3.03m Equilibrium: sum of resolved forces = 0

6 \includegraphics[max width=\textwidth, alt={}, center]{d3bb6702-231d-42a0-830e-9f844dca78d7-3_387_1095_1724_525} A small smooth ring \(R\), of mass 0.6 kg , is threaded on a light inextensible string of length 100 cm . One end of the string is attached to a fixed point \(A\). A small bead \(B\) of mass 0.4 kg is attached to the other end of the string, and is threaded on a fixed rough horizontal rod which passes through \(A\). The system is in equilibrium with \(B\) at a distance of 80 cm from \(A\) (see diagram).

1. Find the tension in the string.
2. Find the frictional and normal components of the contact force acting on \(B\).
3. Given that the equilibrium is limiting, find the coefficient of friction between the bead and the rod.

5 \includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-4_620_623_255_760} A small block of mass 1.25 kg is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle \(\theta\) is such that \(\cos \theta = 0.28\) and \(\sin \theta = 0.96\). A horizontal frictional force also acts on the block, and the block is in equilibrium.

1. Show that the magnitude of the frictional force is 7.5 N and state the direction of this force.
2. Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. The force of magnitude 6.1 N is now replaced by a force of magnitude 8.6 N acting in the same direction, and the block begins to move.
3. Find the magnitude and direction of the acceleration of the block.

7 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-4_529_481_255_831} A small ring of mass 0.2 kg is threaded on a fixed vertical rod. The end \(A\) of a light inextensible string is attached to the ring. The other end \(C\) of the string is attached to a fixed point of the rod above \(A\). A horizontal force of magnitude 8 N is applied to the point \(B\) of the string, where \(A B = 1.5 \mathrm {~m}\) and \(B C = 2 \mathrm {~m}\). The system is in equilibrium with the string taut and \(A B\) at right angles to \(B C\) (see diagram).

1. Find the tension in the part \(A B\) of the string and the tension in the part \(B C\) of the string. The equilibrium is limiting with the ring on the point of sliding up the rod.
2. Find the coefficient of friction between the ring and the rod.

2 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_465_478_479_836} Three coplanar forces of magnitudes \(F \mathrm {~N} , 12 \mathrm {~N}\) and 15 N are in equilibrium acting at a point \(P\) in the directions shown in the diagram. Find \(\alpha\) and \(F\).

4 \includegraphics[max width=\textwidth, alt={}, center]{fa0e0e0d-b0a6-44e0-8b4f-4923e235c6c6-2_168_711_1612_717} A ring of mass 4 kg is attached to one end of a light string. The ring is threaded on a fixed horizontal rod and the string is pulled at an angle of \(25 ^ { \circ }\) below the horizontal (see diagram). With a tension in the string of \(T \mathrm {~N}\) the ring is in equilibrium.

1. Find, in terms of \(T\), the horizontal and vertical components of the force exerted on the ring by the rod. The coefficient of friction between the ring and the rod is 0.4 .
2. Given that the equilibrium is limiting, find the value of \(T\).

2 \includegraphics[max width=\textwidth, alt={}, center]{918b65cc-617d-4942-8d96-b02eef21e417-2_471_621_870_762} A smooth ring \(R\) of mass 0.16 kg is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\). A horizontal force of magnitude 11.2 N acts on \(R\), in the same vertical plane as \(A\) and \(B\). The ring is in equilibrium. The string is taut with angle \(A R B = 90 ^ { \circ }\), and the part \(A R\) of the string makes an angle of \(\theta ^ { \circ }\) with the horizontal (see diagram). The tension in the string is \(T \mathrm {~N}\).

1. Find two simultaneous equations involving \(T \sin \theta\) and \(T \cos \theta\).
2. Hence find \(T\) and \(\theta\).

