3.03v Motion on rough surface: including inclined planes

\includegraphics{figure_3} A fixed wedge has two plane faces, each inclined at \(30°\) to the horizontal. Two particles \(A\) and \(B\), of mass \(3m\) and \(m\) respectively, are attached to the ends of a light inextensible string. Each particle moves on one of the plane faces of the wedge. The string passes over a small smooth light pulley fixed at the top of the wedge. The face on which \(A\) moves is smooth. The face on which \(B\) moves is rough. The coefficient of friction between \(B\) and this face is \(\mu\). Particle \(A\) is held at rest with the string taut. The string lies in the same vertical plane as lines of greatest slope on each plane face of the wedge, as shown in Figure 3. The particles are released from rest and start to move. Particle \(A\) moves downwards and \(B\) moves upwards. The accelerations of \(A\) and \(B\) each have magnitude \(\frac{1}{10}g\).

1. By considering the motion of \(A\), find, in terms of \(m\) and \(g\), the tension in the string. [3]
2. By considering the motion of \(B\), find the value of \(\mu\). [8]
3. Find the resultant force exerted by the string on the pulley, giving its magnitude and direction. [3]

\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

1. Find the value of \(P\). [8]

The tension in the rope is now increased to 150 N.

1. Find the acceleration of the box. [6]

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

1. the acceleration of the particle, [3]
2. the coefficient of friction between the particle and the plane. [5]

The particle is now held on the same rough plane by a horizontal force of magnitude \(X\) newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane. \includegraphics{figure_3}

1. Find the value of \(X\). [7]

A lifeboat slides down a straight ramp inclined at an angle of \(15°\) to the horizontal. The lifeboat has mass 800 kg and the length of the ramp is 50 m. The lifeboat is released from rest at the top of the ramp and is moving with a speed of 12.6 m s\(^{-1}\) when it reaches the end of the ramp. By modelling the lifeboat as a particle and the ramp as a rough inclined plane, find the coefficient of friction between the lifeboat and the ramp. [9]

\includegraphics{figure_5} Figure 5 shows two particles \(A\) and \(B\), of mass \(2m\) and \(4m\) respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{3}{4}\). The coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth pulley \(P\) which is 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 and \(B\) hangs vertically below \(P\). The system is released from rest with the string taut, with \(A\) at the point \(X\) and with \(B\) at a height \(h\) above the ground. For the motion until \(B\) hits the ground,

1. give a reason why the magnitudes of the accelerations of the two particles are the same, [1]
2. write down an equation of motion for each particle, [4]
3. find the acceleration of each particle. [5]

Particle \(B\) does not rebound when it hits the ground and \(A\) continues moving up the plane towards \(P\). Given that \(A\) comes to rest at the point \(Y\), without reaching \(P\),

1. find the distance \(XY\) in terms of \(h\). [6]

\includegraphics{figure_2} A box of mass \(6 \text{ kg}\) lies on a rough plane inclined at an angle of \(30°\) to the horizontal. The box is held in equilibrium by means of a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. The line of action of the force is in the same vertical plane as a line of greatest slope of the plane. The coefficient of friction between the box and the plane is \(0.4\). The box is modelled as a particle. Given that the box is in limiting equilibrium and on the point of moving up the plane, find,

1. the normal reaction exerted on the box by the plane, [4]
2. the value of \(P\). [3]

The horizontal force is removed.

1. Show that the box will now start to move down the plane. [5]

\includegraphics{figure_3} Particles \(A\) and \(B\), of mass \(2m\) and \(m\) respectively, are attached to the ends of a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal table. Particle \(A\) is held on the table, while \(B\) rests on a smooth plane inclined at \(30°\) to the horizontal, as shown in Fig. 3. The string is in the same vertical plane as a line of greatest slope of the inclined plane. The coefficient of friction between \(A\) and the table is \(\mu\). The particle \(A\) is released from rest and begins to move. By writing down an equation of motion for each particle,

1. show that, while both particles move with the string taut. Each particle has an acceleration of magnitude \(\frac{1}{5}(1 - 4\mu)g\). [7]

When each particle has moved a distance \(h\), the string breaks. The particle \(A\) comes to rest before reaching the pulley. Given that \(\mu = 0.2\),

1. find, in terms of \(h\), the total distance moved by \(A\). [6]

For the model described above,

1. state two physical factors, apart from air resistance, which could be taken into account to make the model more realistic. [2]

\includegraphics{figure_3} Figure 3 shows a boat \(B\) of mass \(400\) kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at \(15°\) to the horizontal. The coefficient of friction between \(B\) and the slipway is \(0.2\). The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.

