3.03v Motion on rough surface: including inclined planes

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.

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 \(\mu = \frac { 1 } { 2 }\)

   (Total 15 marks)	Leave blank Q8

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}
   

   END

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 \(A C\).

5. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The points \(A\) and \(B\) are on a line of greatest slope of the ramp, with \(A B = 2.5 \mathrm {~m}\) and \(B\) above \(A\), as shown in Figure 2. A package of mass 1.5 kg is projected up the ramp from \(A\) with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and first comes to instantaneous rest at \(B\). The coefficient of friction between the package and the ramp is \(\frac { 2 } { 7 }\) The package is modelled as a particle.

1. Find the work done against friction as the package moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\). After coming to instantaneous rest at \(B\), the package slides back down the slope.
3. Use the work-energy principle to find the speed of the package at the instant it returns to \(A\).

4. \begin{figure}[h] \end{figure} A fixed rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) A small box of mass \(m\) is at rest on the plane. A force of magnitude \(k m g\), where \(k\) is a constant, is applied to the box. The line of action of the force is at angle \(\alpha\) to the line of greatest slope of the plane through the box, as shown in Figure 1, and lies in the same vertical plane as this line of greatest slope. The coefficient of friction between the box and the plane is \(\mu\). The box is on the point of slipping up the plane. By modelling the box as a particle, find \(k\) in terms of \(\mu\).

6. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 0.1 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 rough horizontal table. The string lies along the table and passes over a small smooth pulley which is fixed to the edge of the table. Particle \(Q\) is at rest on a smooth plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\) 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 2. Particle \(P\) is released from rest with the string taut. During the first 0.5 s of the motion \(P\) does not reach the pulley and \(Q\) moves 0.75 m down the plane.

1. Find the tension in the string during the first 0.5 s of the motion.
2. Find the coefficient of friction between \(P\) and the table. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-19_72_59_2613_1886}

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is held at rest at the point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) lies on the line of greatest slope of the plane that passes through \(A\). The point \(B\) is 5 m down the plane from \(A\), as shown in Figure 3. The coefficient of friction between the plane and \(P\) is 0.3 The particle is released from rest at \(A\) and slides down the plane.

1. Find the speed of \(P\) at the instant it reaches \(B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{ba698f74-a51c-409a-a9d9-e9080fc87be2-10_478_1011_1343_456} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The particle is now returned to \(A\) and is held in equilibrium by a horizontal force of magnitude \(H\) newtons, as shown in Figure 4. The line of action of the force lies in the vertical plane containing the line of greatest slope of the plane through \(A\). The particle is on the point of moving up the plane.
2. Find the value of \(H\).

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}

8. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), have masses 2 kg and 4 kg respectively. The particles are connected by a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a rough plane. The plane is inclined to the horizontal ground at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\). The particle \(A\) is held at rest on the plane at a distance \(d\) metres from the pulley. The particle \(B\) hangs freely at rest, vertically below the pulley, at a distance \(h\) metres above the ground, as shown in Figure 3. The part of the string between \(A\) and the pulley is parallel to a line of greatest slope of the plane. The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 4 }\) The system is released from rest with the string taut and \(B\) descends.

1. Find the tension in the string as \(B\) descends. On hitting the ground, \(B\) immediately comes to rest. Given that \(A\) comes to rest before reaching the pulley,
2. find, in terms of \(h\), the range of possible values of \(d\).
3. State one physical factor, other than air resistance, that could be taken into account to make the model described above more realistic.

5. \begin{figure}[h] \end{figure} A particle of mass \(m\) rests in equilibrium on a fixed rough plane under the action of a force of magnitude \(X\). The force acts up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The coefficient of friction between the particle and the plane is \(\mu\).

• When \(X = 2 P\), the particle is on the point of sliding up the plane.
• When \(X = P\), the particle is on the point of sliding down the plane.

Find the value of \(\mu\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(4 m\) lies on the surface of a fixed rough inclined plane.
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The particle \(P\) is attached to one end of a light inextensible string.
The string passes over a small smooth pulley that is fixed at the top of the plane. The other end of the string is attached to a particle \(Q\) of mass \(m\) which lies on a smooth horizontal plane. The string lies along the horizontal plane and 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, as shown in Figure 4, and \(P\) moves down the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\) For the motion before \(Q\) reaches the pulley

1. write down an equation of motion for \(Q\),
2. find, in terms of \(m\) and \(g\), the tension in the string,
3. find the magnitude of the force exerted on the pulley by the string.
4. State where in your working you have used the information that the string is light.

