3.03u Static equilibrium: on rough surfaces

5 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small ring of weight 12 N is threaded on a fixed rough horizontal rod. A light string is attached to the ring and the string is pulled with a force of 15 N at an angle of \(30 ^ { \circ }\) to the horizontal.

1. When the angle of \(30 ^ { \circ }\) is below the horizontal (see Fig. 1), the ring is in limiting equilibrium. Show that the coefficient of friction between the ring and the rod is 0.666 , correct to 3 significant figures.
2. When the angle of \(30 ^ { \circ }\) is above the horizontal (see Fig. 2), the ring is moving with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the value of \(a\).

1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude 2000 N acts on the block at an angle of \(15 ^ { \circ }\) to the upwards vertical. Find the coefficient of friction between the block and the ground, correct to 2 significant figures.

3 \includegraphics[max width=\textwidth, alt={}, center]{f0200d12-4ab0-4395-804c-e693f7f26507-2_368_853_1503_644} A small smooth pulley is fixed at the highest point \(A\) of a cross-section \(A B C\) of a triangular prism. Angle \(A B C = 90 ^ { \circ }\) and angle \(B C A = 30 ^ { \circ }\). The prism is fixed with the face containing \(B C\) in contact with a horizontal surface. Particles \(P\) and \(Q\) are attached to opposite ends of a light inextensible string, which passes over the pulley. The particles are in equilibrium with \(P\) hanging vertically below the pulley and \(Q\) in contact with \(A C\). The resultant force exerted on the pulley by the string is \(3 \sqrt { } 3 \mathrm {~N}\) (see diagram).

1. Show that the tension in the string is 3 N . The coefficient of friction between \(Q\) and the prism is 0.75 .
2. Given that \(Q\) is in limiting equilibrium and on the point of moving upwards, find its mass.

4 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_449_1273_1829_438} \(A , B\) and \(C\) are three points on a line of greatest slope of a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. \(A\) is higher than \(B\) and \(B\) is higher than \(C\), and the distances \(A B\) and \(B C\) are 1.76 m and 2.16 m respectively. A particle slides down the plane with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The speed of the particle at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). The particle takes 0.8 s to travel from \(A\) to \(B\) and takes 1.4 s to travel from \(A\) to \(C\). Find

1. the values of \(u\) and \(a\),
2. the value of \(\theta\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_188_510_260_388} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{28562a1b-ec9a-40d2-bbb3-729770688971-3_196_570_255_1187} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A block of mass 2 kg is at rest on a horizontal floor. The coefficient of friction between the block and the floor is \(\mu\). A force of magnitude 12 N acts on the block at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). When the applied force acts downwards as in Fig. 1 the block remains at rest.

6 \includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-3_465_410_1891_865} The diagram shows a ring of mass 2 kg threaded on a fixed rough vertical rod. A light string is attached to the ring and is pulled upwards at an angle of \(30 ^ { \circ }\) to the horizontal. The tension in the string is \(T \mathrm {~N}\). The coefficient of friction between the ring and the rod is 0.24 . Find the two values of \(T\) for which the ring is in limiting equilibrium.

5 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-3_250_846_260_648} A small block \(B\) of mass 0.25 kg is attached to the mid-point of a light inextensible string. Particles \(P\) and \(Q\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of the string. The string passes over two smooth pulleys fixed at opposite sides of a rough table, with \(B\) resting in limiting equilibrium on the table between the pulleys and particles \(P\) and \(Q\) and block \(B\) are in the same vertical plane (see diagram).

1. Find the coefficient of friction between \(B\) and the table. \(Q\) is now removed so that \(P\) and \(B\) begin to move.
2. Find the acceleration of \(P\) and the tension in the part \(P B\) of the string.

4 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_383_791_262_678} Forces of magnitude \(X \mathrm {~N}\) and 40 N act on a block \(B\) of mass 15 kg , which is in equilibrium in contact with a horizontal surface between points \(A\) and \(C\) on the surface. The forces act in the same vertical plane and in the directions shown in the diagram.

1. Given that the surface is smooth, find the value of \(X\).
2. It is given instead that the surface is rough and that the block is in limiting equilibrium. The frictional force acting on the block has magnitude 10 N in the direction towards \(A\). Find the coefficient of friction between the block and the surface.

4 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_574_483_260_829} The diagram shows a vertical cross-section \(A B C\) of a surface. The part of the surface containing \(A B\) is smooth and \(A\) is 2.5 m above the level of \(B\). The part of the surface containing \(B C\) is rough and is at \(45 ^ { \circ }\) to the horizontal. The distance \(B C\) is 4 m (see diagram). A particle \(P\) of mass 0.2 kg is released from rest at \(A\) and moves in contact with the curve \(A B\) and then with the straight line \(B C\). The coefficient of friction between \(P\) and the part of the surface containing \(B C\) is 0.4 . Find the speed with which \(P\) reaches \(C\).

6 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small ring of mass 0.024 kg is threaded on a fixed rough horizontal rod. A light inextensible string is attached to the ring and the string is pulled with a force of magnitude 0.195 N at an angle of \(\theta\) with the horizontal, where \(\sin \theta = \frac { 5 } { 13 }\). When the angle \(\theta\) is below the horizontal (see Fig. 1) the ring is in limiting equilibrium.

