane and passes over a small smooth light pulley which is fixed at the top of the plane. Particle \(Q\) hangs freely at rest vertically below the pulley. The string lies in the vertical plane which contains the pulley and a line of greatest slope of the inclined plane, as shown in Figure 4. Particle \(P\) is released from rest with the string taut and slides down the plane. Given that \(Q\) has not hit the pulley, find

1. the tension in the string during the motion,
2. the magnitude of the resultant force exerted by the string on the pulley.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N . The direction of the force is inclined to the plane at an angle of \(30 ^ { \circ }\). The plane is inclined to the horizontal at an angle of \(20 ^ { \circ }\), as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 1.5 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a fixed rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 5 }\). The string is parallel to the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs freely at rest vertically below the pulley, as shown in Figure 3. Particle \(P\) is released from rest with the string taut and slides along the table. Assuming that \(P\) has not reached the pulley, find

1. the tension in the string during the motion,
2. the magnitude and direction of the resultant force exerted on the pulley by the string.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 5 kg is held at rest in equilibrium on a rough inclined plane by a horizontal force of magnitude 10 N . The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 1. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding down the plane, find the value of \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{c809d34e-83db-4a16-a831-001f9f36b1c3-13_2460_72_311_27}

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses \(2 m\) and \(m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(A\) is held at rest on a fixed rough horizontal table at a distance \(d\) from a small smooth light pulley which is fixed at the edge of the table at the point \(P\). The coefficient of friction between \(A\) and the table is \(\mu\), where \(\mu < \frac { 1 } { 2 }\). The string is parallel to the table from \(A\) to \(P\) and passes over the pulley. Particle \(B\) hangs freely at rest vertically below \(P\) with the string taut and at a height \(h\), ( \(h < d\) ), above a horizontal floor, as shown in Figure 3. Particle \(A\) is released from rest with the string taut and slides along the table.

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence show that, until \(B\) hits the floor, the acceleration of \(A\) is \(\frac { g } { 3 } ( 1 - 2 \mu )\).
3. Find, in terms of \(g , h\) and \(\mu\), the speed of \(A\) at the instant when \(B\) hits the floor. After \(B\) hits the floor, \(A\) continues to slide along the table. Given that \(\mu = \frac { 1 } { 3 }\) and that \(A\) comes to rest at \(P\),
4. find \(d\) in terms of \(h\).
5. Describe what would happen if \(\mu = \frac { 1 } { 2 }\)

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2. \begin{figure}[h] \end{figure} A particle of mass 2 kg lies on a rough plane. The plane is inclined to the horizontal at \(30 ^ { \circ }\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons. The force makes an angle of \(20 ^ { \circ }\) with the horizontal and acts in a vertical plane containing a line of greatest slope of the plane, as shown in Figure 1. Find the least possible value of \(P\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(4 m\) is held at rest at the point \(X\) on the surface of a rough inclined plane which is fixed to horizontal ground. The point \(X\) is a distance \(h\) from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The particle \(P\) is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which hangs freely at a distance \(d\), where \(d > h\), below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and \(P\) moves down the plane. For the motion of the particles before \(P\) hits the ground,

1. state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,
2. write down an equation of motion for each particle,
3. find the acceleration of each particle. When \(P\) hits the ground, it immediately comes to rest. Given that \(Q\) comes to instantaneous rest before reaching the pulley,
4. show that \(d > \frac { 28 h } { 25 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{4fd21e83-0bdf-4bb1-8a3f-76beada511ae-27_56_20_109_1950}
   

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5. \section*{Figure 3} A suitcase of mass 10 kg slides down a ramp which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The suitcase is modelled as a particle and the ramp as a rough plane. The top of the plane is \(A\). The bottom of the plane is \(C\) and \(A C\) is a line of greatest slope, as shown in Fig. 3. The point \(B\) is on \(A C\) with \(A B = 5 \mathrm {~m}\). The suitcase leaves \(A\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and passes \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the decleration of the suitcase,
2. the coefficient of friction between the suitcase and the ramp. The suitcase reaches the bottom of the ramp.
3. Find the greatest possible length of \(A C\).

8. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), with masses \(2 m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the edge of a rough horizontal table. Particle \(Q\) is held at rest on the table and particle \(P\) is on the surface of a smooth inclined plane. The top of the plane coincides with the edge of the table. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 4. The string lies in a vertical plane containing the pulley and a line of greatest slope of the plane. The coefficient of friction between \(Q\) and the table is \(\frac { 1 } { 2 }\). Particle \(Q\) is released from rest with the string taut and \(P\) begins to slide down the plane.

