Resultant of coplanar 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
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.

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.

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
   1. parallel to the force of magnitude 10 N ,
   2. perpendicular to the force of magnitude 10 N .
   3. 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.
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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.

6. \begin{figure}[h] \end{figure} Two forces \(\mathbf { P }\) and \(\mathbf { Q }\) act on a particle at \(O\). The angle between the lines of action of \(\mathbf { P }\) and \(\mathbf { Q }\) is \(120 ^ { \circ }\) as shown in Figure 4. The force \(\mathbf { P }\) has magnitude 20 N and the force \(\mathbf { Q }\) has magnitude \(X\) newtons. The resultant of \(\mathbf { P }\) and \(\mathbf { Q }\) is the force \(\mathbf { R }\). Given that the magnitude of \(\mathbf { R }\) is \(3 X\) newtons, find, giving your answers to 3 significant figures

1. the value of \(X\),
2. the magnitude of \(( \mathbf { P } - \mathbf { Q } )\).

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
   1. the \(x\)-direction,
   2. the \(y\)-direction.
   3. Find the magnitude and direction of the resultant.

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
   1. the value of \(P\),
   2. the magnitude of the resultant,
   3. the magnitude of the acceleration of the particle.
   4. 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 \(R\).

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.

1 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_305_295_264_868} Two perpendicular forces of magnitudes \(F \mathrm {~N}\) and 8 N act at a point \(O\) (see diagram). Their resultant has magnitude 17 N .

1. Calculate \(F\) and find the angle which the resultant makes with the 8 N force. A third force of magnitude \(E \mathrm {~N}\), acting in the same plane as the two original forces, is now applied at the point \(O\). The three forces of magnitudes \(E N , F N\) and \(8 N\) are in equilibrium.
2. State the value of \(E\) and the angle between the directions of the \(E \mathrm {~N}\) and 8 N forces.

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
   1. the angle between the resultant and the 12 N force,
   2. the magnitude of the resultant.
   3. 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 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.

1 Two horizontal forces of magnitudes 7 N and 15 N act at a point O .
The 15 N force acts an angle of \(\theta ^ { \circ }\) above the positive \(x\)-axis.
The 7 N force acts at an angle of \(70 ^ { \circ }\) below the negative \(x\)-axis (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-2_606_773_402_239} The resultant of the two forces acts only in the positive \(x\)-direction.

1. Calculate the value of \(\theta\).
2. Calculate the magnitude of the resultant of the two forces.

9 Two forces, of magnitudes 2 N and 5 N , act on a particle in the directions shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-07_323_755_548_283}

1. Calculate the magnitude of the resultant force on the particle.
2. Calculate the angle between this resultant force and the force of magnitude 5 N .

7 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-3_343_401_1439_872} The diagram shows two forces of magnitudes 10 N and 15 N acting in a horizontal plane on a particle \(P\).

1. Find the component of the 15 N force which is parallel to the 10 N force.
2. Write down the component of the 15 N force which is perpendicular to the 10 N force.
3. Hence, or otherwise, calculate the magnitude and direction of the resultant force on \(P\).