3.03p Resultant forces: using vectors

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{9ac08732-e825-473a-943c-8ad8e9e0bc17-08_572_572_262_790} Coplanar forces, of magnitudes \(F \mathrm {~N} , 3 \mathrm {~N} , 6 \mathrm {~N}\) and 4 N , act at a point \(P\), as shown in the diagram.

1. Given that \(\alpha = 60\), and that the resultant of the four forces is in the direction of the 3 N force, find \(F\).
2. Given instead that the four forces are in equilibrium, find the values of \(F\) and \(\alpha\).

3 \includegraphics[max width=\textwidth, alt={}, center]{430f1f9a-7a3a-47a0-b742-daf74e68adfd-2_368_584_1302_794} Two forces, each of magnitude 10 N , act at a point \(O\) in the directions of \(O A\) and \(O B\), as shown in the diagram. The angle between the forces is \(\theta\). The resultant of these two forces has magnitude 12 N .

1. Find \(\theta\).
2. Find the component of the resultant force in the direction of \(O A\).

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 \(\alpha\).
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2 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_549_589_934_778} Two forces, each of magnitude 8 N , act at a point in the directions \(O A\) and \(O B\). The angle between the forces is \(\theta ^ { \circ }\) (see diagram). The resultant of the two forces has component 9 N in the direction \(O A\). Find

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

1 \includegraphics[max width=\textwidth, alt={}, center]{dafc271d-a77b-4401-9170-e13e484d6e5f-2_582_751_255_696} Three coplanar forces act at a point. The magnitudes of the forces are \(5.5 \mathrm {~N} , 6.8 \mathrm {~N}\) and 7.3 N , and the directions in which the forces act are as shown in the diagram. Given that the resultant of the three forces is in the same direction as the force of magnitude 6.8 N , find the value of \(\alpha\) and the magnitude of the resultant.

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.

2 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_318_632_482_753} Forces of magnitudes 13 N and 14 N act at a point \(O\) in the directions shown in the diagram. The resultant of these forces has magnitude 15 N . Find

1. the value of \(\theta\),
2. the component of the resultant in the direction of the force of magnitude 14 N .

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 \(P\).

3 \includegraphics[max width=\textwidth, alt={}, center]{a9f3480e-7a8a-497d-a26a-b2aba9b05512-2_462_721_1672_712} Two forces have magnitudes \(P \mathrm {~N}\) and \(Q \mathrm {~N}\). The resultant of the two forces has magnitude 12 N and acts in a direction \(40 ^ { \circ }\) clockwise from the force of magnitude \(P \mathrm {~N}\) and \(80 ^ { \circ }\) anticlockwise from the force of magnitude \(Q \mathrm {~N}\) (see diagram). Find the value of \(Q\).

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.

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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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 \(\theta\).

4 \includegraphics[max width=\textwidth, alt={}, center]{9fbb63e3-4017-461e-9110-500be2c20778-2_583_862_1343_644} Three coplanar forces of magnitudes \(68 \mathrm {~N} , 75 \mathrm {~N}\) and 100 N act at an origin \(O\), as shown in the diagram. The components of the three forces in the positive \(x\)-direction are \(- 60 \mathrm {~N} , 0 \mathrm {~N}\) and 96 N , respectively. Find

1. the components of the three forces in the positive \(y\)-direction,
2. the magnitude and direction of the resultant of the three forces. \(5 A , B\) and \(C\) are three points on a line of greatest slope of a plane which is inclined at \(\theta ^ { \circ }\) to the horizontal, with \(A\) higher than \(B\) and \(B\) higher than \(C\). Between \(A\) and \(B\) the plane is smooth, and between \(B\) and \(C\) the plane is rough. A particle \(P\) is released from rest on the plane at \(A\) and slides down the line \(A B C\). At time 0.8 s after leaving \(A\), the particle passes through \(B\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 \includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-2_666_953_662_596} Three coplanar forces act at a point. The magnitudes of the forces are \(20 \mathrm {~N} , 25 \mathrm {~N}\) and 30 N , and the directions in which the forces act are as shown in the diagram, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\), and \(\sin \beta = 0.6\) and \(\cos \beta = 0.8\).

1. Show that the resultant of the three forces has a zero component in the \(x\)-direction.
2. Find the magnitude and direction of the resultant of the three forces.
3. The force of magnitude 20 N is replaced by another force. The effect is that the resultant force is unchanged in magnitude but reversed in direction. State the magnitude and direction of the replacement force.

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 \(\theta\).

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.

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]{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 \(\alpha\).