3.03v Motion on rough surface: including inclined planes

7 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-10_336_803_258_671} Two particles \(A\) and \(B\) of masses \(m \mathrm {~kg}\) and 4 kg respectively are connected by a light inextensible string that passes over a fixed smooth pulley. Particle \(A\) is on a rough fixed slope which is at an angle of \(30 ^ { \circ }\) to the horizontal ground. Particle \(B\) hangs vertically below the pulley and is 0.5 m above the ground (see diagram). The coefficient of friction between the slope and particle \(A\) is 0.2 .

1. In the case where the system is in equilibrium with particle \(A\) on the point of moving directly up the slope, show that \(m = 5.94\), correct to 3 significant figures.
2. In the case where \(m = 3\), the system is released from rest with the string taut. Find the total distance travelled by \(A\) before coming to instantaneous rest. You may assume that \(A\) does not reach the pulley.

5 A particle of mass 20 kg is on a rough plane inclined at an angle of \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude \(P \mathrm {~N}\) acting on the particle, in a direction parallel to a line of greatest slope of the plane. The greatest possible value of \(P\) is twice the least possible value of \(P\). Find the value of the coefficient of friction between the particle and the plane.

7
As shown in the diagram, a particle \(A\) of mass 1.6 kg lies on a horizontal plane and a particle \(B\) of mass 2.4 kg lies on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the inclined plane. The distance \(A P\) is 2.5 m and the distance of \(B\) from the bottom of the inclined plane is 1 m . There is a barrier at the bottom of the inclined plane preventing any further motion of \(B\). The part \(B P\) of the string is parallel to a line of greatest slope of the inclined plane. The particles are released from rest with both parts of the string taut.

1. Given that both planes are smooth, find the acceleration of \(A\) and the tension in the string.
2. It is given instead that the horizontal plane is rough and that the coefficient of friction between \(A\) and the horizontal plane is 0.2 . The inclined plane is smooth. Find the total distance travelled by \(A\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 A particle of mass 1.3 kg rests on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 12 } { 5 }\). The coefficient of friction between the particle and the plane is \(\mu\).

1. A force of magnitude 20 N parallel to a line of greatest slope of the plane is applied to the particle and the particle is on the point of moving up the plane. Show that \(\mu = 1.6\).
   The force of magnitude 20 N is now removed.
2. Find the acceleration of the particle.
3. Find the work done against friction during the first 2 s of motion.

5 A particle of mass 18 kg is on a plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The particle is projected up a line of greatest slope of the plane with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Given that the plane is smooth, use an energy method to find the distance the particle moves up the plane before coming to instantaneous rest.
2. Given instead that the plane is rough and the coefficient of friction between the particle and the plane is 0.25 , find the speed of the particle as it returns to its starting point.

4 A particle \(P\) of mass 0.8 kg is placed on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\mu\). A force of magnitude 5 N , acting upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), is applied to \(P\). The particle is on the point of sliding on the table.

1. Find the value of \(\mu\).
2. The magnitude of the force acting on \(P\) is increased to 10 N , with the direction of the force remaining the same. Find the acceleration of \(P\).

6 Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m . The particles are released from rest with both sections of the string taut.

1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
2. Given instead that the surface is rough and the coefficient of friction between \(A\) and the surface is 0.1 , find the speed of \(A\) immediately before it reaches the pulley. \(7 \quad\) A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4 \\ v = k & \text { for } 4 \leqslant t \leqslant 14 \\ v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
3. Find \(k\).
4. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
5. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
6. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).

5 \begin{figure}[h] \end{figure} A force, whose direction is upwards parallel to a line of greatest slope of a plane inclined at \(35 ^ { \circ }\) to the horizontal, acts on a box of mass 15 kg which is at rest on the plane. The normal component of the contact force on the box has magnitude \(R\) newtons (see Fig. 1).

1. Show that \(R = 123\), correct to 3 significant figures. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fcd2b219-d9b4-4972-b8fe-25cf543b9054-3_369_1045_1247_555} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} When the force parallel to the plane acting on the box has magnitude \(X\) newtons the box is about to move down the plane, and when this force has magnitude \(5 X\) newtons the box is about to move up the plane (see Fig. 2).
2. Find the value of \(X\) and the coefficient of friction between the box and the plane.
3. A particle \(P\) of mass 1.2 kg is released from rest at the top of a slope and starts to move. The slope has length 4 m and is inclined at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the slope is \(\frac { 1 } { 4 }\). Find
   1. the frictional component of the contact force on \(P\),
   2. the acceleration of \(P\),
   3. the speed with which \(P\) reaches the bottom of the slope.
   4. After reaching the bottom of the slope, \(P\) moves freely under gravity and subsequently hits a horizontal floor which is 3 m below the bottom of the slope.
      (a) Find the loss in gravitational potential energy of \(P\) during its motion from the bottom of the slope until it hits the floor.
      (b) Find the speed with which \(P\) hits the floor.
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      [0pt] [3] \(7 \quad\) A particle \(P\) starts to move from a point \(O\) and travels in a straight line. At time \(t\) s after \(P\) starts to move its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 0.12 t - 0.0006 t ^ { 2 }\).
      1. Verify that \(P\) comes to instantaneous rest when \(t = 200\), and find the acceleration with which it starts to return towards \(O\).
      2. Find the maximum speed of \(P\) for \(0 \leqslant t \leqslant 200\).
      3. Find the displacement of \(P\) from \(O\) when \(t = 200\).
      4. Find the value of \(t\) when \(P\) reaches \(O\) again.

