Resultant of three coplanar forces

2 \includegraphics[max width=\textwidth, alt={}, center]{cb04a09c-af23-4e9d-b3da-da9e351fe879-2_405_384_550_884} Three coplanar forces of magnitudes \(10 \mathrm {~N} , 10 \mathrm {~N}\) and 6 N act at a point \(P\) in the directions shown in the diagram. \(P Q\) is the bisector of the angle between the two forces of magnitude 10 N .

1. Find the component of the resultant of the three forces
   1. in the direction of \(P Q\),
   2. in the direction perpendicular to \(P Q\).
   3. Find the magnitude of the resultant of the three forces.

2 \includegraphics[max width=\textwidth, alt={}, center]{e060fc3b-ae93-46b5-b476-dcecb14d6d06-2_684_257_1114_945} Coplanar forces of magnitudes \(250 \mathrm {~N} , 100 \mathrm {~N}\) and 300 N act at a point in the directions shown in the diagram. The resultant of the three forces has magnitude \(R \mathrm {~N}\), and acts at an angle \(\alpha ^ { \circ }\) anticlockwise from the force of magnitude 100 N . Find \(R\) and \(\alpha\).
[0pt] [6]

1 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_373_591_260_776} Coplanar forces of magnitudes \(7 \mathrm {~N} , 6 \mathrm {~N}\) and 8 N act at a point in the directions shown in the diagram. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the magnitude and direction of the resultant of the three forces.

2 \includegraphics[max width=\textwidth, alt={}, center]{539be201-7bfc-4ba0-8378-c7aec4473ac7-03_577_691_262_724} Coplanar forces of magnitudes \(12 \mathrm {~N} , 24 \mathrm {~N}\) and 30 N act at a point in the directions shown in the diagram.

1. Find the components of the resultant of the three forces in the \(x\)-direction and in the \(y\)-direction. Component in \(x\)-direction \(\_\_\_\_\) Component in \(y\)-direction. \(\_\_\_\_\)
2. Hence find the direction of the resultant.

3 \includegraphics[max width=\textwidth, alt={}, center]{9a99a969-db40-4d29-bb37-ea7ac15cdc2d-2_476_659_897_742} Coplanar forces of magnitudes \(50 \mathrm {~N} , 40 \mathrm {~N}\) and 30 N act at a point \(O\) in the directions shown in the diagram, where \(\tan \alpha = \frac { 7 } { 24 }\).

1. Find the magnitude and direction of the resultant of the three forces.
2. The force of magnitude 50 N is replaced by a force of magnitude \(P \mathrm {~N}\) acting in the same direction. The resultant of the three forces now acts in the positive \(x\)-direction. Find the value of \(P\).

6 \includegraphics[max width=\textwidth, alt={}, center]{d0fa61eb-f320-427e-8883-de224d293933-4_474_831_269_657} Forces of magnitudes \(P \mathrm {~N}\) and 25 N act at right angles to each other. The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the \(x\)-axis (see diagram). The force of magnitude \(P \mathrm {~N}\) has components - 2.8 N and 9.6 N in the \(x\)-direction and the \(y\)-direction respectively, and makes an angle of \(\alpha ^ { \circ }\) with the negative \(x\)-axis.

1. Find the values of \(P\) and \(R\).
2. Find the value of \(\alpha\), and hence find the components of the force of magnitude 25 N in
   1. the \(x\)-direction,
   2. the \(y\)-direction.
   3. Find the value of \(\theta\).

5 A force of magnitude \(F \mathrm {~N}\) acts in a horizontal plane and has components 27.5 N and - 24 N in the \(x\)-direction and the \(y\)-direction respectively. The force acts at an angle of \(\alpha ^ { \circ }\) below the \(x\)-axis.

1. Find the values of \(F\) and \(\alpha\). A second force, of magnitude 87.6 N , acts in the same plane at \(90 ^ { \circ }\) anticlockwise from the force of magnitude \(F \mathrm {~N}\). The resultant of the two forces has magnitude \(R \mathrm {~N}\) and makes an angle of \(\theta ^ { \circ }\) with the positive \(x\)-axis.
2. Find the values of \(R\) and \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{28562a1b-ec9a-40d2-bbb3-729770688971-2_476_714_744_719} Three coplanar forces of magnitudes \(15 \mathrm {~N} , 12 \mathrm {~N}\) and 12 N act at a point \(A\) in directions as shown in the diagram.

1. Find the component of the resultant of the three forces
   1. in the direction of \(A B\),
   2. perpendicular to \(A B\).
   3. Hence find the magnitude and direction of the resultant of the three forces.

4 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_387_1091_2019_525} Three coplanar forces of magnitudes \(8 \mathrm {~N} , 12 \mathrm {~N}\) and 2 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of the three forces and the resultant are shown in the diagram. Find \(R\) and \(\theta\).

