3.03v Motion on rough surface: including inclined planes

7 A box of mass 50 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal.

1. Find an inequality for the coefficient of friction between the box and the plane. In fact the coefficient of friction between the box and the plane is 0.19 .
2. A girl pushes the box with a force of 50 N , acting down a line of greatest slope of the plane, for a distance of 5 m . She then stops pushing. Use an energy method to find the speed of the box when it has travelled a further 5 m . The box then comes to a plane inclined at \(20 ^ { \circ }\) below the horizontal. The box moves down a line of greatest slope of this plane. The coefficient of friction is still 0.19 and the girl is not pushing the box.
3. Find the acceleration of the box.

3 A particle is released from rest and slides down a line of greatest slope of a rough plane which is inclined at \(25 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.4 .

1. Find the acceleration of the particle.
2. Find the distance travelled by the particle in the first 3 s after it is released.

6 \includegraphics[max width=\textwidth, alt={}, center]{f08a4870-9466-4f8b-bd0f-431fb1803514-08_661_1244_262_452} The diagram shows the velocity-time graphs for two particles, \(P\) and \(Q\), which are moving in the same straight line. The graph for \(P\) consists of four straight line segments. The graph for \(Q\) consists of three straight line segments. Both particles start from the same initial position \(O\) on the line. \(Q\) starts 2 seconds after \(P\) and both particles come to rest at time \(t = T\). The greatest velocity of \(Q\) is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the displacement of \(P\) from \(O\) at \(t = 10\).
2. Find the velocity of \(P\) at \(t = 12\).
3. Given that the total distance covered by \(P\) during the \(T\) seconds of its motion is 49.5 m , find the value of \(T\).
4. Given also that the acceleration of \(Q\) from \(t = 2\) to \(t = 6\) is \(1.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the value of \(V\) and hence find the distance between the two particles when they both come to rest at \(t = T\). \includegraphics[max width=\textwidth, alt={}, center]{f08a4870-9466-4f8b-bd0f-431fb1803514-10_392_529_262_808} A particle \(P\) of mass 0.2 kg rests on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The coefficient of friction between the particle and the plane is 0.3 . A force of magnitude \(T \mathrm {~N}\) acts upwards on \(P\) at \(15 ^ { \circ }\) above a line of greatest slope of the plane (see diagram).
5. Find the least value of \(T\) for which the particle remains at rest.
   The force of magnitude \(T \mathrm {~N}\) is now removed. A new force of magnitude 0.25 N acts on \(P\) up the plane, parallel to a line of greatest slope of the plane. Starting from rest, \(P\) slides down the plane. After moving a distance of \(3 \mathrm {~m} , P\) passes through the point \(A\).
6. Use an energy method to find the speed of \(P\) at \(A\).

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 A particle \(P\) of mass 0.5 kg moves on a horizontal surface along the straight line \(O A\), in the direction from \(O\) to \(A\). The coefficient of friction between \(P\) and the surface is 0.08 . Air resistance of magnitude \(0.2 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\). The particle passes through \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0\).

1. Show that \(2.5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 2 )\) and hence find the value of \(t\) when \(v = 0\).
2. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 6 \mathrm { e } ^ { - 0.4 t } - 2\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance \(O P\) when \(v = 0\).

6 \includegraphics[max width=\textwidth, alt={}, center]{9d377c95-09b8-4893-b29f-8517a5016e8b-3_151_949_1206_598} \(O\) and \(A\) are fixed points on a horizontal surface, with \(O A = 0.5 \mathrm {~m}\). A particle \(P\) of mass 0.2 kg is projected horizontally with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in the direction \(O A\) and moves in a straight line (see diagram). At time \(t \mathrm {~s}\) after projection, the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement from \(O\) is \(x \mathrm {~m}\). The coefficient of friction between the surface and \(P\) is 0.5 , and a force of magnitude \(\frac { 0.4 } { x ^ { 2 } } \mathrm {~N}\) acts on \(P\) in the direction \(P O\).

1. Show that, while the particle is in motion, \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = - \left( 5 + \frac { 2 } { x ^ { 2 } } \right)\).
2. Calculate the distance travelled by \(P\) before it comes to rest, and show that \(P\) does not subsequently move.

6 \includegraphics[max width=\textwidth, alt={}, center]{9c82b387-8e5e-48b9-973d-5337b4e56a66-3_652_618_849_762} A particle \(P\) of mass 0.6 kg is attached to one end of a light elastic string of natural length 1.5 m and modulus of elasticity 9 N . The string passes through a small smooth ring \(R\) fixed at a height of 0.4 m above a rough horizontal surface. The other end of the string is attached to a fixed point \(O\) which is 1.5 m vertically above \(R\). The points \(A\) and \(B\) are on the horizontal surface, and \(B\) is vertically below \(R\). When \(P\) is on the surface between \(A\) and \(B , R P\) makes an acute angle \(\theta ^ { \circ }\) with the horizontal (see diagram).