1. Calculate the tension in the rope. [6]

The boat is \(50\) m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.

1. Calculate the time taken for the boat to slide to the bottom of the slipway. [6]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\includegraphics{figure_2} Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m\) kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough 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 \(\frac{1}{2}\). The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and \(Q\) accelerates vertically downwards at 1.4 m s\(^{-2}\). Find

1. the magnitude of the normal reaction of the inclined plane on \(P\), [2]
2. the value of \(m\). [8]

When the particles have been moving for 0.5 s, the string breaks. Assuming that \(P\) does not reach the pulley,

1. find the further time that elapses until \(P\) comes to instantaneous rest. [6]

\includegraphics{figure_2} A fixed rough plane is inclined at 30° to the horizontal. A small smooth pulley \(P\) is fixed at the top of the plane. Two particles \(A\) and \(B\), of mass 2 kg and 4 kg respectively, are attached to the ends of a light inextensible string which passes over the pulley \(P\). The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane and \(B\) hangs freely below \(P\), as shown in Figure 2. The coefficient of friction between \(A\) and the plane is \(\frac{1}{\sqrt{3}}\). Initially \(A\) is held at rest on the plane. The particles are released from rest with the string taut and \(A\) moves up the plane. Find the tension in the string immediately after the particles are released. [9]

\includegraphics{figure_3} A particle \(P\) of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points \(A\) and \(B\), where \(AB = 10\) m, as shown in Figure 3. The speed of \(P\) at \(A\) is 2 m s\(^{-1}\). The particle \(P\) takes 3.5 s to move from \(A\) to \(B\). Find

1. the speed of \(P\) at \(B\), [3]
2. the acceleration of \(P\), [2]
3. the coefficient of friction between \(P\) and the plane. [5]

\includegraphics{figure_3} A small parcel of mass \(2\) kg moves on a rough plane inclined at an angle of \(30°\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30°\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is \(0.4\). Given that the tension in the rope is \(24\) N,

1. find, to 2 significant figures, the acceleration of the parcel. [8]

The rope now breaks. The parcel slows down and comes to rest.

1. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again. [4]
2. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest. [3]

\includegraphics{figure_1} A heavy suitcase \(S\) of mass 50 kg is moving along a horizontal floor under the action of a force of magnitude \(P\) newtons. The force acts at 30° to the floor, as shown in Fig. 1, and \(S\) moves in a straight line at constant speed. The suitcase is modelled as a particle and the floor as a rough horizontal plane. The coefficient of friction between \(S\) and the floor is \(\frac{3}{4}\). Calculate the value of \(P\). [9]

\includegraphics{figure_3} A sledge has mass 30 kg. The sledge is pulled in a straight line along horizontal ground by means of a rope. The rope makes an angle \(20°\) with the horizontal, as shown in Figure 3. The coefficient of friction between the sledge and the ground is 0.2. The sledge is modelled as a particle and the rope as a light inextensible string. The tension in the rope is 150 N. Find, to 3 significant figures,

1. the normal reaction of the ground on the sledge, [3]
2. the acceleration of the sledge. [3]

When the sledge is moving at \(12 \text{ m s}^{-1}\), the rope is released from the sledge.

1. Find, to 3 significant figures, the distance travelled by the sledge from the moment when the rope is released to the moment when the sledge comes to rest. [6]

\includegraphics{figure_4} A heavy package is held in equilibrium on a slope by a rope. The package is attached to one end of the rope, the other end being held by a man standing at the top of the slope. The package is modelled as a particle of mass 20 kg. The slope is modelled as a rough plane inclined at \(60°\) to the horizontal and the rope as a light inextensible string. The string is assumed to be parallel to a line of greatest slope of the plane, as shown in Figure 4. At the contact between the package and the slope, the coefficient of friction is 0.4.