1. By writing down an equation of motion for each particle,
   1. find the initial acceleration of the system,
   2. find the tension in the string. Suppose now that the coefficient of friction between \(Q\) and the table is \(\mu\) and when \(Q\) is released it remains at rest.
2. Find the smallest possible value of \(\mu\).

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   Q8

4. A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). A particle of mass 2 kg is projected with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is 0.25

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(A\).
2. Find the distance \(O A\). The particle now moves down the plane from \(A\).
3. Find the speed of \(P\) as it passes through \(O\).

3. \begin{figure}[h] \end{figure} A particle of mass 10 kg is placed on a fixed rough inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\). The particle is held in equilibrium by a force of magnitude \(P\) newtons, which acts up the plane, as shown in Figure 1. The line of action of the force lies in a vertical plane that contains a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).

1. Find the normal reaction between the particle and the plane.
2. Find the greatest possible value of \(P\).
3. Find the least possible value of \(P\). DO NOT WRITEIN THIS AREA
   

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1. A fixed rough plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\)

A particle of mass 6 kg is projected with speed \(5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) on the plane, up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\frac { 1 } { 4 }\)

1. Find the magnitude of the frictional force acting on the particle as it moves up the plane. The particle comes to instantaneous rest at the point \(B\).
2. Find the distance \(A B\). The particle now slides down the plane from \(B\). At the instant when the particle passes through the point \(C\) on the plane, the speed of the particle is again \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the distance \(B C\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-23_2647_1835_118_116}

4. \begin{figure}[h] \end{figure} A small block of mass 5 kg lies at rest on a rough horizontal plane.
The coefficient of friction between the block and the plane is \(\frac { 3 } { 7 }\) A force of magnitude \(P\) newtons is applied to the block in a direction which makes an angle of \(30 ^ { \circ }\) with the plane, as shown in Figure 1. The block is modelled as a particle.
Given that \(P = 14\)

1. find the magnitude of the frictional force exerted on the block by the plane and describe what happens to the block, justifying your answer.
   (6) The value of \(P\) is now changed so that the block is on the point of slipping along the plane.
2. Find the value of \(P\)

6. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons lies at rest on a rough horizontal surface, as shown in Figure 3.
A force of magnitude \(P\) newtons is applied to the particle.
The force acts at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\) The coefficient of friction between the particle and the surface is \(\frac { 1 } { 4 }\) Given that the particle does not move, show that $$P \leqslant \frac { 5 W } { 8 }$$

6. \begin{figure}[h] \end{figure} A box of mass \(m\) lies on a rough horizontal plane. The box is pulled along the plane in a straight line at constant speed by a light rope. The rope is inclined at an angle \(\theta\) to the plane, as shown in Figure 3.
The coefficient of friction between the box and the plane is \(\frac { 1 } { 3 }\) The box is modelled as a particle.
Given that \(\tan \theta = \frac { 3 } { 4 }\)

1. find, in terms of \(m\) and \(g\), the tension in the rope. The rope is now removed and the box is placed at rest on the plane.
   The box is then projected horizontally along the plane with speed \(u\).
   The box is again modelled as a particle.
   When the box has moved a distance \(d\) along the plane, the speed of the box is \(\frac { 1 } { 2 } u\).
2. Find \(d\) in terms of \(u\) and \(g\).

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8. \begin{figure}[h] \end{figure} One end of a light inextensible string is attached to a particle \(A\) of mass \(2 m\). The other end of the string is attached to a particle \(B\) of mass \(3 m\). Particle \(A\) is held at rest on a rough plane which is inclined to horizontal ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 5 } { 12 }\) The string passes over a small smooth pulley \(P\) which is fixed at the top of the plane. Particle \(B\) hangs vertically below \(P\) with the string taut, at a height \(h\) above the ground, as shown in Figure 4. The part of the string between \(A\) and \(P\) lies along a line of greatest slope of the plane. The two particles, the string and the pulley all lie in the same vertical plane.
The coefficient of friction between \(A\) and the plane is \(\frac { 11 } { 36 }\) The particle \(A\) is released from rest and begins to move up the plane.

1. Show that the frictional force acting on \(A\) as it moves up the plane is \(\frac { 22 m g } { 39 }\)
2. Write down an equation of motion for \(B\).
3. Show that the acceleration of \(A\) immediately after its release is \(\frac { 1 } { 3 } g\) In the subsequent motion, \(A\) comes to rest before it reaches the pulley.
4. Find, in terms of \(h\), the total distance travelled by \(A\) from when it was released from rest to when it first comes to rest again.