8. \begin{figure}[h] \end{figure} A parcel of mass 2 kg is pulled up a rough inclined plane by the action of a constant force. The force has magnitude 18 N and acts at an angle of \(40 ^ { \circ }\) to the plane.
The line of action of the force lies in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 5.
The coefficient of friction between the plane and the parcel is 0.3
The parcel is modelled as a particle \(P\)

1. Find the acceleration of \(P\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, where \(A B = 5 \mathrm {~m}\) and \(B\) is above \(A\). Particle \(P\) passes through \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\).
2. Find the speed of \(P\) as it passes through \(B\). The force of 18 N is removed at the instant \(P\) passes through \(B\). As a result, \(P\) comes to rest at the point \(C\).
3. Determine whether \(P\) will remain at rest at \(C\). You must show all stages of your working clearly.

8. \begin{figure}[h] \end{figure} A fixed rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) A small smooth pulley is fixed at the top of the plane.
One end of a light inextensible string is attached to a particle \(P\) which is at rest on the plane. The string passes over the pulley and the other end of the string is attached to a particle \(Q\) which hangs vertically below the pulley, as shown in Figure 5. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(0.5 m\) The string from \(P\) to the pulley lies along a line of greatest slope of the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\).
The system is in limiting equilibrium with the string taut and \(P\) is on the point of slipping up the plane.

1. Find the value of \(\mu\). The string breaks and \(P\) begins to move down the plane.
   When particle \(P\) has travelled a distance of 0.8 m down the plane, the speed of \(P\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find the value of \(V\).

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is pushed by a constant horizontal force of magnitude 30 N up a line of greatest slope of a rough plane. 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 the line of greatest slope of the plane. The particle \(P\) starts from rest. The coefficient of friction between \(P\) and the plane is \(\mu\). After 2 seconds, \(P\) has travelled a distance of 5.5 m up the plane.

1. Find the acceleration of \(P\) up the plane.
2. Find the value of \(\mu\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses \(m\) and \(4 m\) respectively. The particles are attached to the ends of a light inextensible string. Particle \(P\) is held at rest on a rough horizontal table. The string lies along the table and passes over a small smooth light pulley which is fixed at the edge of the table. Particle \(Q\) hangs at rest vertically below the pulley, at a height \(h\) above a horizontal plane, as shown in Figure 3. The coefficient of friction between \(P\) and the table is 0.5 . Particle \(P\) is released from rest with the string taut and slides along the table.

1. Find, in terms of \(m g\), the tension in the string while both particles are moving. The particle \(P\) does not reach the pulley before \(Q\) hits the plane.
2. Show that the speed of \(Q\) immediately before it hits the plane is \(\sqrt { 1.4 g h }\) When \(Q\) hits the plane, \(Q\) does not rebound and \(P\) continues to slide along the table. Given that \(P\) comes to rest before it reaches the pulley,
3. show that the total length of the string must be greater than 2.4 h

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 \(\mu\).

   Leave
   blank
   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
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

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}

7. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4. The road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The driving force produced by the engine of the car is 3000 N .
The car moves with acceleration \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The non-gravitational resistance to motion of

• the car is modelled as a constant force of magnitude \(2 R\) newtons
• the trailer is modelled as a constant force of magnitude \(R\) newtons

The car and the trailer are modelled as particles.
The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,

1. show that the value of \(R\) is 60
2. find the tension in the tow bar. When the car and trailer are moving at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow bar breaks.
   Given that the non-gravitational resistance to motion of the trailer remains unchanged,
3. use the model to find the further distance moved by the trailer before it first comes to rest.

2. A particle \(P\) of mass \(2 m\) is moving on a rough horizontal plane when it collides directly with a particle \(Q\) of mass \(4 m\) which is at rest on the plane. The speed of \(P\) immediately before the collision is \(3 u\). The speed of \(Q\) immediately after the collision is \(2 u\).

1. Find, in terms of \(u\), the speed of \(P\) immediately after the collision.
2. State clearly the direction of motion of \(P\) immediately after the collision. Following the collision, \(Q\) comes to rest after travelling a distance \(\frac { 6 u ^ { 2 } } { g }\) along the plane. The coefficient of friction between \(Q\) and the plane is \(\mu\).
3. Find the value of \(\mu\).

5. Two small balls \(A\) and \(B\) have masses 0.6 kg and 0.2 kg respectively. They are moving towards each other in opposite directions on a horizontal table when they collide directly. Immediately before the collision, the speed of \(A\) is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision, \(A\) and \(B\) move in the same direction and the speed of \(B\) is twice the speed of \(A\). By modelling the balls as particles, find

1. the speed of \(B\) immediately after the collision,
2. the magnitude of the impulse exerted on \(B\) in the collision, stating the units in which your answer is given. The table is rough. After the collision, \(B\) moves a distance of 2 m on the table before coming to rest.
3. Find the coefficient of friction between \(B\) and the table.