3.03m Equilibrium: sum of resolved forces = 0

3 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-2_597_616_888_762} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes \(6 \mathrm {~N} , 5 \mathrm {~N} , F \mathrm {~N}\) and \(F \mathrm {~N}\) acting in the directions shown. Find the values of \(\alpha\) and \(F\).

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]{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.

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.

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.

4 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-2_396_880_1996_630} A particle \(P\) of weight 21 N is attached to one end of each of two light inextensible strings, \(S _ { 1 }\) and \(S _ { 2 }\), of lengths 0.52 m and 0.25 m respectively. The other end of \(S _ { 1 }\) is attached to a fixed point \(A\), and the other end of \(S _ { 2 }\) is attached to a fixed point \(B\) at the same horizontal level as \(A\). The particle \(P\) hangs in equilibrium at a point 0.2 m below the level of \(A B\) with both strings taut (see diagram). Find the tension in \(S _ { 1 }\) and the tension in \(S _ { 2 }\).

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 \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_291_591_255_776} A particle \(P\) of mass 0.3 kg is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(X\). A horizontal force of magnitude \(F \mathrm {~N}\) is applied to the particle, which is in equilibrium when the string is at an angle \(\alpha\) to the vertical, where \(\tan \alpha = \frac { 8 } { 15 }\) (see diagram). Find the tension in the string and the value of \(F\).

2 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

1 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_346_583_255_781} A small block of weight 5.1 N rests on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 8 } { 17 }\). The block is held in equilibrium by means of a light inextensible string. The string makes an angle \(\beta\) above the line of greatest slope on which the block rests, where \(\sin \beta = \frac { 7 } { 25 }\) (see diagram). Find the tension in the string.

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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3 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_307_857_1695_644} A particle \(P\) of mass 1.05 kg is attached to one end of each of two light inextensible strings, of lengths 2.6 m and 1.25 m . The other ends of the strings are attached to fixed points \(A\) and \(B\), which are at the same horizontal level. \(P\) hangs in equilibrium at a point 1 m below the level of \(A\) and \(B\) (see diagram). Find the tensions in the strings.

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.

2 \includegraphics[max width=\textwidth, alt={}, center]{ffefbc81-402f-4048-8741-23c8bae30d5a-2_385_621_488_762} Small blocks \(A\) and \(B\) are held at rest on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Each is held in equilibrium by a force of magnitude 18 N . The force on \(A\) acts upwards parallel to a line of greatest slope of the plane, and the force on \(B\) acts horizontally in the vertical plane containing a line of greatest slope (see diagram). Find the weight of \(A\) and the weight of \(B\).

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.

2 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).

1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
2. Find the magnitude and direction of the resultant of the three forces.
3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.

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]{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.

5 \includegraphics[max width=\textwidth, alt={}, center]{2a91fb7a-0eaf-4c50-8a2c-4755c0b44c17-3_355_1048_255_552} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(A B\) of length 3 m . Three horizontal forces of magnitudes \(F \mathrm {~N} , 10 \mathrm {~N}\) and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R \mathrm {~N}\) in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\).
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of \(11.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the mass of the bead.

1 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-2_558_529_258_808} Four horizontal forces act at a point \(O\) and are in equilibrium. The magnitudes of the forces are \(F \mathrm {~N}\), \(G \mathrm {~N} , 15 \mathrm {~N}\) and \(F \mathrm {~N}\), and the forces act in directions as shown in the diagram.

1. Show that \(F = 41.0\), correct to 3 significant figures.
2. Find the value of \(G\).

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 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_446_497_258_826} A small ball \(B\) of mass 4 kg is attached to one end of a light inextensible string. A particle \(P\) of mass 3 kg is attached to the other end of the string. The string passes over a fixed smooth pulley. The system is in equilibrium with the string taut and its straight parts vertical. \(B\) is at rest on a rough plane inclined to the horizontal at an angle of \(\alpha\), where \(\cos \alpha = 0.8\) (see diagram). State the tension in the string and find the normal component of the contact force exerted on \(B\) by the plane.

2 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_195_719_1114_712} A ring of mass 0.2 kg is threaded on a fixed rough horizontal rod and a light inextensible string is attached to the ring at an angle \(\alpha\) above the horizontal, where \(\cos \alpha = 0.96\). The ring is in limiting equilibrium with the tension in the string \(T \mathrm {~N}\) (see diagram). Given that the coefficient of friction between the ring and the rod is 0.25 , find the value of \(T\).
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4 \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-2_334_832_1617_660} Three coplanar forces of magnitudes \(F \mathrm {~N} , 2 F \mathrm {~N}\) and 15 N act at a point \(P\), as shown in the diagram. Given that the forces are in equilibrium, find the values of \(F\) and \(\alpha\).