3.03u Static equilibrium: on rough surfaces

A particle \(P\) of mass 0.2 kg lies at rest on a rough horizontal plane. A horizontal force of 1.2 N is applied to \(P\).

1. Given that \(P\) is in limiting equilibrium, find the coefficient of friction between \(P\) and the plane. [3]
2. Given instead that the coefficient of friction between \(P\) and the plane is 0.3, find the distance travelled by \(P\) in the third second of its motion. [4]

\includegraphics{figure_6} A particle of mass \(1.2\) kg is placed on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). The particle is kept in equilibrium by a horizontal force of magnitude \(P\) N acting in a vertical plane containing a line of greatest slope (see diagram). The coefficient of friction between the particle and the plane is \(0.15\). Find the least possible value of \(P\). [6]

\includegraphics{figure_5} A particle of mass 12 kg is going to be pulled across a rough horizontal plane by a light inextensible string. The string is at an angle of 30° above the plane and has tension \(T\) N (see diagram). The coefficient of friction between the particle and the plane is 0.5.

1. Given that the particle is on the point of moving, find the value of \(T\). [5]
2. Given instead that the particle is accelerating at 0.2 ms\(^{-2}\), find the value of \(T\). [3]

\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_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_7} The diagram shows a triangular block with sloping faces inclined to the horizontal at \(45°\) and \(30°\). Particle \(A\) of mass \(0.8 \text{ kg}\) lies on the face inclined at \(45°\) and particle \(B\) of mass \(1.2 \text{ kg}\) lies on the face inclined at \(30°\). The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the faces. The parts \(AP\) and \(BP\) of the string are parallel to lines of greatest slope of the respective faces. The particles are released from rest with both parts of the string taut. In the subsequent motion neither particle reaches the pulley and neither particle reaches the bottom of a face.

1. Given that both faces are smooth, find the speed of \(A\) after each particle has travelled a distance of \(0.4 \text{ m}\). [6]
2. It is given instead that both faces are rough. The coefficient of friction between each particle and a face of the block is \(\mu\). Find the value of \(\mu\) for which the system is in limiting equilibrium. [6]

A particle of mass \(3\text{ kg}\) is on a rough plane inclined at an angle of \(20°\) to the horizontal. A force of magnitude \(P\text{ N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is \(0.35\). Show that the least possible value of \(P\) is \(0.394\), correct to 3 significant figures, and find the greatest possible value of \(P\). [6]

\includegraphics{figure_3} A particle of mass \(0.6\) kg is placed on a rough plane which is inclined at an angle of \(21°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.3\). Show that the least possible value of \(P\) is \(0.470\), correct to \(3\) significant figures, and find the greatest possible value of \(P\). [6]

\includegraphics{figure_1} A small ring \(P\) of mass \(0.03\) kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15°\) to the horizontal. The tension in the string is \(2.5\) N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod. [4]

\includegraphics{figure_5} A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude \(T\) N acting at an angle of \(60°\) to the downward vertical (see diagram). The ring is in equilibrium.

1. The normal and frictional components of the contact force exerted on the ring by the rod are \(R\) N and \(F\) N respectively. Find \(R\) and \(F\) in terms of \(T\). [4]
2. The coefficient of friction between the rod and the ring is 0.7. Find the value of \(T\) for which the ring is about to slip. [3]

A particle of mass 0.2 kg is resting in equilibrium on a rough plane inclined at \(20°\) to the horizontal.

1. Show that the friction force acting on the particle is 0.684 N, correct to 3 significant figures. [1]

The coefficient of friction between the particle and the plane is 0.6. A force of magnitude 0.9 N is applied to the particle down a line of greatest slope of the plane. The particle accelerates down the plane.

1. Find this acceleration. [4]

\includegraphics{figure_6} Two particles \(P\) and \(Q\), each of mass \(m\) kg, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The plane is inclined at an angle \(α\) to the horizontal, where \(\tan α = \frac{4}{3}\). Particle \(P\) rests on the plane and particle \(Q\) hangs vertically, as shown in the diagram. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane. The system is in limiting equilibrium.

1. Show that the coefficient of friction between \(P\) and the plane is \(\frac{4}{3}\). [5]

A force of magnitude 10 N is applied to \(P\), acting up a line of greatest slope of the plane, and \(P\) accelerates at 2.5 m s\(^{-2}\).

1. Find the value of \(m\). [5]

A block of mass \(5\) kg is being pulled by a rope up a rough plane inclined at \(6°\) to the horizontal. The rope is parallel to a line of greatest slope of the plane and the block is moving at constant speed. The coefficient of friction between the block and the plane is \(0.3\). Find the tension in the rope. [4]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of masses \(0.4\) kg and \(0.7\) kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley which is attached to the edge of a rough plane. The coefficient of friction between \(P\) and the plane is \(0.5\). The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac{3}{4}\). Particle \(P\) lies on the plane and particle \(Q\) hangs vertically. The string between \(P\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). A force of magnitude \(X\) N, acting directly down the plane, is applied to \(P\).

1. Show that the greatest value of \(X\) for which \(P\) remains stationary is \(6.2\). [4]
2. Given instead that \(X = 0.8\), find the acceleration of \(P\). [4]

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]

\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_4} A uniform solid cone has base radius \(0.4\) m and height \(4.4\) m. A uniform solid cylinder has radius \(0.4\) m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_4} A uniform rod \(AB\) has weight \(15\) N and length \(1.2\) m. The end \(A\) of the rod is in contact with a rough plane inclined at \(30°\) to the horizontal, and the rod is perpendicular to the plane. The rod is held in equilibrium in this position by means of a horizontal force applied at \(B\), acting in the vertical plane containing the rod (see diagram).

1. Show that the magnitude of the force applied at \(B\) is \(4.33\) N, correct to \(3\) significant figures. [3]
2. Find the magnitude of the frictional force exerted by the plane on the rod. [2]
3. Given that the rod is in limiting equilibrium, calculate the coefficient of friction between the rod and the plane. [3]

\includegraphics{figure_1} A non-uniform rod \(AB\), of length 0.6 m and weight 9 N, has its centre of mass 0.4 m from \(A\). The end \(A\) of the rod is in contact with a rough vertical wall. The rod is held in equilibrium, perpendicular to the wall, by means of a light string attached to \(B\). The string is inclined at \(30°\) to the horizontal. The tension in the string is \(T\) N (see diagram).

1. Calculate \(T\). [2]
2. Find the least possible value of the coefficient of friction at \(A\). [3]

\includegraphics{figure_2} A uniform rod \(AB\) has weight \(6\) N and length \(0.8\) m. The rod rests in limiting equilibrium with \(B\) in contact with a rough horizontal surface and \(AB\) inclined at \(60°\) to the horizontal. Equilibrium is maintained by a force, in the vertical plane containing \(AB\), acting at \(A\) at an angle of \(45°\) to \(AB\) (see diagram). Calculate

1. the magnitude of the force applied at \(A\), [3]
2. the least possible value of the coefficient of friction at \(B\). [4]

\includegraphics{figure_3} A uniform square lamina of side \(2a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(AB\). The other end of the string is attached to a fixed point \(P\) on a rough vertical wall. The vertex \(B\) of the lamina is in contact with the wall. The string \(EP\) is perpendicular to the side \(AB\) and makes an angle \(\theta\) with the wall (see diagram). The string and the lamina are in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the lamina is \(\frac{1}{2}\). Given that the vertex \(B\) is about to slip up the wall, find the value of \(\tan\theta\). [8]

\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]