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

4 A particle \(P\) of mass 0.8 kg is placed on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\mu\). A force of magnitude 5 N , acting upwards at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\), is applied to \(P\). The particle is on the point of sliding on the table.

1. Find the value of \(\mu\).
2. The magnitude of the force acting on \(P\) is increased to 10 N , with the direction of the force remaining the same. Find the acceleration of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{38ece0f6-1c29-4e7a-9d66-16c3e2b695f9-3_330_572_1037_788} Two identical boxes, each of mass 400 kg , are at rest, with one on top of the other, on horizontal ground. A horizontal force of magnitude \(P\) newtons is applied to the lower box (see diagram). The coefficient of friction between the lower box and the ground is 0.75 and the coefficient of friction between the two boxes is 0.4 .

1. Show that the boxes will remain at rest if \(P \leqslant 6000\). The boxes start to move with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that no sliding takes place between the boxes, show that \(a \leqslant 4\) and deduce the maximum possible value of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_276_570_264_790} A stone slab of mass 320 kg rests in equilibrium on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts upwards on the slab at an angle of \(\theta\) to the vertical, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram).

1. Find, in terms of \(X\), the normal component of the force exerted on the slab by the ground.
2. Given that the coefficient of friction between the slab and the ground is \(\frac { 3 } { 8 }\), find the value of \(X\) for which the slab is about to slip.

2 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-2_701_323_1244_913} A small ring of mass 0.6 kg is threaded on a rough rod which is fixed vertically. The ring is in equilibrium, acted on by a force of magnitude 5 N pulling upwards at \(30 ^ { \circ }\) to the vertical (see diagram).

1. Show that the frictional force acting on the ring has magnitude 1.67 N , correct to 3 significant figures.
2. The ring is on the point of sliding down the rod. Find the coefficient of friction between the ring and the rod.

7 A particle of mass \(m \mathrm {~kg}\) moves up a line of greatest slope of a rough plane inclined at \(21 ^ { \circ }\) to the horizontal. The frictional and normal components of the contact force on the particle have magnitudes \(F \mathrm {~N}\) and \(R \mathrm {~N}\) respectively. The particle passes through the point \(P\) with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 2 s later it reaches its highest point on the plane.

1. Show that \(R = 9.336 m\) and \(F = 1.416 m\), each correct to 4 significant figures.
2. Find the coefficient of friction between the particle and the plane. After the particle reaches its highest point it starts to move down the plane.
3. Find the speed with which the particle returns to \(P\).

2 A block of mass 20 kg is at rest on a plane inclined at \(10 ^ { \circ }\) to the horizontal. A force acts on the block parallel to a line of greatest slope of the plane. The coefficient of friction between the block and the plane is 0.32 . Find the least magnitude of the force necessary to move the block,

1. given that the force acts up the plane,
2. given instead that the force acts down the plane.

4 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-3_335_751_264_696} A particle \(P\) of weight 5 N is attached to one end of each of two light inextensible strings of lengths 30 cm and 40 cm . The other end of the shorter string is attached to a fixed point \(A\) of a rough rod which is fixed horizontally. A small ring \(S\) of weight \(W \mathrm {~N}\) is attached to the other end of the longer string and is threaded on to the rod. The system is in equilibrium with the strings taut and \(A S = 50 \mathrm {~cm}\) (see diagram).

1. By resolving the forces acting on \(P\) in the direction of \(P S\), or otherwise, find the tension in the longer string.
2. Find the magnitude of the frictional force acting on \(S\).
3. Given that the coefficient of friction between \(S\) and the rod is 0.75 , and that \(S\) is in limiting equilibrium, find the value of \(W\).

5 A particle \(P\) of mass 0.6 kg moves upwards along a line of greatest slope of a plane inclined at \(18 ^ { \circ }\) to the horizontal. The deceleration of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the frictional and normal components of the force exerted on \(P\) by the plane. Hence find the coefficient of friction between \(P\) and the plane, correct to 2 significant figures. After \(P\) comes to instantaneous rest it starts to move down the plane with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(a\).