7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.

1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
3. Find the speed with which the particle returns to \(P\).

2 A block of mass 20 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal. A force acts on the block parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is 0.32 . Find the least magnitude of the force necessary to move the block,

1. given that the force acts up the plane,
2. given instead that the force acts down the plane.

5 A particle \(P\) of mass 0.6 kg moves upwards along a line of greatest slope of a plane inclined at \(18 ^ { \circ }\) to the horizontal. The deceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the frictional and normal components of the force exerted on \(P\) by the plane. Hence find the coefficient of friction between \(P\) and the plane, correct to 2 significant figures. After \(P\) comes to instantaneous rest it starts to move down the plane with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(a\).

4 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\) and \(\sin \alpha = 0.28\).

1. Given that the normal component of the contact force acting on the particle has magnitude 1.2 N , find the mass of the particle.
2. Given also that the frictional component of the contact force acting on the particle has magnitude 0.4 N , find the deceleration of the particle. The particle comes to rest on reaching the point \(X\).
3. Determine whether the particle remains at \(X\) or whether it starts to move down the plane.

5 A particle of mass 0.8 kg slides down a rough inclined plane along a line of greatest slope \(A B\). The distance \(A B\) is 8 m . The particle starts at \(A\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and moves with constant acceleration \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the speed of the particle at the instant it reaches \(B\).
2. Given that the work done against the frictional force as the particle moves from \(A\) to \(B\) is 7 J , find the angle of inclination of the plane. When the particle is at the point \(X\) its speed is the same as the average speed for the motion from \(A\) to \(B\).
3. Find the work done by the frictional force for the particle's motion from \(A\) to \(X\).

2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .

1. Find the deceleration of the block.
2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.

4 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_489_1041_258_552} \(A B C\) is a vertical cross-section of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 4 m higher than \(B\). The part of the surface containing \(B C\) is horizontal and the distance \(B C\) is 5 m (see diagram). A particle of mass 0.8 kg is released from rest at \(A\) and slides along \(A B C\). Find the speed of the particle at \(C\) in each of the following cases.

1. The horizontal part of the surface is smooth.
2. The coefficient of friction between the particle and the horizontal part of the surface is 0.3 .

5 An object of mass 12 kg slides down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. The object passes through points \(A\) and \(B\) with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Find the increase in kinetic energy of the object as it moves from \(A\) to \(B\).
2. Hence find the distance \(A B\), assuming there is no resisting force acting on the object. The object is now pushed up the plane from \(B\) to \(A\), with constant speed, by a horizontal force.
3. Find the magnitude of this force.

6 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_255_511_794_817} The diagram shows a particle of mass 0.6 kg on a plane inclined at \(25 ^ { \circ }\) to the horizontal. The particle is acted on by a force of magnitude \(P \mathrm {~N}\) directed up the plane parallel to a line of greatest slope. The coefficient of friction between the particle and the plane is 0.36 . Given that the particle is in equilibrium, find the set of possible values of \(P\).

2 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

4 Particles \(P\) and \(Q\) are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.

1. Find the deceleration of each of the particles given that the coefficient of friction between \(P\) and the plane is 0.2 , and between \(Q\) and the plane is 0.25 . At a certain instant, \(P\) passes through the point \(A\) and \(Q\) passes through the point \(B\). The distance \(A B\) is 5 m . The velocities of \(P\) and \(Q\) at \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively, both in the direction \(A B\).
2. Find the speeds of \(P\) and \(Q\) immediately before they collide.

4 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_365_493_1749_826} A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = 2.4\). A small block of mass 0.6 kg is held at rest on the plane by a horizontal force of magnitude \(P \mathrm {~N}\). This force acts in a vertical plane through a line of greatest slope (see diagram). The coefficient of friction between the block and the plane is 0.4 . The block is on the point of slipping down the plane. By resolving forces parallel to and perpendicular to the inclined plane, or otherwise, find the value of \(P\).
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1 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).

1. Show that the acceleration of the particle is \(- 6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the particle's initial speed is \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance that the particle travels up the plane.

4 A box of mass 30 kg is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\), acted on by a force of magnitude 40 N . The force acts upwards and parallel to a line of greatest slope of the plane. The box is on the point of slipping up the plane.

1. Find the coefficient of friction between the box and the plane. The force of magnitude 40 N is removed.
2. Determine, giving a reason, whether or not the box remains in equilibrium.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A block of weight 7.5 N is at rest on a plane which is inclined to the horizontal at angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\). The coefficient of friction between the block and the plane is \(\mu\). A force of magnitude 7.2 N acting parallel to a line of greatest slope is applied to the block. When the force acts up the plane (see Fig. 1) the block remains at rest.

1. Show that \(\mu \geqslant \frac { 17 } { 24 }\). When the force acts down the plane (see Fig. 2) the block slides downwards.
2. Show that \(\mu < \frac { 31 } { 24 }\).

4 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).

2 A particle of mass 0.1 kg is released from rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal. It is given that, 5 seconds after release, the particle has a speed of \(2 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the particle and hence show that the magnitude of the frictional force acting on the particle is 0.302 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.