Resultant of coplanar forces

6
\includegraphics[max width=\textwidth, alt={}, center]{f14ab5b6-f9eb-46a5-b5a6-3d34433313c6-08_615_693_260_726} Three coplanar forces of magnitudes \(10 \mathrm {~N} , 25 \mathrm {~N}\) and 20 N act at a point \(O\) in the directions shown in the diagram.

1. Given that the component of the resultant force in the \(x\)-direction is zero, find \(\alpha\), and hence find the magnitude of the resultant force.
2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces.

2
\includegraphics[max width=\textwidth, alt={}, center]{41e63d05-d109-47dc-80a6-927953e3e607-03_659_655_258_744} Coplanar forces of magnitudes \(34 \mathrm {~N} , 30 \mathrm {~N}\) and 26 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 5 } { 13 }\) and \(\sin \theta = \frac { 8 } { 17 }\), find the magnitude and direction of the resultant of the three forces.

2
\includegraphics[max width=\textwidth, alt={}, center]{4ee2568e-5902-442f-9af1-3663fa1d59c7-03_680_636_255_756} Coplanar forces of magnitudes \(60 \mathrm {~N} , 20 \mathrm {~N} , 16 \mathrm {~N}\) and 14 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force.

2
\includegraphics[max width=\textwidth, alt={}, center]{2af7fd9a-aa78-4d77-aa4e-c01604c8b0ae-04_558_606_276_715} Two forces of magnitudes 20 N and \(F \mathrm {~N}\) act at a point \(P\) in the directions shown in the diagram.

1. Given that the resultant force has no component in the \(y\)-direction, calculate the value of \(F\).
2. Given instead that \(F = 10\), find the magnitude and direction of the resultant force.

3
\includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-2_492_606_1763_772} Forces of magnitudes \(7 \mathrm {~N} , 10 \mathrm {~N}\) and 15 N act on a particle in the directions shown in the diagram.

1. Find the component of the resultant of the three forces
   (a) in the \(x\)-direction,
   (b) in the \(y\)-direction.
2. Hence find the direction of the resultant.
   \includegraphics[max width=\textwidth, alt={}, center]{af19f1e0-4cdf-407b-a0d6-cb0272066c30-3_414_833_267_657} A block of mass 8 kg is at rest on a plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is connected to a vertical wall at the top of the plane by a string. The string is taut and parallel to a line of greatest slope of the plane (see diagram).
3. Given that the tension in the string is 13 N , find the frictional and normal components of the force exerted on the block by the plane. The string is cut; the block remains at rest, but is on the point of slipping down the plane.
4. Find the coefficient of friction between the block and the plane.

1
\includegraphics[max width=\textwidth, alt={}, center]{a4cb105b-55d2-4793-95d2-3d791990a1f6-2_341_929_269_609} Forces of magnitudes 10 N and 8 N act in directions as shown in the diagram.

1. Write down in terms of \(\theta\) the component of the resultant of the two forces
   (a) parallel to the force of magnitude 10 N ,
   (b) perpendicular to the force of magnitude 10 N .
2. The resultant of the two forces has magnitude 8 N . Show that \(\cos \theta = \frac { 5 } { 8 }\).

2
\includegraphics[max width=\textwidth, alt={}, center]{155bc571-80e4-4c93-859f-bb150a109211-2_652_493_1457_826} Coplanar forces of magnitudes \(58 \mathrm {~N} , 31 \mathrm {~N}\) and 26 N act at a point in the directions shown in the diagram. Given that \(\tan \alpha = \frac { 5 } { 12 }\), find the magnitude and direction of the resultant of the three forces.
[0pt] [6]

3
\includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-2_296_735_1685_705} Three horizontal forces of magnitudes \(150 \mathrm {~N} , 100 \mathrm {~N}\) and \(P \mathrm {~N}\) have directions as shown in the diagram. The resultant of the three forces is shown by the broken line in the diagram. This resultant has magnitude 120 N and makes an angle \(75 ^ { \circ }\) with the 150 N force. Find the values of \(P\) and \(\theta\).

6
\includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-08_529_606_260_767} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 F \mathrm {~N} , G \mathrm {~N}\) and 50 N , act at a point \(P\), as shown in the diagram.

1. Given that \(F = 0 , G = 75\) and \(\alpha = 60 ^ { \circ }\), find the magnitude and direction of the resultant force.
2. Given instead that \(G = 0\) and the forces are in equilibrium, find the values of \(F\) and \(\alpha\).

5
\includegraphics[max width=\textwidth, alt={}, center]{98a5537b-d503-4a42-bbfe-0bd221084ee0-06_449_654_260_742} Coplanar forces, of magnitudes \(15 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , act at a point \(B\) on the line \(A B C\) in the directions shown in the diagram.

1. Find the magnitude and direction of the resultant force.
2. The force of magnitude 15 N is now replaced by a force of magnitude \(F \mathrm {~N}\) acting in the same direction. The new resultant force has zero component in the direction \(B C\). Find the value of \(F\), and find also the magnitude and direction of the new resultant force.

3
\includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-3_638_839_269_653} Three coplanar forces of magnitudes \(5 \mathrm {~N} , 8 \mathrm {~N}\) and 8 N act at the origin \(O\) of rectangular coordinate axes. The directions of the forces are as shown in the diagram.

1. Find the component of the resultant of the three forces in
   (a) the \(x\)-direction,
   (b) the \(y\)-direction.
2. Find the magnitude and direction of the resultant.

2
\includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_583_785_676_680} Three horizontal forces of magnitudes \(15 \mathrm {~N} , 11 \mathrm {~N}\) and 13 N act on a particle \(P\) in the directions shown in the diagram. The angles \(\alpha\) and \(\beta\) are such that \(\sin \alpha = 0.28 , \cos \alpha = 0.96 , \sin \beta = 0.8\) and \(\cos \beta = 0.6\).

