3.03t Coefficient of friction: F <= mu*R model

\includegraphics{figure_2} A uniform square lamina \(ABCD\) of side \(4a\) and weight \(W\) rests in a vertical plane with the edge \(AB\) inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac{1}{4}\). The vertex \(B\) is in contact with a rough horizontal surface for which the coefficient of friction is \(\mu\). The lamina is supported by a smooth peg at the point \(E\) on \(AB\), where \(BE = 3a\) (see diagram).

1. Find expressions in terms of \(W\) for the normal reaction forces at \(E\) and \(B\). [5]
2. Given that the lamina is about to slip, find the value of \(\mu\). [3]

The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}

1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
2. Find the acceleration of the ring when \(T = 3\). [3]

A particle \(P\) of mass \(0.3\text{ kg}\), lying on a smooth plane inclined at \(30°\) to the horizontal, is released from rest. \(P\) slides down the plane for a distance of \(2.5\text{ m}\) and then reaches a horizontal plane. There is no change in speed when \(P\) reaches the horizontal plane. A particle \(Q\) of mass \(0.2\text{ kg}\) lies at rest on the horizontal plane \(1.5\text{ m}\) from the end of the inclined plane (see diagram). \(P\) collides directly with \(Q\). \includegraphics{figure_7}

1. It is given that the horizontal plane is smooth and that, after the collision, \(P\) continues moving in the same direction, with speed \(2\text{ m s}^{-1}\). Find the speed of \(Q\) after the collision. [5]
2. It is given instead that the horizontal plane is rough and that when \(P\) and \(Q\) collide, they coalesce and move with speed \(1.2\text{ m s}^{-1}\). Find the coefficient of friction between \(P\) and the horizontal plane. [5]

\includegraphics{figure_3} A particle of mass 2.5 kg is held in equilibrium on a rough plane inclined at 20° to the horizontal by a force of magnitude \(T\) N making an angle of 60° with a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.3. Find the greatest and least possible values of \(T\). [8]

\includegraphics{figure_7} A slide in a playground descends at a constant angle of 30° for 2.5 m. It then has a horizontal section in the same vertical plane as the sloping section. A child of mass 35 kg, modelled as a particle \(P\), starts from rest at the top of the slide and slides straight down the sloping section. She then continues along the horizontal section until she comes to rest (see diagram). There is no instantaneous change in speed when the child goes from the sloping section to the horizontal section. The child experiences a resistance force on the horizontal section of the slide, and the work done against the resistance force on the horizontal section of the slide is 250 J per metre.

1. It is given that the sloping section of the slide is smooth.
   1. Find the speed of the child when she reaches the bottom of the sloping section. [3]
   2. Find the distance that the child travels along the horizontal section of the slide before she comes to rest. [2]
2. It is given instead that the sloping section of the slide is rough and that the child comes to rest on the slide 1.05 m after she reaches the horizontal section. Find the coefficient of friction between the child and the sloping section of the slide. [6]

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

\includegraphics{figure_4} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.3 \text{ kg}\) respectively, are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. Particle \(A\) hangs freely and particle \(B\) is in contact with the table (see diagram).

1. The system is in limiting equilibrium with the string taut and \(A\) about to move downwards. Find the coefficient of friction between \(B\) and the table. [4]

A force now acts on particle \(B\). This force has a vertical component of \(1.8 \text{ N}\) upwards and a horizontal component of \(X \text{ N}\) directed away from the pulley.

1. The system is now in limiting equilibrium with the string taut and \(A\) about to move upwards. Find \(X\). [3]

\includegraphics{figure_4} A block of mass 8 kg is at rest on a plane inclined at 20° to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).

1. Given that the tension in the string is 13 N, find the frictional and normal components of the force exerted on the block by the plane. [4]

The string is cut; the block remains at rest, but is on the point of slipping down the plane.

1. Find the coefficient of friction between the block and the plane. [2]

\includegraphics{figure_3} A small ring of mass 0.8 kg is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude 7 N pulling upwards at 45° to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude 3.05 N, correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\includegraphics{figure_3} A small ring of mass \(0.8 \text{ kg}\) is threaded on a rough rod which is fixed horizontally. The ring is in equilibrium, acted on by a force of magnitude \(7 \text{ N}\) pulling upwards at \(45°\) to the horizontal (see diagram).

1. Show that the normal component of the contact force acting on the ring has magnitude \(3.05 \text{ N}\), correct to 3 significant figures. [2]
2. The ring is in limiting equilibrium. Find the coefficient of friction between the ring and the rod. [3]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).

