3.03p Resultant forces: using vectors

[In this question, \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors.] A particle \(A\) of mass 0.5 kg is at rest on a smooth horizontal plane. At time \(t = 0\), two forces, \(\mathbf{F}_1 = (-3\mathbf{i} + 2\mathbf{j})\) N and \(\mathbf{F}_2 = (p\mathbf{i} + q\mathbf{j})\) N, where \(p\) and \(q\) are constants, are applied to \(A\). Given that \(A\) moves in the direction of the vector \((\mathbf{i} - 2\mathbf{j})\),

1. show that \(2p + q - 4 = 0\) [4] Given that \(p = 5\)
2. Find the speed of \(A\) at time \(t = 4\) seconds. [5]

Two forces \(\mathbf{P}\) and \(\mathbf{Q}\) act on a particle. The force \(\mathbf{P}\) has magnitude \(7\) N and acts due north. The resultant of \(\mathbf{P}\) and \(\mathbf{Q}\) is a force of magnitude \(10\) N acting in a direction with bearing \(120°\). Find

1. the magnitude of \(\mathbf{Q}\),
2. the direction of \(\mathbf{Q}\), giving your answer as a bearing.

[9]

A particle is acted upon by two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\), given by \(\mathbf{F}_1 = (\mathbf{i} - 3\mathbf{j})\) N, \(\mathbf{F}_2 = (p\mathbf{i} + 2p\mathbf{j})\) N, where \(p\) is a positive constant.

1. Find the angle between \(\mathbf{F}_2\) and \(\mathbf{j}\). [2]

The resultant of \(\mathbf{F}_1\) and \(\mathbf{F}_2\) is \(\mathbf{R}\). Given that \(\mathbf{R}\) is parallel to \(\mathbf{i}\),

1. find the value of \(p\). [4]

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}

Forces of magnitude \(4\) N, \(5\) N and \(8\) N act on a particle in directions whose bearings are \(000°\), \(090°\) and \(210°\) respectively. Find the magnitude of the resultant force and the bearing of the direction in which it acts. [7 marks] \includegraphics{figure_2}

Two forces, both of magnitude 5 N, act on a particle in the directions with bearings 000° and 070°, as shown. \includegraphics{figure_1} Calculate

1. the magnitude of the resultant force on the particle, [3 marks]
2. the bearing on which this resultant force acts. [2 marks]

Two forces \(\mathbf{F}\) and \(\mathbf{G}\) are given by \(\mathbf{F} = (6\mathbf{i} - 5\mathbf{j})\) N, \(\mathbf{G} = (3\mathbf{i} + 17\mathbf{j})\) N, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the \(x\) and \(y\) directions respectively and the unit of length on each axis is 1 cm.

1. Find the magnitude of \(\mathbf{R}\), the resultant of \(\mathbf{F}\) and \(\mathbf{G}\). [3 marks]
2. Find the angle between the direction of \(\mathbf{R}\) and the positive \(x\)-axis. [2 marks]

\(\mathbf{R}\) acts through the point \(P(-4, 3)\). \(O\) is the origin \((0, 0)\).

1. Use the fact that \(OP\) is perpendicular to the line of action of \(\mathbf{R}\) to calculate the moment of \(\mathbf{R}\) about an axis through the origin and perpendicular to the \(x\)-\(y\) plane. [3 marks]

\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]

A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).

1. Show that \(3a + b = 10\). [3]

It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).

1. Determine the value of \(m\). [4]

Three forces \(\mathbf{L}\), \(\mathbf{M}\) and \(\mathbf{N}\) are given by $$\mathbf{L} = 2\mathbf{i} + 5\mathbf{j},$$ $$\mathbf{M} = 3\mathbf{i} - 22\mathbf{j},$$ $$\mathbf{N} = 4\mathbf{i} - 23\mathbf{j}.$$ Find the magnitude and direction of the resultant of the three forces. [6]

Three coplanar horizontal forces of magnitude \(21\) N, \(11\) N and \(8\) N act on a particle \(P\) in the directions shown in the diagram. \includegraphics{figure_7}

1. Given that \(\tan\alpha = \frac{3}{4}\), calculate the magnitude of the resultant force. [5]
2. Explain why the forces cannot be in equilibrium whatever the value of \(\alpha\). [1]

Two forces, of magnitudes 2 N and 5 N, act on a particle in the directions shown in the diagram below. \includegraphics{figure_9}

1. Calculate the magnitude of the resultant force on the particle. [3]
2. Calculate the angle between this resultant force and the force of magnitude 5 N. [1]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. Distance is measured in metres and time in seconds. A ship of mass 100 000 kg is being towed by two tug boats. • The cables attaching each tug to the ship are horizontal. • One tug produces a force of \((350\mathbf{i} + 400\mathbf{j})\) N. • The other tug produces a force of \((250\mathbf{i} - 400\mathbf{j})\) N. • The total resistance to motion is 200 N. • At the instant when the tugs begin to tow the ship, it is moving east at a speed of 1.5 m s\(^{-1}\).

1. Explain why the ship continues to move directly east. [2]
2. Find the acceleration of the ship. [2]
3. Find the time which the ship takes to move 400 m while it is being towed. Find its speed after moving that distance. [6]