6 \includegraphics[max width=\textwidth, alt={}, center]{2c628138-0729-4160-a95c-d6ab0f199cc5-3_639_939_1260_603} A particle \(P\) of mass 0.5 kg lies on a smooth horizontal plane. Horizontal forces of magnitudes \(F \mathrm {~N}\), 2.5 N and 2.6 N act on \(P\). The directions of the forces are as shown in the diagram, where \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \beta = \frac { 7 } { 24 }\).

1. Given that \(P\) is in equilibrium, find the values of \(F\) and \(\tan \theta\).
2. The force of magnitude \(F \mathrm {~N}\) is removed. Find the magnitude and direction of the acceleration with which \(P\) starts to move.

6 \includegraphics[max width=\textwidth, alt={}, center]{ceb367ee-4e12-4cb2-9020-078ea5724d6e-3_703_700_255_721} A small box of mass 40 kg is moved along a rough horizontal floor by three men. Two of the men apply horizontal forces of magnitudes 100 N and 120 N , making angles of \(30 ^ { \circ }\) and \(60 ^ { \circ }\) respectively with the positive \(x\)-direction. The third man applies a horizontal force of magnitude \(F \mathrm {~N}\) making an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-direction (see diagram). The resultant of the three horizontal forces acting on the box is in the positive \(x\)-direction and has magnitude 136 N .

1. Find the values of \(F\) and \(\alpha\).
2. Given that the box is moving with constant speed, state the magnitude of the frictional force acting on the box and hence find the coefficient of friction between the box and the floor.

3 \includegraphics[max width=\textwidth, alt={}, center]{77976dad-c055-45fd-93fe-e37fa8e9ae22-2_520_719_1137_712} \(A\) and \(B\) are fixed points of a vertical wall with \(A\) vertically above \(B\). A particle \(P\) of mass 0.7 kg is attached to \(A\) by a light inextensible string of length \(3 \mathrm {~m} . P\) is also attached to \(B\) by a light inextensible string of length \(2.5 \mathrm {~m} . P\) is maintained in equilibrium at a distance of 2.4 m from the wall by a horizontal force of magnitude 10 N acting on \(P\) (see diagram). Both strings are taut, and the 10 N force acts in the plane \(A P B\) which is perpendicular to the wall. Find the tensions in the strings. [6]

1 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_426_424_258_863} A block \(B\) of mass 7 kg is at rest on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts on \(B\) at an angle of \(15 ^ { \circ }\) to the upward vertical (see diagram).

1. Given that \(B\) is in equilibrium find, in terms of \(X\), the normal component of the force exerted on \(B\) by the ground.
2. The coefficient of friction between \(B\) and the ground is 0.4 . Find the value of \(X\) for which \(B\) is in limiting equilibrium.

3 \includegraphics[max width=\textwidth, alt={}, center]{139371b7-e142-4ed6-bff3-3ca4c32b9c6b-2_657_913_1450_616} A particle \(P\) of weight 1.4 N is attached to one end of a light inextensible string \(S _ { 1 }\) of length 1.5 m , and to one end of another light inextensible string \(S _ { 2 }\) of length 1.3 m . The other end of \(S _ { 1 }\) is attached to a wall at the point 0.9 m vertically above a point \(O\) of the wall. The other end of \(S _ { 2 }\) is attached to the wall at the point 0.5 m vertically below \(O\). The particle is held in equilibrium, at the same horizontal level as \(O\), by a horizontal force of magnitude 2.24 N acting away from the wall and perpendicular to it (see diagram). Find the tensions in the strings.
[0pt] [6]

1 A block \(B\) of mass 2.7 kg is pulled at constant speed along a straight line on a rough horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(\theta\) above the horizontal. The normal component of the contact force acting on \(B\) has magnitude 20 N .

