Resultant of coplanar forces

6 \includegraphics[max width=\textwidth, alt={}, center]{9ddae838-2639-4952-bbc0-3944a81e5762-3_401_1224_1315_456} The diagram shows a barge being towed along a canal by a force of 240 N at an angle of \(25 ^ { \circ }\) to its direction of motion. A force, \(F \mathrm {~N}\), perpendicular to the direction of motion, is applied to the barge to keep it moving in the direction shown.

1. Find the magnitude of \(F\).
2. The mass of the barge is 1100 kg and there is a resistance force of 100 N parallel to the direction of motion. Find the acceleration of the barge.

6 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-4_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.

Three coplanar forces of magnitudes \(100\text{ N}\), \(50\text{ N}\) and \(50\text{ N}\) act at a point \(A\), as shown in the diagram. The value of \(\cos \alpha\) is \(\frac{4}{5}\). \includegraphics{figure_1} Find the magnitude of the resultant of the three forces and state its direction. [3]

\includegraphics{figure_6} Three coplanar forces of magnitudes 10 N, 25 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. [4]
2. Given instead that \(\alpha = 45\), find the magnitude and direction of the resultant of the three forces. [5]

\includegraphics{figure_2} Coplanar forces of magnitudes \(60\text{N}\), \(20\text{N}\), \(16\text{N}\) and \(14\text{N}\) act at a point in the directions shown in the diagram. Find the magnitude and direction of the resultant force. [6]

\includegraphics{figure_2} Coplanar forces of magnitudes 16 N, 12 N, 24 N and 8 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]

\includegraphics{figure_2} Three coplanar forces act at a point. The magnitudes of the forces are \(5 \text{ N}\), \(6 \text{ N}\) and \(7 \text{ N}\), and the directions in which the forces act are shown in the diagram. Find the magnitude and direction of the resultant of the three forces. [6]

\includegraphics{figure_3} Forces of magnitudes 7 N, 10 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
   1. in the \(x\)-direction,
   2. in the \(y\)-direction.
   [3]
2. Hence find the direction of the resultant. [2]

\includegraphics{figure_4} Coplanar forces of magnitudes 250 N, 160 N and 370 N act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\includegraphics{figure_4} Coplanar forces of magnitudes \(250 \text{ N}\), \(160 \text{ N}\) and \(370 \text{ N}\) act at a point \(O\) in the directions shown in the diagram, where the angle \(\alpha\) is such that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). Calculate the magnitude of the resultant of the three forces. Calculate also the angle that the resultant makes with the \(x\)-direction. [7]

\includegraphics{figure_1} Four coplanar forces of magnitudes 4 N, 8 N, 12 N and 16 N act at a point. The directions in which the forces act are shown in Fig. 1.

1. Find the magnitude and direction of the resultant of the four forces. [5]

\includegraphics{figure_2} The forces of magnitudes 4 N and 16 N exchange their directions and the forces of magnitudes 8 N and 12 N also exchange their directions (see Fig. 2).

1. State the magnitude and direction of the resultant of the four forces in Fig. 2. [2]

\includegraphics{figure_3} Coplanar forces of magnitudes 8 N, 12 N and 18 N act at a point in the directions shown in the diagram. Find the magnitude and direction of the single additional force acting at the same point which will produce equilibrium. [6]

\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\). [5]
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]

Forces of magnitude 4 N and 6 N act in directions which make an angle of \(40°\) with each other, as shown. Calculate

1. the magnitude of the resultant of the two forces, [4 marks]
2. the angle, in degrees, between the resultant and the 4 N force. [2 marks]

\includegraphics{figure_1}

Briefly define the following terms used in modelling in Mechanics:

1. lamina,
2. uniform rod,
3. smooth surface,
4. particle.

[4 marks]

A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) 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 \(\alpha = 20°\) find
   1. the value of \(P\), [3]
   2. the magnitude of the resultant, [2]
   3. the magnitude of the acceleration of the particle. [2]
2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]

\includegraphics{figure_2} 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 \(R\) N and acts at right angles to the force of magnitude \(2.5\) N (see diagram).

1. Show that \(\theta = 22.6°\), correct to 3 significant figures. [3]
2. Find the value of \(R\). [3]

\includegraphics{figure_2} Three horizontal forces of magnitudes \(15\) N, \(11\) 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. [4]
2. Find the magnitude of the resultant of the three forces. [3]
3. State the direction of the resultant of the three forces. [1]

\includegraphics{figure_1} 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]
2. Hence find the magnitude of this resultant, and the angle the resultant makes with the positive \(x\)-axis. [4]

\includegraphics{figure_2} Three horizontal forces of magnitudes 15 N, 11 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. [4]
2. Find the magnitude of the resultant of the three forces. [3]
3. State the direction of the resultant of the three forces. [1]

\includegraphics{figure_1} Two perpendicular forces have magnitudes \(x\) N and \(3x\) N (see diagram). Their resultant has magnitude \(6\) N.

1. Calculate \(x\). [3]
2. Find the angle the resultant makes with the smaller force. [3]

Two forces each of magnitude \(4\text{ N}\) have a resultant of magnitude \(6\text{ N}\).

1. Calculate the angle between the two \(4\text{ N}\) forces. [4]

The two given forces of magnitude \(4\text{ N}\) act on a particle of mass \(m\text{ kg}\) which remains at rest on a smooth horizontal surface. The surface exerts a force of magnitude \(3\text{ N}\) on the particle.

1. Find \(m\), and give the acute angle between the surface and one of the \(4\text{ N}\) forces. [3]

Three forces act on a particle. The first force has magnitude \(P\text{ N}\) and acts horizontally due east. The second force has magnitude \(5\text{ N}\) and acts horizontally due west. The third force has magnitude \(2P\text{ N}\) and acts vertically upwards. The resultant of these three forces has magnitude \(25\text{ N}\).

1. Calculate \(P\) and the angle between the resultant and the vertical. [7]

The particle has mass \(3\text{ kg}\) and rests on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.15\).

1. Find the acceleration of the particle, and state the direction in which it moves. [5]

\includegraphics{figure_2} Forces of magnitudes 8 N and 5 N act on a particle. The angle between the directions of the two forces is \(30°\), as shown in Fig. 1. The resultant of the two forces has magnitude \(R\) N and acts at an angle \(\theta°\) to the force of magnitude 8 N, as shown in Fig. 2. Find \(R\) and \(\theta\). [7]

Jackie says: "A person's weight on Earth is directly proportional to their mass." Tom says: "A person's weight on Earth is different to their weight on the moon." Only one of the statements below is correct. Identify the correct statement. Tick (✓) one box. [1 mark] Jackie and Tom are both wrong. \(\square\) Jackie is right but Tom is wrong. \(\square\) Jackie is wrong but Tom is right. \(\square\) Jackie and Tom are both right. \(\square\)