3.03n Equilibrium in 2D: particle under forces

3 \includegraphics[max width=\textwidth, alt={}, center]{2a680bda-4ba2-44eb-8592-95b4e1aed263-04_337_661_262_740} A smooth ring \(R\) of mass 0.2 kg is threaded on a light string \(A R B\). The ends of the string are attached to fixed points \(A\) and \(B\) with \(A\) vertically above \(B\). The string is taut and angle \(A B R = 90 ^ { \circ }\). The angle between the part \(A R\) of the string and the vertical is \(60 ^ { \circ }\). The ring is held in equilibrium by a force of magnitude \(X \mathrm {~N}\), acting on the ring in a direction perpendicular to \(A R\) (see diagram). Calculate the tension in the string and the value of \(X\).

3 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-05_518_616_255_767} A particle \(Q\) of mass 0.2 kg is held in equilibrium by two light inextensible strings \(P Q\) and \(Q R . P\) is a fixed point on a vertical wall and \(R\) is a fixed point on a horizontal floor. The angles which strings \(P Q\) and \(Q R\) make with the horizontal are \(60 ^ { \circ }\) and \(30 ^ { \circ }\) respectively (see diagram). Find the tensions in the two strings.

5 \includegraphics[max width=\textwidth, alt={}, center]{19a41291-2692-48f4-86af-bb4930353959-08_645_611_258_767} Four coplanar forces act at a point. The magnitudes of the forces are \(10 \mathrm {~N} , F \mathrm {~N} , G \mathrm {~N}\) and \(2 F \mathrm {~N}\). The directions of the forces are as shown in the diagram.

1. Given that the forces are in equilibrium, find the values of \(F\) and \(G\).
2. Given instead that \(F = 3\), find the value of \(G\) for which the resultant of the forces is perpendicular to the 10 N force.

6 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-08_412_588_260_776} A block of mass 5 kg is held in equilibrium near a vertical wall by two light strings and a horizontal force of magnitude \(X \mathrm {~N}\), as shown in the diagram. The two strings are both inclined at \(60 ^ { \circ }\) to the vertical.

1. Given that \(X = 100\), find the tension in the lower string.
2. Find the least value of \(X\) for which the block remains in equilibrium in the position shown. [4]

2 A particle of mass 8 kg is suspended in equilibrium by two light inextensible strings which make angles of \(60 ^ { \circ }\) and \(45 ^ { \circ }\) above the horizontal.

1. Draw a diagram showing the forces acting on the particle.
2. Find the tensions in the strings.

6 \includegraphics[max width=\textwidth, alt={}, center]{cb2cec83-6f8d-4c13-90a1-03bbf4e4452f-10_451_1315_258_415} The diagram shows a particle of mass 5 kg on a rough horizontal table, and two light inextensible strings attached to it passing over smooth pulleys fixed at the edges of the table. Particles of masses 4 kg and 6 kg hang freely at the ends of the strings. The particle of mass 6 kg is 0.5 m above the ground. The system is in limiting equilibrium.

1. Show that the coefficient of friction between the 5 kg particle and the table is 0.4 .
   The 6 kg particle is now replaced by a particle of mass 8 kg and the system is released from rest.
2. Find the acceleration of the 4 kg particle and the tensions in the strings.
3. In the subsequent motion the 8 kg particle hits the ground and does not rebound. Find the time that elapses after the 8 kg particle hits the ground before the other two particles come to instantaneous rest. (You may assume this occurs before either particle reaches a pulley.)
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

3 \includegraphics[max width=\textwidth, alt={}, center]{4a2bad7c-6720-414c-b336-060afb2255e9-05_610_591_257_778} A ring of mass 4 kg is threaded on a smooth circular rigid wire with centre \(C\). The wire is fixed in a vertical plane and the ring is kept at rest by a light string connected to \(A\), the highest point of the circle. The string makes an angle of \(25 ^ { \circ }\) to the vertical (see diagram). Find the tension in the string and the magnitude of the normal reaction of the wire on the ring.

2 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-03_280_588_264_774} A particle of mass 2.4 kg is held in equilibrium by two light inextensible strings, one of which is attached to point \(A\) and the other attached to point \(B\). The strings make angles of \(35 ^ { \circ }\) and \(40 ^ { \circ }\) with the horizontal (see diagram). Find the tension in each of the two strings.

5 \includegraphics[max width=\textwidth, alt={}, center]{99f20949-471d-4da3-a680-ec24abf6baa5-06_438_463_264_840} A light string \(A B\) is fixed at \(A\) and has a particle of weight 80 N attached at \(B\). A horizontal force of magnitude \(P \mathrm {~N}\) is applied at \(B\) such that the string makes an angle \(\theta ^ { \circ }\) to the vertical (see diagram).