1. Calculate the maximum frictional force that can act on the block. [2]
2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]

A small box of mass 5 kg is pulled at a constant speed of \(2.5 \text{ m s}^{-1}\) down a line of greatest slope of a rough plane inclined at \(10°\) to the horizontal. The pulling force has magnitude 20 N and acts downwards parallel to a line of greatest slope of the plane.

1. Find the coefficient of friction between the box and the plane. [5]

The pulling force is removed while the box is moving at \(2.5 \text{ m s}^{-1}\).

1. Find the distance moved by the box after the instant at which the pulling force is removed. [4]

\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.

1. Find the coefficient of friction between the block and the plane. [5]

The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.

1. Find the greatest value of \(X\) for which the block does not move. [2]

A block of mass 3 kg is initially at rest on a rough horizontal plane. A force of magnitude 6 N is applied to the block at an angle of \(\theta\) above the horizontal, where \(\cos \theta = \frac{24}{25}\). The force is applied for a period of 5 s, during which time the block moves a distance of 4.5 m.

1. Find the magnitude of the frictional force on the block. [4]
2. Show that the coefficient of friction between the block and the plane is 0.165, correct to 3 significant figures. [3]
3. When the block has moved a distance of 4.5 m, the force of magnitude 6 N is removed and the block then decelerates to rest. Find the total time for which the block is in motion. [4]

\includegraphics{figure_4} Blocks \(P\) and \(Q\), of mass \(m\) kg and 5 kg respectively, 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 rough plane inclined at 35° to the horizontal. Block \(P\) is at rest on the plane and block \(Q\) hangs vertically below the pulley (see diagram). The coefficient of friction between block \(P\) and the plane is 0.2. Find the set of values of \(m\) for which the two blocks remain at rest. [6]

\includegraphics{figure_1} A block of mass 50 kg lies on a rough plane which is inclined to the horizontal at an angle \(\alpha\), where \(\tan\alpha = \frac{7}{24}\). The block is held at rest by a vertical rope, as shown in Figure 1, and is on the point of sliding down the plane. The block is modelled as a particle and the rope is modelled as a light inextensible string. Given that the friction force acting on the block has magnitude 65.8 N, find

1. the tension in the rope, [4]
2. the coefficient of friction between the block and the plane. [4]

\includegraphics{figure_2} A particle \(P\) of mass 2 kg is pushed up a line of greatest slope of a rough plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in the vertical plane which contains \(P\) and a line of greatest slope of the 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 0.5 Given that the acceleration of \(P\) is 1.45 m s\(^{-2}\), find the value of \(X\). [10]

\includegraphics{figure_3} A particle \(A\) of mass \(3m\) is held at rest on a rough horizontal table. The particle is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a particle \(B\) of mass \(2m\), which hangs freely, vertically below \(P\). The system is released from rest, with the string taut, when \(A\) is 1.3 m from \(P\) and \(B\) is 1 m above the horizontal floor, as shown in Figure 3. Given that \(B\) hits the floor 2 s after release and does not rebound,

1. find the acceleration of \(A\) during the first two seconds, [2]
2. find the coefficient of friction between \(A\) and the table, [8]
3. determine whether \(A\) reaches the pulley. [6]

\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

\includegraphics{figure_3} A particle \(P\) of mass 2 kg is attached to one end of a light inextensible string. A particle \(Q\) of mass 5 kg is attached to the other end of the string. The string passes over a small smooth light pulley. The pulley is fixed at a point on the intersection of a rough horizontal table and a fixed smooth inclined plane. The string lies along the table and also lies in a vertical plane which contains a line of greatest slope of the inclined plane. This plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) is at rest on the table, a distance \(d\) metres from the pulley. Particle \(Q\) is on the inclined plane with the string taut, as shown in Figure 3. The coefficient of friction between \(P\) and the table is \(\frac{1}{4}\). The system is released from rest and \(P\) slides along the table towards the pulley. Assuming that \(P\) has not reached the pulley and that \(Q\) remains on the inclined plane,

1. write down an equation of motion for \(P\), [2]
2. write down an equation of motion for \(Q\), [2]
3. 1. find the acceleration of \(P\),
   2. find the tension in the string. [5]

When \(P\) has moved a distance 0.5 m from its initial position, the string breaks. Given that \(P\) comes to rest just as it reaches the pulley,

1. find the value of \(d\). [7]

\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]