3.03a Force: vector nature and diagrams

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

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]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

\includegraphics{figure_1} A small box of mass 15 kg rests on a rough horizontal plane. The coefficient of friction between the box and the plane is 0.2. A force of magnitude \(P\) newtons is applied to the box at 50° to the horizontal, as shown in Figure 1. The box is on the point of sliding along the plane. Find the value of \(P\), giving your answer to 2 significant figures. [9]

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

1. the acceleration of the car and trailer, [3]
2. the magnitude of the tension in the towbar. [3]

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

1. find the value of \(F\). [7]

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]

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]

A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2m\), with a particle of mass \(3m\) attached to the circumference of the disc at the point \(P\). The line \(PQ\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(QP\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(PQ\), of the force acting on the body at \(Q\) at the instant when it is released. [You may assume that the moment of inertia of the body about \(L\) is \(15mr^2\).] [6]

A uniform circular disc, of mass \(m\), radius \(a\) and centre \(O\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point \(A\) of a radius of the disc.

1. Find an equation of motion for the disc when the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\). [5]
2. Hence find the period of small oscillations of the disc about its position of stable equilibrium. [2]

When the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\), the force acting on the disc at \(A\) is \(\mathbf{F}\).

1. Find the magnitude of the component of \(\mathbf{F}\) perpendicular to \(AO\). [5]

The points \(P\) and \(Q\) have position vectors \(4\mathbf{i} - 6\mathbf{j} - 12\mathbf{k}\) and \(2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k}\) respectively, relative to a fixed origin \(O\). Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), act along \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\) and \(\overrightarrow{QP}\) respectively, and have magnitudes 7 N, 3 N and \(3\sqrt{10}\) N respectively.

1. Express \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) in vector form. [3]
2. Show that the resultant of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) is \((2\mathbf{i} - 10\mathbf{j} - 16\mathbf{k})\) N. [2]
3. Find a vector equation of the line of action of this resultant, giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are constant vectors and \(\lambda\) is a parameter. [5]

Two points \(A\) and \(B\) lie in a horizontal plane and have coordinates \((-2, 7)\) and \((3, 19)\) respectively. A particle moves in a straight line from \(A\) to \(B\) under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form. [3 marks]

Two forces, \(\mathbf{F}_1 = 3\mathbf{i} + 2\mathbf{j}\) newtons and \(\mathbf{F}_2 = \mathbf{i} - 3\mathbf{j}\) newtons, are added together to find a resultant force, \(\mathbf{R}\) newtons. This vector addition can be represented using a diagram. Identify the diagram below which correctly represents this vector addition. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_14}

In this question use \(g = 9.8\) m s⁻². A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope. The rope passes over a light smooth pulley, as shown in the diagram. The rope between the pulley and the van is horizontal. \includegraphics{figure_17} Initially, the van is at rest and the crate rests on the lower level. The rope is taut. The van moves away from the pulley to lift the crate from the lower level. The van's engine produces a constant driving force of 5000 N. A constant resistance force of magnitude 780 N acts on the van. Assume there is no resistance force acting on the crate.

1. 1. Draw a diagram to show the forces acting on the crate while it is being lifted. [1 mark]
   2. Draw a diagram to show the forces acting on the van while the crate is being lifted. [1 mark]
2. Show that the acceleration of the van is 0.80 m s⁻² [4 marks]
3. Find the tension in the rope. [2 marks]
4. Suggest how the assumption of a constant resistance force could be refined to produce a better model. [1 mark]

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]

A cylindrical object with mass 8 kg rests on two cylindrical bars of equal radius. The lines connecting the centre of each of the bars to the centre of the object make an angle of \(40°\) to the vertical. \includegraphics{figure_2}

1. Draw a diagram showing all the forces acting on the object. Describe each of the forces using words. [2]
2. Calculate the magnitude of the force on each of the bars due to the cylindrical object. [7]

A particle of mass \(m\) is placed on a rough inclined plane. The plane makes an angle \(\theta\) with the horizontal. The coefficient of friction between the particle and the plane is \(\mu\) where \(\mu < \tan \theta\). The particle is released from rest and accelerates down the plane.

1. Draw a fully labelled diagram to show the forces acting on the particle. [1]
2. Find an expression in terms of \(g\), \(\theta\) and \(\mu\) for the acceleration of the particle. [5]
3. Explain what would happen to the particle if \(\mu > \tan \theta\). [1]