3.03r Friction: concept and vector form

\includegraphics{figure_3} A particle \(P\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass \(m\) 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 \(m\), [10]
2. find the magnitude of the force exerted on the pulley by the string, [4]
3. find the direction of the force exerted on the pulley by the string. [1]

A suitcase of mass 40 kg is being dragged in a straight line along a rough horizontal floor at constant speed using a thin strap. The strap is inclined at \(20°\) above the horizontal. The coefficient of friction between the suitcase and the floor is \(\frac{3}{4}\). The strap is modelled as a light inextensible string and the suitcase is modelled as a particle. Find the tension in the strap. [7]

\includegraphics{figure_2} Figure 2 shows two particles \(A\) and \(B\), of masses \(3m\) and \(4m\) respectively, attached to the ends of a light inextensible string. Initially \(A\) is held at rest 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 coefficient of friction between \(A\) and the plane is \(\frac{1}{4}\). The string passes over a small smooth light 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. The particle \(B\) hangs freely and is vertically below \(P\). The system is released from rest with the string taut and with \(B\) at a height of 1.75 m above the ground. In the subsequent motion, \(A\) does not hit the pulley. For the period before \(B\) hits the ground,

1. write down an equation of motion for each particle. [4]
2. Hence show that the acceleration of \(B\) is \(\frac{8}{35}g\). [5]
3. Explain how you have used the fact that the string is inextensible in your calculation. [1]

When \(B\) hits the ground, \(B\) does not rebound and comes immediately to rest.

1. Find the distance travelled by \(A\) from the instant when the system is released to the instant when \(A\) first comes to rest. [7]

\includegraphics{figure_2} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) A particle \(P\) of mass 2 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

1. Show that when \(X = 14.7\) there is no frictional force acting on \(P\) [3] The coefficient of friction between \(P\) and the plane is 0.5
2. Find the smallest possible value of \(X\). [8]

\includegraphics{figure_4} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string. Another particle \(Q\), also of mass \(m\), is attached to the other end of the 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 \(P\) hangs vertically below the pulley with the string taut, as shown in Figure 4. The pulley, \(P\) and \(Q\) all lie in the same vertical plane. The coefficient of friction between \(Q\) and the table is \(\mu\), where \(\mu < 1\) Particle \(Q\) is released from rest. The tension in the string before \(Q\) hits the pulley is \(kmg\), where \(k\) is a constant.

1. Find \(k\) in terms of \(\mu\). [7] Given that \(Q\) is initially a distance \(d\) from the pulley,
2. find, in terms of \(d\), \(g\) and \(\mu\), the time taken by \(Q\), after release, to reach the pulley. [4]
3. Describe what would happen if \(\mu \geqslant 1\), giving a reason for your answer. [2]

\includegraphics{figure_1} A small box is pushed along a floor. The floor is modelled as a rough horizontal plane and the box is modelled as a particle. The coefficient of friction between the box and the floor is \(\frac{1}{2}\). The box is pushed by a force of magnitude 100 N which acts at an angle of 30° with the floor, as shown in Figure 1. Given that the box moves with constant speed, find the mass of the box. [7]

\includegraphics{figure_4} Two particles \(P\) and \(Q\) have masses \(3m\) and \(5m\) respectively. They are connected by a light inextensible string which passes over a small smooth light pulley fixed at the edge of a rough horizontal table. Particle \(P\) lies on the table and particle \(Q\) hangs freely below the pulley, as shown in Fig. 4. The coefficient of friction between \(P\) and the table is 0.6. The system is released from rest with the string taut. For the period before \(Q\) hits the floor or \(P\) reaches the pulley,

1. write down an equation of motion for each particle separately, [4]
2. find, in terms of \(g\), the acceleration of \(Q\), [4]
3. find, in terms of \(m\) and \(g\), the tension in the string. [2]

When \(Q\) has moved a distance \(h\), it hits the floor and the string becomes slack. Given that \(P\) remains on the table during the subsequent motion and does not reach the pulley,

1. find, in terms of \(h\), the distance moved by \(P\) after the string becomes slack until \(P\) comes to rest. [6]

\includegraphics{figure_4} A particle \(A\) of mass 0.8 kg rests on a horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass 1.2 kg which hangs freely below the pulley, as shown in Fig. 4. The system is released from rest with the string taut and with \(B\) at a height of 0.6 m above the ground. In the subsequent motion \(A\) does not reach \(P\) before \(B\) reaches the ground. In an initial model of the situation, the table is assumed to be smooth. Using this model, find

1. the tension in the string before \(B\) reaches the ground, [5]
2. the time taken by \(B\) to reach the ground. [3]

In a refinement of the model, it is assumed that the table is rough and that the coefficient of friction between \(A\) and the table is \(\frac{1}{4}\). Using this refined model,

1. find the time taken by \(B\) to reach the ground. [8]

The tile on a roof becomes loose and slides from rest down the roof. The roof is modelled as a plane surface inclined at 30° to the horizontal. The coefficient of friction between the tile and the roof is 0.4. The tile is modelled as a particle of mass \(m\) kg.

1. Find the acceleration of the tile as it slides down the roof. [7]

The tile moves a distance 3 m before reaching the edge of the roof.

1. Find the speed of the tile as it reaches the edge of the roof. [2]
2. Write down the answer to part (a) if the tile had mass \(2m\) kg. [1]

\includegraphics{figure_3} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at 20° to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4. The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

1. the normal reaction of the plane on \(P\), [2]
2. the value of \(X\). [4]

The force of magnitude \(X\) newtons is now removed.

1. Show that \(P\) remains in equilibrium on the plane. [4]

\includegraphics{figure_4} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s. Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\), [3]
2. the tension in the string, [4]
3. the value of \(\mu\). [5]
4. State how in your calculations you have used the information that the string is inextensible. [1]

\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find

1. the value of \(P\), [4]
2. the value of \(\mu\). [5]

The horizontal force is removed.

1. Determine whether or not the parcel moves. [5]

\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]

\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_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]

\includegraphics{figure_1} A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 1. The coefficient of friction between the particle and the plane is \(\frac{1}{2}\). Given that the particle is on the point of sliding down the plane,

1. show that the magnitude of the normal reaction between the particle and the plane is 20 N,
2. find the value of \(W\). [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_1} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), and the plane is at an angle of \(30°\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac{1}{2}\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

A particle \(P\) of mass \(m\) is held at a point \(A\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac{2}{3}\). The particle is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4mg\). The other end of the string is attached to a fixed point \(O\) on the plane, where \(OA = \frac{3}{4}a\). The particle \(P\) is released from rest and comes to rest at a point \(B\), where \(OB < a\). Using the work-energy principle, or otherwise, calculate the distance \(AB\). [6]

A force of magnitude \(F\) N is applied to a block of mass \(M\) kg which is initially at rest on a horizontal plane. The block starts to move with acceleration 3 ms\(^{-2}\). Modelling the block as a particle, \includegraphics{figure_4}

1. if the plane is smooth, find an expression for \(F\) in terms of \(M\). [2 marks]

If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),

1. express \(F\) in terms of \(M\), \(\mu\) and \(g\). [2 marks]
2. Calculate the value of \(\mu\) if \(F = \frac{1}{2}Mg\). [3 marks]