1. Find the coefficient of friction between the ring and the rod. When the angle \(\theta\) is above the horizontal (see Fig. 2) the ring moves.
2. Find the acceleration of the ring.

1 A particle of mass 2 kg is initially at rest on a rough horizontal plane. A force of magnitude 10 N is applied to the particle at \(15 ^ { \circ }\) above the horizontal. It is given that 10 s after the force is applied, the particle has a speed of \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the magnitude of the frictional force is 8.96 N , correct to 3 significant figures.
2. Find the coefficient of friction between the particle and the plane.

5 A particle of mass \(m \mathrm {~kg}\) is resting on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. A force of magnitude 10 N applied to the particle up a line of greatest slope of the plane is just sufficient to stop the particle sliding down the plane. When a force of 75 N is applied to the particle up a line of greatest slope of the plane, the particle is on the point of sliding up the plane. Find \(m\) and the coefficient of friction between the particle and the plane.

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.

7 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-10_212_1029_255_557} Two particles \(A\) and \(B\) of masses 0.9 kg and 0.4 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 top of two inclined planes. The particles are initially at rest with \(A\) on a smooth plane inclined at angle \(\theta ^ { \circ }\) to the horizontal and \(B\) on a plane inclined at angle \(25 ^ { \circ }\) to the horizontal. The string is taut and the particles can move on lines of greatest slope of the two planes. A force of magnitude 2.5 N is applied to \(B\) acting down the plane (see diagram).

1. For the case where \(\theta = 15\) and the plane on which \(B\) rests is smooth, find the acceleration of \(B\).
2. For a different value of \(\theta\), the plane on which \(B\) rests is rough with coefficient of friction between the plane and \(B\) of 0.8 . The system is in limiting equilibrium with \(B\) on the point of moving in the direction of the 2.5 N force. Find the value of \(\theta\).

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 \(A\).

1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.

1 \includegraphics[max width=\textwidth, alt={}, center]{36259e2a-aa9b-4655-b0c2-891f96c3f5a4-2_549_775_269_685} A particle \(A\) and a block \(B\) are attached to opposite ends of a light elastic string of natural length 2 m and modulus of elasticity 6 N . The block is at rest on a rough horizontal table. The string passes over a small smooth pulley \(P\) at the edge of the table, with the part \(B P\) of the string horizontal and of length 1.2 m . The frictional force acting on \(B\) is 1.5 N and the system is in equilibrium (see diagram). Find the distance \(P A\).

2
A uniform solid cone has height 30 cm and base radius \(r \mathrm {~cm}\). The cone is placed with its axis vertical on a rough horizontal plane. The plane is slowly tilted and the cone remains in equilibrium until the angle of inclination of the plane reaches \(35 ^ { \circ }\), when the cone topples. The diagram shows a cross-section of the cone.

1. Find the value of \(r\).
2. Show that the coefficient of friction between the cone and the plane is greater than 0.7 .

3 \includegraphics[max width=\textwidth, alt={}, center]{d6cb7a28-e8d7-4239-b9d3-120a284d7353-2_373_759_1119_694} A uniform object \(A B C\) is formed from two rods \(A B\) and \(B C\) joined rigidly at right angles at \(B\). The rod \(A B\) has length 0.3 m and the rod \(B C\) has length 0.2 m . The object rests with the end \(A\) on a rough horizontal surface and the \(\operatorname { rod } A B\) vertical. The object is held in equilibrium by a horizontal force of magnitude 4 N applied at \(B\) and acting in the direction \(C B\) (see diagram).

1. Find the distance of the centre of mass of the object from \(A B\).
2. Calculate the weight of the object.
3. Find the least possible value of the coefficient of friction between the surface and the object.

5 \includegraphics[max width=\textwidth, alt={}, center]{c85aa042-7b8c-44cc-b579-a5deef91e7e5-3_341_529_260_808} A block \(B\) of mass 3 kg is attached to one end of a light elastic string of modulus of elasticity 70 N and natural length 1.4 m . The other end of the string is attached to a particle \(P\) of mass 0.3 kg . \(B\) is at rest 0.9 m from the edge of a horizontal table and the string passes over a small smooth pulley at the edge of the table. \(P\) is released from rest at a point next to the pulley and falls vertically. At the first instant when \(P\) is 0.8 m below the pulley and descending, \(B\) is in limiting equilibrium with the part of the string attached to \(B\) horizontal (see diagram).

1. Calculate the speed of \(P\) when \(B\) is first in limiting equilibrium.
2. Find the coefficient of friction between \(B\) and the table.

2 \includegraphics[max width=\textwidth, alt={}, center]{334b4bdf-6d9c-4208-9032-572eb7c5f9ee-2_295_805_484_671} A uniform solid hemisphere of weight 60 N and radius 0.8 m rests in limiting equilibrium with its curved surface on a rough horizontal plane. The axis of symmetry of the hemisphere is inclined at an angle of \(\theta\) to the horizontal, where \(\cos \theta = 0.28\). Equilibrium is maintained by a horizontal force of magnitude \(P\) N applied to the lowest point of the circular rim of the hemisphere (see diagram).

