Equilibrium of particle under coplanar forces

3 \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-04_586_1003_260_571} Four coplanar forces of magnitudes \(40 \mathrm {~N} , 20 \mathrm {~N} , 50 \mathrm {~N}\) and \(F \mathrm {~N}\) act at a point in the directions shown in the diagram. The four forces are in equilibrium. Find \(F\) and \(\alpha\).

3 \includegraphics[max width=\textwidth, alt={}, center]{e5ee28f2-5876-4149-9a77-18c5792c1bd8-04_442_636_264_758} Coplanar forces of magnitudes \(30 \mathrm {~N} , 15 \mathrm {~N} , 33 \mathrm {~N}\) and \(P \mathrm {~N}\) act at a point in the directions shown in the diagram, where \(\tan \alpha = \frac { 4 } { 3 }\). The system is in equilibrium.

1. Show that \(\left( \frac { 14.4 } { 30 - P } \right) ^ { 2 } + \left( \frac { 28.8 } { P + 30 } \right) ^ { 2 } = 1\).
2. Verify that \(P = 6\) satisfies this equation and find the value of \(\theta\).

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]{e1b91e54-a3ae-436c-a4f7-7095891f7034-04_519_616_260_762} Coplanar forces of magnitudes \(24 \mathrm {~N} , P \mathrm {~N} , 20 \mathrm {~N}\) and 36 N act at a point in the directions shown in the diagram. The system is in equilibrium. Given that \(\sin \alpha = \frac { 3 } { 5 }\), find the values of \(P\) and \(\theta\).

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]

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 \(F\).

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]{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.

3 \includegraphics[max width=\textwidth, alt={}, center]{16640429-198d-4ea9-a2f6-6e2ef6ac1b4a-05_535_616_260_762} The three coplanar forces shown in the diagram have magnitudes \(3 \mathrm {~N} , 2 \mathrm {~N}\) and \(P \mathrm {~N}\). Given that the three forces are in equilibrium, find the values of \(\theta\) and \(P\).

1 \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-03_563_503_262_820} Given that \(\tan \alpha = \frac { 12 } { 5 }\) and \(\tan \theta = \frac { 4 } { 3 }\), show that the coplanar forces shown in the diagram are in equilibrium.

3 \includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-2_479_771_1356_687} Each of three light strings has a particle attached to one of its ends. The other ends of the strings are tied together at a point \(A\). The strings are in equilibrium with two of them passing over fixed smooth horizontal pegs, and with the particles hanging freely. The weights of the particles, and the angles between the sloping parts of the strings and the vertical, are as shown in the diagram. Find the values of \(W _ { 1 }\) and \(W _ { 2 }\).
[0pt] [6]

3 \includegraphics[max width=\textwidth, alt={}, center]{5125fab5-0be5-4904-afdf-93e91b16e773-2_606_843_1731_651} The diagram shows three particles \(A , B\) and \(C\) hanging freely in equilibrium, each being attached to the end of a string. The other ends of the three strings are tied together and are at the point \(X\). The strings carrying \(A\) and \(C\) pass over smooth fixed horizontal pegs \(P _ { 1 }\) and \(P _ { 2 }\) respectively. The weights of \(A , B\) and \(C\) are \(5.5 \mathrm {~N} , 7.3 \mathrm {~N}\) and \(W \mathrm {~N}\) respectively, and the angle \(P _ { 1 } X P _ { 2 }\) is a right angle. Find the angle \(A P _ { 1 } X\) and the value of \(W\).

3 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-2_597_616_888_762} A particle \(P\) is in equilibrium on a smooth horizontal table under the action of four horizontal forces of magnitudes \(6 \mathrm {~N} , 5 \mathrm {~N} , F \mathrm {~N}\) and \(F \mathrm {~N}\) acting in the directions shown. Find the values of \(\alpha\) and \(F\).