1. Show that \(\sin \theta = 0.28\).
2. Find the work done by the pulling force in moving the block a distance of 5 m .

2 \includegraphics[max width=\textwidth, alt={}, center]{f4f2996b-5382-4b0d-9804-b5f5945946b3-2_636_519_664_813} Three horizontal forces of magnitudes \(F \mathrm {~N} , 63 \mathrm {~N}\) and 25 N act at \(O\), the origin of the \(x\)-axis and \(y\)-axis. The forces are in equilibrium. The force of magnitude \(F \mathrm {~N}\) makes an angle \(\theta\) anticlockwise with the positive \(x\)-axis. The force of magnitude 63 N acts along the negative \(y\)-axis. The force of magnitude 25 N acts at \(\tan ^ { - 1 } 0.75\) clockwise from the negative \(x\)-axis (see diagram). Find the value of \(F\) and the value of \(\tan \theta\).

7 \includegraphics[max width=\textwidth, alt={}, center]{d5f48bef-2518-4abd-b3e1-5e48ce56cf62-4_657_618_255_760} A small ring \(R\) is attached to one end of a light inextensible string of length 70 cm . A fixed rough vertical wire passes through the ring. The other end of the string is attached to a point \(A\) on the wire, vertically above \(R\). A horizontal force of magnitude 5.6 N is applied to the point \(J\) of the string 30 cm from \(A\) and 40 cm from \(R\). The system is in equilibrium with each of the parts \(A J\) and \(J R\) of the string taut and angle \(A J R\) equal to \(90 ^ { \circ }\) (see diagram).

1. Find the tension in the part \(A J\) of the string, and find the tension in the part \(J R\) of the string. The ring \(R\) has mass 0.2 kg and is in limiting equilibrium, on the point of moving up the wire.
2. Show that the coefficient of friction between \(R\) and the wire is 0.341 , correct to 3 significant figures. A particle of mass \(m \mathrm {~kg}\) is attached to \(R\) and \(R\) is now in limiting equilibrium, on the point of moving down the wire.
3. Given that the coefficient of friction is unchanged, find the value of \(m\). {www.cie.org.uk} after the live examination series. }

4 \includegraphics[max width=\textwidth, alt={}, center]{099c81e0-a95a-4f98-801c-32d905ef7c7d-2_446_752_1521_699} Coplanar forces of magnitudes \(50 \mathrm {~N} , 48 \mathrm {~N} , 14 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\tan \alpha = \frac { 7 } { 24 }\), find the values of \(P\) and \(\theta\).

5 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_394_531_260_806} A block of mass 2.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The block is kept in equilibrium by a light string making an angle of \(20 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\), as shown in the diagram. The coefficient of friction between the block and plane is \(\frac { 1 } { 4 }\). The block is in limiting equilibrium and is about to move up the plane. Find the value of \(T\).

3 \includegraphics[max width=\textwidth, alt={}, center]{aaf655c6-47f0-4f17-9a57-58aaf48728df-2_586_611_1171_767} The coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{4941e074-2f93-4a0c-80ba-0ca96a48389e-04_442_584_255_778} Two light inextensible strings are attached to a particle of weight 25 N . The strings pass over two smooth fixed pulleys and have particles of weights \(A \mathrm {~N}\) and \(B \mathrm {~N}\) hanging vertically at their ends. The sloping parts of the strings make angles of \(30 ^ { \circ }\) and \(40 ^ { \circ }\) respectively with the vertical (see diagram). The system is in equilibrium. Find the values of \(A\) and \(B\).

2 \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-03_522_604_262_769} The four coplanar forces shown in the diagram are in equilibrium. Find the values of \(P\) and \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-05_535_616_260_762} The three coplanar forces shown in the diagram have magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and \(P \mathrm {~N}\). Given that the three forces are in equilibrium, find the values of \(\theta\) and \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
[0pt] [6]

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 \(W\).

1 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} A small block of weight 12 N is at rest on a smooth plane inclined at \(40 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a force of magnitude \(P \mathrm {~N}\). Find the value of \(P\) when

1. the force is parallel to the plane as in Fig. 1,
2. the force is horizontal as in Fig. 2.

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 \(a\).

3 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).