1. Show that \(P = 8.75\).
2. Find the coefficient of friction between the hemisphere and the plane.

2 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 3 } \mathrm { mg }\). The other end of the string is attached to a fixed point \(O\) on a rough horizontal surface. The particle is at rest on the surface with the string at its natural length. The coefficient of friction between \(P\) and the surface is \(\frac { 1 } { 3 }\). The particle is projected along the surface in the direction \(O P\) with a speed of \(\frac { 1 } { 2 } \sqrt { \mathrm { ga } }\). Find the greatest extension of the string during the subsequent motion.

2 A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is fixed to a point \(O\) on a smooth horizontal surface. A particle \(P\) of mass \(m\) is attached to the other end of the string. The particle \(P\) is projected along the surface in the direction \(O P\). When the length of the string is \(\frac { 5 } { 4 } a\), the speed of \(P\) is \(v\). When the length of the string is \(\frac { 3 } { 2 } a\), the speed of \(P\) is \(\frac { 1 } { 2 } v\).

1. Find an expression for \(v\) in terms of \(a\) and \(g\).
2. Find, in terms of \(g\), the acceleration of \(P\) when the stretched length of the string is \(\frac { 3 } { 2 } a\). \includegraphics[max width=\textwidth, alt={}, center]{5e95e0c9-d47d-4f2b-89da-ab949b9661f4-04_552_1059_264_502} A smooth cylinder is fixed to a rough horizontal surface with its axis of symmetry horizontal. A uniform rod \(A B\), of length \(4 a\) and weight \(W\), rests against the surface of the cylinder. The end \(A\) of the rod is in contact with the horizontal surface. The vertical plane containing the rod \(A B\) is perpendicular to the axis of the cylinder. The point of contact between the rod and the cylinder is \(C\), where \(A C = 3 a\). The angle between the rod and the horizontal surface is \(\theta\) where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram). The coefficient of friction between the rod and the horizontal surface is \(\frac { 6 } { 7 }\). A particle of weight \(k W\) is attached to the rod at \(B\). The rod is about to slip. The normal reaction between the rod and the cylinder is \(N\).

2 A ball of mass 2 kg is projected vertically downwards with speed \(5 \mathrm {~ms} ^ { - 1 }\) through a liquid. At time \(t \mathrm {~s}\) after projection, the velocity of the ball is \(v \mathrm {~ms} ^ { - 1 }\) and its displacement from its starting point is \(x \mathrm {~m}\). The forces acting on the ball are its weight and a resistive force of magnitude \(0.2 v ^ { 2 } \mathrm {~N}\).

1. Find an expression for \(v\) in terms of \(t\).
2. Deduce what happens to \(v\) for large values of \(t\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-06_803_652_251_703} A uniform square lamina of side \(2 a\) and weight \(W\) is suspended from a light inextensible string attached to the midpoint \(E\) of the side \(A B\). 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 \(E P\) is perpendicular to the side \(A B\) 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 \(\tan \theta\). \includegraphics[max width=\textwidth, alt={}, center]{e7091f6c-af72-49f3-b825-cdce9fb2c06f-08_581_576_269_731} A light elastic string has natural length \(8 a\) and modulus of elasticity \(5 m g\). A particle \(P\) of mass \(m\) is attached to the midpoint of the string. The ends of the string are attached to points \(A\) and \(B\) which are a distance \(12 a\) apart on a smooth horizontal table. The particle \(P\) is held on the table so that \(A P = B P = L\) (see diagram). The particle \(P\) is released from rest. When \(P\) is at the midpoint of \(A B\) it has speed \(\sqrt { 80 a g }\).
   1. Find \(L\) in terms of \(a\).
   2. Find the initial acceleration of \(P\) in terms of \(g\).

6. \begin{figure}[h] \end{figure} A particle of weight 120 N is placed on a fixed rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\).
The particle is held at rest in equilibrium by a horizontal force of magnitude 30 N , which acts in the vertical plane containing the line of greatest slope of the plane through the particle, as shown in Figure 2.

1. Show that the normal reaction between the particle and the plane has magnitude 114 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4878b6c2-0c62-4398-8a8f-913139bc8a14-10_433_774_1464_604} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The horizontal force is removed and replaced by a force of magnitude \(P\) newtons acting up the slope along the line of greatest slope of the plane through the particle, as shown in Figure 3. The particle remains in equilibrium.
2. Find the greatest possible value of \(P\).
3. Find the magnitude and direction of the frictional force acting on the particle when \(P = 30\).

4. \begin{figure}[h] \end{figure} A small parcel of mass 3 kg is held in equilibrium on a rough plane by the action of a horizontal force of magnitude 30 N acting in a vertical plane through a line of greatest slope. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Fig. 3. The parcel is modelled as a particle. The parcel is on the point of moving up the slope.

1. Draw a diagram showing all the forces acting on the parcel.
2. Find the normal reaction on the parcel.
3. Find the coefficient of friction between the parcel and the plane.