6. \begin{figure}[h] \end{figure} A boat is pulled along a river at a constant speed by two ropes.
The banks of the river are parallel and the boat travels horizontally in a straight line, parallel to the riverbanks.

• The tension in the first rope is 500 N acting at an angle of \(40 ^ { \circ }\) to the direction of motion, as shown in Figure 3.
• The tension in the second rope is \(P\) newtons, acting at an angle of \(\alpha ^ { \circ }\) to the direction of motion, also shown in Figure 3.
• The resistance to motion of the boat as it moves through the water is a constant force of magnitude 900 N

The boat is modelled as a particle. The ropes are modelled as being light and lying in a horizontal plane. Use the model to find

1. the value of \(\alpha\)
2. the value of \(P\)

4 A box of mass 2.5 kg is on a smooth horizontal table, as shown in Fig. 4. A light string AB is attached to the table at A and the box at B . AB is at an angle of \(50 ^ { \circ }\) to the vertical. Another light string is attached to the box at C ; this string is inclined at \(15 ^ { \circ }\) above the horizontal and the tension in it is 20 N . The box is in equilibrium. \begin{figure}[h] \end{figure}

1. Calculate the horizontal component of the force exerted on the box by the string at C .
2. Calculate the tension in the string AB .
3. Calculate the normal reaction of the table on the box. The string at C is replaced by one inclined at \(15 ^ { \circ }\) below the horizontal with the same tension of 20 N .
4. Explain why this has no effect on the tension in string AB .

1 Fig. 1 shows four forces in equilibrium. \begin{figure}[h] \end{figure}

1. Find the value of \(P\).
2. Hence find the value of \(Q\).

1 Fig. 1 shows four forces acting at a point. The forces are in equilibrium. \begin{figure}[h] \end{figure} Show that \(P = 14\). Find \(Q\), giving your answer correct to 3 significant figures.

1 Fig. 1 shows four forces acting at a point. The forces are in equilibrium. \begin{figure}[h] \end{figure} Show that \(P = 14\).
Find \(Q\), giving your answer correct to 3 significant figures.

6 Fig. 1 shows four forces in equilibrium. \begin{figure}[h] \end{figure}

1. Find the value of \(P\).
2. Hence find the value of \(Q\).

2 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-02_597_885_676_630} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate

1. \(\alpha\),
2. \(F\).

10 Three forces, of magnitudes \(4 \mathrm {~N} , 6 \mathrm {~N}\) and \(P \mathrm {~N}\), act at a point in the directions shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-7_604_601_306_724} The forces are in equilibrium.

1. Show that \(\theta = 41.4 ^ { \circ }\), correct to 3 significant figures.
2. Hence find the value of \(P\). The force of magnitude 4 N is now removed and the force of magnitude 6 N is replaced by a force of magnitude 3 N acting in the same direction.
3. Find
   1. the magnitude of the resultant of the two remaining forces,
   2. the direction of the resultant of the two remaining forces.

9 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-07_609_894_495_242} A body remains at rest when subjected to the horizontal and vertical forces shown in the diagram.
Determine the value of \(F _ { 1 }\) and the value of \(F _ { 2 }\).

3 The diagram shows three forces which act in the same plane and are in equilibrium. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_419_516_383_761}

1. Find \(F\).
2. Find \(\alpha\).

2 A particle is in equilibrium under the action of four horizontal forces of magnitudes 5 newtons, 8 newtons, \(P\) newtons and \(Q\) newtons, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-2_355_357_1146_852}

1. Show that \(P = 9\).
2. Find the value of \(Q\).

4 A particle, of mass \(m \mathrm {~kg}\), remains in equilibrium under the action of three forces, which act in a vertical plane, as shown in the diagram. The force with magnitude 60 N acts at \(48 ^ { \circ }\) above the horizontal and the force with magnitude 50 N acts at an angle \(\theta\) above the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-08_576_647_548_701}

1. By resolving horizontally, find \(\theta\).
2. Find \(m\).
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-09_2484_1709_223_153}