4 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\) and \(\sin \alpha = 0.28\).

1. Given that the normal component of the contact force acting on the particle has magnitude 1.2 N , find the mass of the particle.
2. Given also that the frictional component of the contact force acting on the particle has magnitude 0.4 N , find the deceleration of the particle. The particle comes to rest on reaching the point \(X\).
3. Determine whether the particle remains at \(X\) or whether it starts to move down the plane.

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.

2 A block of mass 6 kg is sliding down a line of greatest slope of a plane inclined at \(8 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 .

1. Find the deceleration of the block.
2. Given that the initial speed of the block is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find how far the block travels.

4 \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-2_608_723_1247_712} A particle \(P\) has weight 10 N and is in limiting equilibrium on a rough horizontal table. The forces shown in the diagram represent the weight of \(P\), an applied force of magnitude 4 N acting on \(P\) in a direction at \(30 ^ { \circ }\) above the horizontal, and the contact force exerted on \(P\) by the table (the resultant of the frictional and normal components) of magnitude \(C \mathrm {~N}\).

1. Find the value of \(C\).
2. Find the coefficient of friction between \(P\) and the table.

3 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_241_535_1247_806} A particle \(P\) of mass 0.5 kg rests on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). A force of magnitude 0.6 N , acting upwards on \(P\) at angle \(\alpha\) from a line of greatest slope of the plane, is just sufficient to prevent \(P\) sliding down the plane (see diagram). Find

1. the normal component of the contact force on \(P\),
2. the frictional component of the contact force on \(P\),
3. the coefficient of friction between \(P\) and the plane.

6 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_255_511_794_817} The diagram shows a particle of mass 0.6 kg on a plane inclined at \(25 ^ { \circ }\) to the horizontal. The particle is acted on by a force of magnitude \(P \mathrm {~N}\) directed up the plane parallel to a line of greatest slope. The coefficient of friction between the particle and the plane is 0.36 . Given that the particle is in equilibrium, find the set of possible values of \(P\).

4 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_365_493_1749_826} A rough plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = 2.4\). A small block of mass 0.6 kg is held at rest on the plane by a horizontal force of magnitude \(P \mathrm {~N}\). This force acts in a vertical plane through a line of greatest slope (see diagram). The coefficient of friction between the block and the plane is 0.4 . The block is on the point of slipping down the plane. By resolving forces parallel to and perpendicular to the inclined plane, or otherwise, find the value of \(P\).
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4 A box of mass 30 kg is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\), acted on by a force of magnitude 40 N . The force acts upwards and parallel to a line of greatest slope of the plane. The box is on the point of slipping up the plane.

1. Find the coefficient of friction between the box and the plane. The force of magnitude 40 N is removed.
2. Determine, giving a reason, whether or not the box remains in equilibrium.

3 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A block of weight 7.5 N is at rest on a plane which is inclined to the horizontal at angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\). The coefficient of friction between the block and the plane is \(\mu\). A force of magnitude 7.2 N acting parallel to a line of greatest slope is applied to the block. When the force acts up the plane (see Fig. 1) the block remains at rest.

1. Show that \(\mu \geqslant \frac { 17 } { 24 }\). When the force acts down the plane (see Fig. 2) the block slides downwards.
2. Show that \(\mu < \frac { 31 } { 24 }\).

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_289_567_1233_788} Particles \(A\) and \(B\), each of mass 0.3 kg , are connected by a light inextensible string. The string passes over a small smooth pulley fixed at the edge of a rough horizontal surface. Particle \(A\) hangs freely and particle \(B\) is held at rest in contact with the surface (see diagram). The coefficient of friction between \(B\) and the surface is 0.7 . Particle \(B\) is released and moves on the surface without reaching the pulley.

1. Find, for the first 0.9 m of \(B\) 's motion,

4 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-2_499_784_1617_685} Blocks \(P\) and \(Q\), of mass \(m \mathrm {~kg}\) 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 \(35 ^ { \circ }\) 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 \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.