1. Show that the component, in the \(y\)-direction, of the resultant of the three forces is zero.
2. Find the magnitude of the resultant of the three forces.
3. State the direction of the resultant of the three forces.
   \includegraphics[max width=\textwidth, alt={}, center]{102e108b-2a36-4765-9990-78e2dd4276c0-2_348_711_1804_717} A block \(B\) of mass 0.4 kg and a particle \(P\) of mass 0.3 kg are connected by a light inextensible string. The string passes over a smooth pulley at the edge of a rough horizontal table. \(B\) is in contact with the table and the part of the string between \(B\) and the pulley is horizontal. \(P\) hangs freely below the pulley (see diagram).
4. The system is in limiting equilibrium with the string taut and \(P\) on the point of moving downwards. Find the coefficient of friction between \(B\) and the table.
5. A horizontal force of magnitude \(X \mathrm {~N}\), acting directly away from the pulley, is now applied to \(B\). The system is again in limiting equilibrium with the string taut, and with \(P\) now on the point of moving upwards. Find the value of \(X\).

3
\includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-2_570_679_1512_731} Three horizontal forces act at the point \(O\). One force has magnitude 7 N and acts along the positive \(x\)-axis. The second force has magnitude 9 N and acts along the positive \(y\)-axis. The third force has magnitude 5 N and acts at an angle of \(30 ^ { \circ }\) below the negative \(x\)-axis (see diagram).

1. Find the magnitudes of the components of the 5 N force along the two axes.
2. Calculate the magnitude of the resultant of the three forces. Calculate also the angle the resultant makes with the positive \(x\)-axis.

6 A particle of mass 0.04 kg is acted on by a force of magnitude \(P \mathrm {~N}\) in a direction at an angle \(\alpha\) to the upward vertical.

1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20 ^ { \circ }\) find
   (a) the value of \(P\),
   (b) the magnitude of the resultant,
   (c) the magnitude of the acceleration of the particle.
2. It is given instead that \(P = 0.08\) and \(\alpha = 90 ^ { \circ }\). Find the magnitude and direction of the resultant force on the particle.

2
\includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-2_620_711_543_717} Forces of magnitudes 6.5 N and 2.5 N act at a point in the directions shown. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at right angles to the force of magnitude 2.5 N (see diagram).

1. Show that \(\theta = 22.6 ^ { \circ }\), correct to 3 significant figures.
2. Find the value of \(R\).

2 \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30 ^ { \circ }\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts at an angle \(\theta ^ { \circ }\) to the force of magnitude 8 N , as shown in Fig. 2. Find \(R\) and \(\theta\).

4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.

4 Fig. 4 shows forces of magnitudes 20 N and 16 N inclined at \(60 ^ { \circ }\). \begin{figure}[h] \end{figure}

1. Calculate the component of the resultant of these two forces in the direction of the 20 N force.
2. Calculate the magnitude of the resultant of these two forces. These are the only forces acting on a particle of mass 2 kg .
3. Find the magnitude of the acceleration of the particle and the angle the acceleration makes with the 20 N force.

4
\includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-2_325_481_1699_792} Three horizontal forces of magnitudes \(8 \mathrm {~N} , 15 \mathrm {~N}\) and 20 N act at a point. The 8 N and 15 N forces are at right angles. The 20 N force makes an angle of \(150 ^ { \circ }\) with the 8 N force and an angle of \(120 ^ { \circ }\) with the 15 N force (see diagram).

1. Calculate the components of the resultant force in the directions of the 8 N and 15 N forces.
2. Calculate the magnitude of the resultant force, and the angle it makes with the direction of the 8 N force. The directions in which the three horizontal forces act can be altered.
3. State the greatest and least possible magnitudes of the resultant force.

1 Two perpendicular forces have magnitudes 8 N and 15 N . Calculate the magnitude of the resultant force, and the angle which the resultant makes with the larger force.

3 Two forces of magnitudes 8 N and 12 N act at a point \(O\).

1. Given that the two forces are perpendicular to each other, find
   (a) the angle between the resultant and the 12 N force,
   (b) the magnitude of the resultant.
2. It is given instead that the resultant of the two forces has magnitude \(R \mathrm {~N}\) and acts in a direction perpendicular to the 8 N force (see diagram).
   \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-2_248_388_1877_826}
   (a) Calculate the angle between the resultant and the 12 N force.
   (b) Find \(R\).

4
\includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.

3 Two forces each of magnitude 4 N have a resultant of magnitude 6 N .

1. Calculate the angle between the two 4 N forces. The two given forces of magnitude 4 N act on a particle of mass \(m \mathrm {~kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude 3 N on the particle.
2. Find \(m\), and give the acute angle between the surface and one of the 4 N forces.

3 A car is travelling in a straight line on a horizontal road. A driving force, of magnitude 3000 N , acts in the direction of motion and a resistance force, of magnitude 600 N , opposes the motion of the car. Assume that no other horizontal forces act on the car.

1. Find the magnitude of the resultant force on the car.
2. The mass of the car is 1200 kg . Find the acceleration of the car. \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_38_118_440_159}
   \includegraphics[max width=\textwidth, alt={}, center]{c022c936-72bc-4cf9-8f98-285f12c1d479-07_40_118_529_159}

4 Two forces, acting at a point, have magnitudes of 40 newtons and 70 newtons. The angle between the two forces is \(30 ^ { \circ }\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{cb5001b1-1744-439f-aa35-8dd01bc90421-3_213_531_400_759}

1. Find the magnitude of the resultant of these two forces.
2. Find the angle between the resultant force and the 70 newton force.