1. Show that the normal force exerted on \(P\) by the surface has magnitude 3.6 N , for all values of \(\theta\). \(P\) is projected with speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) towards \(B\) from its initial position at \(A\) where \(\theta = 30\). The speed of \(P\) when it passes through \(B\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Calculate the value of the coefficient of friction between \(P\) and the surface.

7 A force of magnitude \(0.4 t \mathrm {~N}\), applied at an angle of \(30 ^ { \circ }\) above the horizontal, acts on a particle \(P\), where \(t \mathrm {~s}\) is the time since the force starts to act. \(P\) is at rest on rough horizontal ground when \(t = 0\). The mass of \(P\) is 0.2 kg and the coefficient of friction between \(P\) and the ground is \(\mu\).

1. Given that \(P\) is about to slip when \(t = 2\), find \(\mu\) and the value of \(t\) for the instant when \(P\) loses contact with the ground.
2. While \(P\) is moving on the ground, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 2.165 t - 4.330$$ where the coefficients are correct to 4 significant figures.
3. Calculate the speed of \(P\) when it loses contact with the ground. {www.cie.org.uk} after the live examination series. }

5 A particle \(P\) of mass 0.4 kg is placed at rest at a point \(A\) on a rough horizontal surface. A horizontal force, directed away from \(A\) and with magnitude \(0.6 t \mathrm {~N}\), acts on \(P\), where \(t \mathrm {~s}\) is the time after \(P\) is placed at \(A\). The coefficient of friction between \(P\) and the surface is 0.3 , and \(P\) has displacement from \(A\) of \(x \mathrm {~m}\) at time \(t \mathrm {~s}\).

1. Show that \(P\) starts to move when \(t = 2\). Show also that when \(P\) is in motion it has acceleration \(( 1.5 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Express the velocity of \(P\) in terms of \(t\), for \(t \geqslant 2\).
3. Express \(x\) in terms of \(t\), for \(t \geqslant 2\).

5 A particle of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 6 N . The other end of the string is attached to a fixed point \(O\). The particle is projected vertically downwards from \(O\) with initial speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the greatest speed of the particle during its descent.
2. Find the greatest distance of the particle below \(O\). \includegraphics[max width=\textwidth, alt={}, center]{2b0425b2-2f8f-491a-996c-3d3b589bd7df-12_558_554_260_794} The end \(A\) of a non-uniform rod \(A B\) of length 0.6 m and weight 8 N rests on a rough horizontal plane, with \(A B\) inclined at \(60 ^ { \circ }\) to the horizontal. Equilibrium is maintained by a force of magnitude 3 N applied to the rod at \(B\). This force acts at \(30 ^ { \circ }\) above the horizontal in the vertical plane containing the rod (see diagram).
3. Find the distance of the centre of mass of the rod from \(A\).
   The 3 N force is removed, and the rod is held in equilibrium by a force of magnitude \(P \mathrm {~N}\) applied at \(B\), acting in the vertical plane containing the rod, at an angle of \(30 ^ { \circ }\) below the horizontal.
4. Calculate \(P\).
   In one of the two situations described, the \(\operatorname { rod } A B\) is in limiting equilibrium.
5. Find the coefficient of friction at \(A\). \(7 \quad\) A particle \(P\) is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\) after projection the horizontal and vertically upwards displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively. The equation of the trajectory of \(P\) is \(y = 2 x - \frac { 25 x ^ { 2 } } { V ^ { 2 } }\).
6. Write down the value of \(\tan \theta\), where \(\theta\) is the angle of projection of \(P\).
   When \(t = 4 , P\) passes through the point \(A\) where \(x = y = a\).
7. Calculate \(V\) and \(a\).
8. Find the direction of motion of \(P\) when it passes through \(A\).

6 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a plane inclined at \(30 ^ { \circ }\) to the horizontal. At time \(t \mathrm {~s}\) after its release, \(P\) has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and displacement \(x \mathrm {~m}\) down the plane from \(O\). The coefficient of friction between \(P\) and the plane increases as \(P\) moves down the plane, and equals \(0.1 x ^ { 2 }\).

1. Show that \(2 v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 10 - ( \sqrt { } 3 ) x ^ { 2 }\).
2. Calculate the maximum speed of \(P\).
3. Find the value of \(x\) at the point where \(P\) comes to rest.

7 A particle \(P\) is projected horizontally from a point \(O\) on a rough horizontal surface. The coefficient of friction between the particle and the surface is 0.2 . A horizontal force of magnitude \(0.06 t \mathrm {~N}\) directed away from \(O\) acts on \(P\), where \(t \mathrm {~s}\) is the time after projection. \(P\) comes to rest when \(t = 4\).