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5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion before \(Q\) strikes the ground, find the tension in the string.
2. Find the value of \(\mu\).

   END

8. \begin{figure}[h] \end{figure} A rough plane is inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.5 kg is held at rest on the plane by a horizontal force of magnitude 5 N , as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The particle is on the point of moving up the plane.

1. Find the magnitude of the normal reaction of the plane on \(P\).
2. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 5 N is now removed and \(P\) accelerates from rest down the plane.
3. Find the speed of \(P\) after it has travelled 3 m down the plane.

5. \begin{figure}[h] \end{figure} A small bead of mass 0.2 kg is attached to the end \(P\) of a light rod \(P Q\). The bead is threaded onto a fixed vertical rough wire. The bead is held in equilibrium with the \(\operatorname { rod } P Q\) inclined to the wire at an angle \(\alpha\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 2. The thrust in the rod is \(T\) newtons.
The bead is modelled as a particle.

1. Find the magnitude and direction of the friction force acting on the bead when \(T = 2.5\) The coefficient of friction between the bead and the wire is \(\mu\).
   Given that the greatest possible value of \(T\) is 6.125
2. find the value of \(\mu\).

5. \begin{figure}[h] \end{figure} A small ring of mass 0.2 kg is attached to one end of a light inextensible string.
The ring is threaded onto a fixed rough vertical rod.
The string is taut and makes an angle \(\theta\) with the rod, as shown in Figure 3, where \(\tan \theta = \frac { 12 } { 5 }\) Given that the ring is in equilibrium and that the tension in the string is 10 N ,

1. find the magnitude of the frictional force acting on the ring,
2. state the direction of the frictional force acting on the ring. The coefficient of friction between the ring and the rod is \(\frac { 1 } { 4 }\) Given that the ring is in equilibrium, and that the tension in the string, \(T\) newtons, can now vary,
3. 1. find the minimum value of \(T\)
   2. find the maximum value of \(T\)

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string. The string passes over a small smooth pulley which is fixed at a point on the intersection of two fixed inclined planes. The string lies in a vertical plane that contains a line of greatest slope of each of the two inclined planes. The first plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) and the second plane is inclined to the horizontal at an angle \(\beta\), where \(\tan \beta = \frac { 4 } { 3 }\). Particle \(P\) is on the first plane and particle \(Q\) is on the second plane with the string taut, as shown in Figure 3. The first plane is rough and the coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\). The second plane is smooth. The system is in limiting equilibrium. Given that \(P\) is on the point of slipping down the first plane,

1. find the value of \(m\),
2. find the magnitude of the force exerted on the pulley by the string,
3. find the direction of the force exerted on the pulley by the string. \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-21_2258_50_314_37}
   

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   \includegraphics[max width=\textwidth, alt={}, center]{6ab8838f-d6f8-4761-8def-1022d97d4e82-23_2258_50_314_37}
   
   \includegraphics[max width=\textwidth, alt={}]{6ab8838f-d6f8-4761-8def-1022d97d4e82-24_2655_1830_105_121}

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7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass \(m\) and \(2 m\) respectively, are attached to the ends of a light inextensible string. The particle \(A\) lies on a rough horizontal table. The string passes over a small smooth pulley \(P\) fixed on the edge of the table. The particle \(B\) hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between \(A\) and the table is \(\mu\). The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of \(A\) and \(B\) is \(\frac { 4 } { 9 } g\). By writing down separate equations of motion for \(A\) and \(B\),

1. find the tension in the string immediately after the particles begin to move,
2. show that \(\mu = \frac { 2 } { 3 }\). When \(B\) has fallen a distance \(h\), it hits the ground and does not rebound. Particle \(A\) is then a distance \(\frac { 1 } { 3 } h\) from \(P\).
3. Find the speed of \(A\) as it reaches \(P\).
4. State how you have used the information that the string is light.

5. \section*{Figure 2} A small package of mass 1.1 kg is held in equilibrium on a rough plane by a horizontal force. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The force acts in a vertical plane containing a line of greatest slope of the plane and has magnitude \(P\) newtons, as shown in Figure 2. The coefficient of friction between the package and the plane is 0.5 and the package is modelled as a particle. The package is in equilibrium and on the point of slipping down the plane.

1. Draw, on Figure 2, all the forces acting on the package, showing their directions clearly.
2. 1. Find the magnitude of the normal reaction between the package and the plane.
   2. Find the value of \(P\).