1. It is given that \(P = 32\) and the system is in equilibrium. Find the tension in the string and the value of \(\theta\).
2. It is given instead that the tension in the string is 120 N and that the particle attached at \(B\) still has weight 80 N . Find the value of \(P\) and the value of \(\theta\).

4 \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_389_1134_258_468} The diagram shows two particles, \(A\) and \(B\), of masses 0.2 kg and 0.1 kg respectively. The particles are suspended below a horizontal ceiling by two strings, \(A P\) and \(B Q\), attached to fixed points \(P\) and \(Q\) on the ceiling. The particles are connected by a horizontal string, \(A B\). Angle \(A P Q = 45 ^ { \circ }\) and \(B Q P = \theta ^ { \circ }\). Each string is light and inextensible. The particles are in equilibrium.

1. Find the value of the tension in the string \(A B\). \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-06_2715_44_110_2006} \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-07_2721_34_101_20}
2. Find the value of \(\theta\) and the tension in the string \(B Q\).

1 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_200_588_267_781} A ring of mass 1.1 kg 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 magnitude 13 N at an angle \(\alpha\) below the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) (see diagram). The ring is in equilibrium.

1. Find the frictional component of the contact force on the ring.
2. Find the normal component of the contact force on the ring.
3. Given that the equilibrium of the ring is limiting, find the coefficient of friction between the ring and the rod.

3 \includegraphics[max width=\textwidth, alt={}, center]{b5873699-d207-4cad-9518-1321dc429c15-2_508_1011_1096_568} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of horizontal forces of magnitudes \(F\) N, \(F\) N, \(G\) N and 12 N acting in the directions shown. Find the values of \(F\) and \(G\). [6]

7 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_414_865_1512_641} Two light strings are attached to a block of mass 20 kg . The block is in equilibrium on a horizontal surface \(A B\) with the strings taut. The strings make angles of \(60 ^ { \circ }\) and \(30 ^ { \circ }\) with the horizontal, on either side of the block, and the tensions in the strings are \(T \mathrm {~N}\) and 75 N respectively (see diagram).

1. Given that the surface is smooth, find the value of \(T\) and the magnitude of the contact force acting on the block.
2. It is given instead that the surface is rough and that the block is on the point of slipping. The frictional force on the block has magnitude 25 N and acts towards \(A\). Find the coefficient of friction between the block and the surface.

3 \includegraphics[max width=\textwidth, alt={}, center]{ee138c3f-51e1-4a69-9750-9eb49ac87e22-2_520_565_1009_792} Three horizontal forces of magnitudes \(F \mathrm {~N} , 13 \mathrm {~N}\) and 10 N act at a fixed point \(O\) and are in equilibrium. The directions of the forces are as shown in the diagram. Find, in either order, the value of \(\theta\) and the value of \(F\).

7 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-4_246_665_253_739} Two rectangular boxes \(A\) and \(B\) are of identical size. The boxes are at rest on a rough horizontal floor with \(A\) on top of \(B\). Box \(A\) has mass 200 kg and box \(B\) has mass 250 kg . A horizontal force of magnitude \(P\) N is applied to \(B\) (see diagram). The boxes remain at rest if \(P \leqslant 3150\) and start to move if \(P > 3150\).

1. Find the coefficient of friction between \(B\) and the floor. The coefficient of friction between the two boxes is 0.2 . Given that \(P > 3150\) and that no sliding takes place between the boxes,
2. show that the acceleration of the boxes is not greater than \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. find the maximum possible value of \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{d5acfe31-8614-4508-ac5b-865e15a1f539-2_661_565_1069_790} A small smooth ring \(R\) of weight 8.5 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\), with \(A\) vertically above \(B\). A horizontal force of magnitude 15.5 N acts on \(R\) so that the ring is in equilibrium with angle \(A R B = 90 ^ { \circ }\). The part \(A R\) of the string makes an angle \(\theta\) with the horizontal and the part \(B R\) makes an angle \(\theta\) with the vertical (see diagram). The tension in the string is \(T \mathrm {~N}\). Show that \(T \sin \theta = 12\) and \(T \cos \theta = 3.5\) and hence find \(\theta\).

4
The three coplanar forces shown in the diagram act at a point \(P\) and are in equilibrium.

1. Find the values of \(F\) and \(\theta\).
2. State the magnitude and direction of the resultant force at \(P\) when the force of magnitude 12 N is removed.

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.

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]