1. The particle begins to move again when \(t = 8\). Show that the mass of \(P\) is 0.24 kg .
2. Show that, for \(0 \leqslant t \leqslant 4 , \frac { \mathrm {~d} v } { \mathrm {~d} t } = 0.25 t - 2\), and find the speed of projection of \(P\).
3. Find the distance from \(O\) at which \(P\) comes to rest.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 One end of a light elastic string of natural length 0.4 m and modulus of elasticity 16 N is attached to a fixed point \(O\) of a horizontal table. A particle \(P\) of mass 0.8 kg is attached to the other end of the string. The particle \(P\) is released from rest on the table, at a point which is 0.5 m from \(O\). The coefficient of friction between the particle and the table is 0.2 . By considering work and energy, find the speed of \(P\) at the instant the string becomes slack.

6 \includegraphics[max width=\textwidth, alt={}, center]{e30ba526-db21-4904-96dc-c12a1f67c81a-4_238_725_258_712} Two particles \(P\) and \(Q\), of masses 0.4 kg and 0.2 kg respectively, are attached to opposite ends of a light inextensible string. \(P\) is placed on a horizontal table and the string passes over a small smooth pulley at the edge of the table. The string is taut and the part of the string attached to \(Q\) is vertical (see diagram). The coefficient of friction between \(P\) and the table is 0.5 . \(Q\) is projected vertically downwards with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at time \(t \mathrm {~s}\) after the instant of projection the speed of the particles is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The motion of each particle is opposed by a resisting force of magnitude \(0.9 v \mathrm {~N}\). The particle \(P\) does not reach the pulley.

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 3 v\).
2. Find the value of \(t\) when the particles have speed \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance that each particle has travelled in this time.

7 A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string of natural length 0.8 m and modulus of elasticity 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
2. Calculate the speed of \(P\) immediately before it collides with the plane.
3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.

7 A particle \(P\) of mass 0.2 kg is released from rest at a point \(O\) on a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, and travels down a line of greatest slope. The coefficient of friction between \(P\) and the plane is 0.3 . A force of magnitude \(0.6 x \mathrm {~N}\) acts on \(P\) in the direction \(P O\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\).

1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 5 \sqrt { } 3 - 1.5 - 3 x\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at a displacement \(x \mathrm {~m}\) from \(O\).
2. Find the value of \(x\) for which \(P\) reaches its maximum velocity, and calculate this maximum velocity.
3. Calculate the magnitude of the acceleration of \(P\) immediately after it has first come to instantaneous rest.

7. \begin{figure}[h] \end{figure} Two particles \(A\) and \(B\), of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially \(B\) is held at rest on a rough fixed plane inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The part of the string from \(B\) to \(P\) is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, \(P\), fixed at the top of the plane. The particle \(A\) hangs freely below \(P\), as shown in Figure 4. The coefficient of friction between \(B\) and the plane is \(\frac { 2 } { 3 }\). The particles are released from rest with the string taut and \(B\) moves up the plane.

1. Find the magnitude of the acceleration of \(B\) immediately after release.
2. Find the speed of \(B\) when it has moved 1 m up the plane. When \(B\) has moved 1 m up the plane the string breaks. Given that in the subsequent motion \(B\) does not reach \(P\),
3. find the time between the instants when the string breaks and when \(B\) comes to instantaneous rest.

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.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is on a rough plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held at rest on the plane by the action of a force of magnitude 4 N acting up the plane in a direction parallel to a line of greatest slope of the plane, as shown in Figure 2. The particle is on the point of slipping up the plane.

1. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 4 N is removed.
2. Find the acceleration of \(P\) down the plane.

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force \(\mathbf { F }\) of magnitude 30 N is applied to \(Q\) in the direction \(P Q\), as shown in Figure 4. The force is applied for 3 s and during this time \(Q\) travels a distance of 6 m . The coefficient of friction between each particle and the plane is \(\mu\). Find

1. the acceleration of \(Q\),
2. the value of \(\mu\),
3. the tension in the string.
4. State how in your calculation you have used the information that the string is inextensible. When the particles have moved for 3 s , the force \(\mathbf { F }\) is removed.
5. Find the time between the instant that the force is removed and the instant that \(Q\) comes to rest.

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.

2. \begin{figure}[h] \end{figure} A rough plane is inclined at \(40 ^ { \circ }\) to the horizontal. Two points \(A\) and \(B\) are 3 metres apart and lie on a line of greatest slope of the inclined plane, with \(A\) above \(B\), as shown in Figure 2. A particle \(P\) of mass \(m \mathrm {~kg}\) is held at rest on the plane at \(A\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is released.

1. Find the acceleration of \(P\) down the plane.
2. Find the speed of \(P\) at \(B\).

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.

3. A particle \(P\) of mass 0.4 kg is moving on rough horizontal ground when it hits a fixed vertical plane wall. Immediately before hitting the wall, \(P\) is moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a direction perpendicular to the wall. The particle rebounds from the wall and comes to rest at a distance of 5 m from the wall. The coefficient of friction between \(P\) and the ground is \(\frac { 1 } { 8 }\). Find the magnitude of the impulse exerted on \(P\) by the wall.