2.1 Forces and their interactions — Learn

Prerequisites: multiplying and dividing numbers; substituting numbers into a formula; the units kilogram, metre and newton; the idea of a gravitational store from Chapter 1.

By the end of this section you can: describe a force as an interaction between two objects; sort forces into contact and non-contact forces; represent a force as a vector; calculate weight from mass; and find the resultant of two forces that act in a straight line.

1. A force is a push or a pull

A force is a push or a pull that acts on an object because of an interaction with another object. Forces come in pairs. If you push a wall, the wall pushes back on you.

A force is measured in newtons (N). You measure a force with a newtonmeter, which is a spring balance with a scale.

2. Contact and non-contact forces

Forces are sorted into two groups.

Contact forces act when the objects touch.
Non-contact forces act when the objects are apart.

Contact force	What it does
Friction	Opposes sliding between two surfaces.
Air resistance	Opposes motion through the air.
Tension	A pull along a rope, cable or string.
Normal contact force	A push at right angles to a surface.

Non-contact force	What it does
Gravitational force	Pulls objects towards each other.
Electrostatic force	Acts between electric charges.
Magnetic force	Acts between magnets, or between a magnet and a magnetic material.

3. Forces are vectors

A vector is a quantity that has both a size and a direction. Force is a vector. Mass and speed have a size only, so they are scalars.

We draw a force as an arrow. The length of the arrow shows the size of the force, and the arrow points in the direction of the force.

4. Mass and weight

Mass is the amount of matter in an object. It is measured in kilograms (kg) and does not change when the object moves to another place.

Weight is the force of gravity on an object. It is a force, so it is measured in newtons (N). The weight of an object depends on the gravitational field strength, $g$ , at that place. On the Earth, $g$ is about $9.8\,\text{N/kg}$ .

W = m g

where $W$ is the weight in newtons, $m$ is the mass in kilograms and $g$ is the gravitational field strength in newtons per kilogram.

Weight is measured with a newtonmeter. The weight of an object acts at a single point, called the centre of mass. The weight and the mass of an object are directly proportional, so a graph of weight against mass is a straight line through the origin.

Worked example 1. A pupil of mass 70 kg stands on the ground. Take $g = 9.8\,\text{N/kg}$ . Calculate the weight of the pupil.

\begin{aligned} W &= m g \\ &= 70 \times 9.8 \\ &= 686\,\text{N} \\ &= 690\,\text{N} \quad (2\ \text{significant figures}) \end{aligned}

5. Resultant forces

Several forces can act on the same object. A single force that has the same effect as all the others together is called the resultant force.

When the forces act in a straight line, you can combine them.

Forces in the same direction add together.
Forces in opposite directions subtract.
If the forces are equal and opposite, the resultant force is zero. The forces are balanced.

Worked example 2. A force of 250 N acts to the right and a force of 100 N acts to the left. Calculate the resultant force.

The forces act in opposite directions, so subtract.

250 - 100 = 150\,\text{N}

The resultant force is 150 N to the right.

Worked example 3. Two forces of 35 N and 50 N act in the same direction. Calculate the resultant force.

The forces act in the same direction, so add.

35 + 50 = 85\,\text{N}

Watch out. You can add and subtract forces only when they act along the same straight line. Two forces at an angle do not add in the same way. A force of 3 N and a force of 4 N at right angles give a resultant of 5 N, not 7 N. Use a vector diagram, and see the Higher tier part below.

Try it: forces in a line

Two people pull on a trolley, one to the left and one to the right. The figure lets you set both forces.

Predict first. Do not move the sliders yet. Write down your answers.

If both people pull with 100 N, what is the resultant force?
If one pulls with 150 N and the other with 100 N, what is the resultant force, and in which direction?

Test. Set both forces to 100 N and read the resultant. Then set one force to 150 N and read the resultant again. Finally, make the two forces equal and watch what happens.

Explain. Why is the resultant zero when the two forces are equal? Why does the resultant point towards the larger force when they are not equal? Write your answer in two or three sentences.

Resolving a force into two components

A single force can be split into two components that act at right angles. The two components together have the same effect as the single force.

To find the components of a force $F$ at an angle $\theta$ to the horizontal:

F_{\text{horizontal}} = F \cos\theta \qquad F_{\text{vertical}} = F \sin\theta

Worked example 4. A force of 50 N acts at $30^\circ$ to the horizontal. Resolve the force into a horizontal component and a vertical component.

\begin{aligned} F_{\text{horizontal}} &= 50 \cos 30^\circ = 43.3\,\text{N} \\ F_{\text{vertical}} &= 50 \sin 30^\circ = 25.0\,\text{N} \end{aligned}

Free body diagrams

A free body diagram shows one object on its own, with arrows for every force that acts on it. The arrows show the direction and the relative size of each force.

For an object that is not moving, or is moving at a constant speed, the forces are balanced, so the resultant force is zero. A lift hanging still has its weight pulling down and the tension in the cable pulling up. The two forces are equal and opposite.

Vector diagrams

A vector diagram is a scale drawing of two or more forces. Draw the arrows tip to tail, keeping their directions. The arrow from the start of the first force to the tip of the last force is the resultant. You can use a vector diagram to find the resultant of two forces that are not in a straight line, and to find a missing force when three forces are in equilibrium.

Worked example 5. Two forces of 30 N and 40 N act at right angles. Calculate the magnitude and direction of the resultant force.

\begin{aligned} R &= \sqrt{30^2 + 40^2} = 50\,\text{N} \\ \tan\theta &= \frac{40}{30} \qquad \theta = 53^\circ \end{aligned}

The resultant is 50 N, at $53^\circ$ to the 30 N force.

Progress check

Return to the chapter problem. The bike and rider have a mass of 110 kg. Work out their weight, using $g = 9.8\,\text{N/kg}$ .

You should now be able to

describe a force as an interaction between a pair of objects;
sort forces into contact forces and non-contact forces;
explain why force is a vector, and draw a force as an arrow;
calculate weight from mass, using $W = mg$ ;
find the resultant of two forces acting in a straight line;
resolve a force into two components at right angles, and use a free body diagram or a vector diagram (Higher tier).