1. Find the minimum tension in the rope for the package to stay in equilibrium on the slope. [8]

The man now pulls the package up the slope. Given that the package moves at constant speed,

1. find the tension in the rope. [4]
2. State how you have used, in your answer to part (b), the fact that the package moves
   1. up the slope,
   2. at constant speed.
   [2]

\includegraphics{figure_4} A particle of mass \(m\) rests on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). The particle is attached to one end of a light inextensible string which lies in a line of greatest slope of the plane and passes over a small light smooth pulley \(P\) fixed at the top of the plane. The other end of the string is attached to a particle \(B\) of mass \(3m\), and \(B\) hangs freely below \(P\), as shown in Fig. 4. The particles are released from rest with the string taut. The particle \(B\) moves down with acceleration of magnitude \(\frac{1}{3}g\). Find

1. the tension in the string, [4]
2. the coefficient of friction between \(A\) and the plane. [9]

\includegraphics{figure_2} Figure 2 shows a uniform rod \(AB\), of mass \(m\) and length \(2a\), with the end \(B\) resting on rough horizontal ground. The rod is held in equilibrium at an angle \(\theta\) to the vertical by a light inextensible string. One end of the string is attached to the rod at the point \(C\), where \(AC = \frac{2}{3}a\). The other end of the string is attached to the point \(D\), which is vertically above \(B\), where \(BD = 2a\).

1. By taking moments about \(D\), show that the magnitude of the frictional force acting on the rod at \(B\) is \(\frac{1}{2}mg \sin \theta\) [3]
2. Find the magnitude of the normal reaction on the rod at \(B\). [5]

The rod is in limiting equilibrium when \(\tan \theta = \frac{4}{3}\).

1. Find the coefficient of friction between the rod and the ground. [3]

\includegraphics{figure_1} Figure 1 A particle \(P\) of mass 6.5 kg is projected up a fixed rough plane with initial speed 6 m s\(^{-1}\) from a point \(X\) on the plane, as shown in Figure 1. The particle moves up the plane along the line of greatest slope through \(X\) and comes to instantaneous rest at the point \(Y\), where \(XY = d\) metres. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{5}{12}\). The coefficient of friction between \(P\) and the plane is \(\frac{1}{3}\).

1. Use the work-energy principle to show that, to 2 significant figures, \(d = 2.7\) [7]

After coming to rest at \(Y\), the particle \(P\) slides back down the plane.

1. Find the speed of \(P\) as it passes through \(X\). [4]

\includegraphics{figure_2} Figure 2 A uniform rod \(AB\) has length \(4a\) and weight \(W\). A particle of weight \(kW\), \(k < 1\), is attached to the rod at \(B\). The rod rests in equilibrium against a fixed smooth horizontal peg. The end \(A\) of the rod is on rough horizontal ground, as shown in Figure 2. The rod rests on the peg at \(C\), where \(AC = 3a\), and makes an angle \(\alpha\) with the ground, where \(\tan \alpha = \frac{1}{3}\). The peg is perpendicular to the vertical plane containing \(AB\).

1. Give a reason why the force acting on the rod at \(C\) is perpendicular to the rod. [1]
2. Show that the magnitude of the force acting on the rod at \(C\) is $$\frac{\sqrt{10}}{5}W(1 + 2k)$$ [4]

The coefficient of friction between the rod and the ground is \(\frac{3}{4}\).

1. Show that for the rod to remain in equilibrium \(k \leq \frac{2}{11}\). [7]

A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(0.15\) and \(B\) of the ladder rests against a smooth vertical wall. The ladder rests in equilibrium in a vertical plane perpendicular to the wall, and makes an angle of \(30°\) with the wall. A man of mass \(5m\) stands on the ladder, which remains in equilibrium. The ladder is modelled as a uniform rod and the man as a particle. The greatest possible distance of the man from \(A\) is \(6a\). Find the value of \(k\). [9]

A child is playing with a small model of a fire-engine of mass \(0.5\) kg and a straight, rigid plank. The plank is inclined at an angle \(α\) to the horizontal. The fire-engine is projected up the plank along a line of greatest slope. The non-gravitational resistance to the motion of the fire-engine is constant and has magnitude \(R\) newtons. When \(α = 20°\) the fire-engine is projected with an initial speed of \(5\) m s\(^{-1}\) and first comes to rest after travelling 2 m.

1. Find, to 3 significant figures, the value of \(R\). [7]

When \(α = 40°\) the fire-engine is again projected with an initial speed of \(5\) m s\(^{-1}\).

1. Find how far the fire-engine travels before first coming to rest. [3]

A brick of mass 3 kg slides in a straight line on a horizontal floor. The brick is modelled as a particle and the floor as a rough plane. The initial speed of the brick is 8 m s\(^{-1}\). The brick is brought to rest after moving 12 m by the constant frictional force between the brick and the floor.

1. Calculate the kinetic energy lost by the brick in coming to rest, stating the units of your answer. [2]
2. Calculate the coefficient of friction between the brick and the floor. [4]