3 \includegraphics[max width=\textwidth, alt={}, center]{221f995d-2ef2-4be8-935b-49f8acf1cbe0-2_317_1104_1050_518} A boat is being pulled along a river by two people. One of the people walks along a path on one side of the river and the other person walks along a path on the opposite side of the river. The first person exerts a horizontal force of 60 N at an angle of \(25 ^ { \circ }\) to the direction of the river. The second person exerts a horizontal force of 50 N at an angle of \(15 ^ { \circ }\) to the direction of the river (see diagram).

1. Find the total force exerted by the two people in the direction of the river.
2. Find the magnitude and direction of the resultant force exerted by the two people.

3 \includegraphics[max width=\textwidth, alt={}, center]{60a41d3b-62a0-40d9-a30d-0560903429af-05_479_647_264_749} Three coplanar forces of magnitudes \(50 \mathrm {~N} , 60 \mathrm {~N}\) and 100 N act at a point. The resultant of the forces has magnitude \(R \mathrm {~N}\). The directions of these forces are shown in the diagram. Find the values of \(R\) and \(\alpha\).

5. A particle is acted upon by two forces \(\mathbf { F }\) and \(\mathbf { G }\). The force \(\mathbf { F }\) has magnitude 8 N and acts in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find

1. the magnitude of \(\mathbf { R }\),
2. the direction of \(\mathbf { R }\), giving your answer as a bearing to the nearest degree. in a direction with a bearing of \(240 ^ { \circ }\). The force \(\mathbf { G }\) has magnitude 10 N and acts due South. Given that \(\mathbf { R } = \mathbf { F } + \mathbf { G }\), find

1. Two forces, \(\mathbf { P }\) and \(\mathbf { Q }\), act on a particle.

• \(\mathbf { P }\) has magnitude 10 N and acts due west
• Q has magnitude 8 N and acts on a bearing of \(330 ^ { \circ }\)

Given that \(\mathbf { F } = \mathbf { P } + \mathbf { Q }\), find the magnitude of \(\mathbf { F }\).

3 Two horizontal forces \(\mathbf { X }\) and \(\mathbf { Y }\) act at a point \(O\) and are at right angles to each other. \(\mathbf { X }\) has magnitude 12 N and acts along a bearing of \(090 ^ { \circ } . \mathbf { Y }\) has magnitude 15 N and acts along a bearing of \(000 ^ { \circ }\).

1. Calculate the magnitude and bearing of the resultant of \(\mathbf { X }\) and \(\mathbf { Y }\).
2. A third force \(\mathbf { E }\) is now applied at \(O\). The three forces \(\mathbf { X } , \mathbf { Y }\) and \(\mathbf { E }\) are in equilibrium. State the magnitude of \(\mathbf { E }\), and give the bearing along which it acts.

1 \includegraphics[max width=\textwidth, alt={}, center]{ae5d1e27-5853-48aa-9046-86ce1c1a154a-2_415_823_264_660} Two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) act at the origin O of rectangular coordinates Oxy (see diagram). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 14 N and 5 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 9 N and 7 N respectively.

1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis.

2 Fig. 2 shows two forces acting at A. The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h] \end{figure}

1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.

1 Fig. 2 shows two forces acting at A . The figure also shows the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) which are respectively horizontal and vertically upwards. The resultant of the two forces is \(\mathbf { F } \mathbf { N }\). \begin{figure}[h] \end{figure}

1. Find \(\mathbf { F }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\), giving your answer correct to three significant figures.
2. Calculate the magnitude of \(\mathbf { F }\) and the angle that \(\mathbf { F }\) makes with the upward vertical.

2 Two forces \(\mathbf { F } _ { 1 } \mathrm {~N}\) and \(\mathbf { F } _ { 2 } \mathrm {~N}\) are given by \(\mathbf { F } _ { 1 } = - 6 \mathbf { i } + 2 \mathbf { j }\) and \(\mathbf { F } _ { 2 } = - 8 \mathbf { i } + \mathbf { j }\).
Show that the magnitude of the resultant of these two forces is \(\sqrt { 205 } \mathrm {~N}\).

2 The diagram shows three forces and the perpendicular unit vectors \(\mathbf { i }\) and \(\mathbf { j }\), which all lie in the same plane. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_415_398_1507_605} \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-2_172_166_1567_1217}

1. Express the resultant of the three forces in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of the resultant force.
3. Draw a diagram to show the direction of the resultant force, and find the angle that it makes with the unit vector \(\mathbf { i }\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 5 N and 7 N respectively.

1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.

The three forces \(\mathbf{F_1}\), \(\mathbf{F_2}\) and \(\mathbf{F_3}\) are acting on a particle. \(\mathbf{F_1} = (25\mathbf{i} + 12\mathbf{j})\) N \(\mathbf{F_2} = (-7\mathbf{i} + 5\mathbf{j})\) N \(\mathbf{F_3} = (15\mathbf{i} - 28\mathbf{j})\) N The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The resultant of these three forces is \(\mathbf{F}\) newtons.

The fourth force, \(\mathbf{F_4}\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf{F_4}\), giving your answer in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1 mark]

\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are \(-5\) N and 7 N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]

\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are -5 N and 7 N respectively.

1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]