Common forces: weight, normal reaction, tension and the spring force

The forces of everyday life

Most problems in mechanics use only a handful of forces. Learn them well, and you can draw any situation.

Weight

Weight is the pull of the Earth's gravity on a body: $W = mg$, straight down, with $g = 9.8\ \text{m/s}^2$ at the Earth's surface.

Watch out. Mass and weight are different. Mass, in kilograms, is the amount of matter and the same everywhere. Weight, in newtons, depends on where you are: on the Moon, where $g$ is about $1.6\ \text{m/s}^2$, a $60\ \text{kg}$ person weighs only $96\ \text{N}$.

The normal reaction

A surface pushes on anything pressed against it, at right angles to the surface. This push is the normal reaction $N$ ("normal" means perpendicular).

It is a self-adjusting force: it is just as big as it needs to be. On level ground with nothing else acting, $N = mg$. But pull a box up at an angle and $N$ is less; press down on it and $N$ is more. On a slope, $N = mg\cos\theta$.

Tension

A string, rope or chain pulls along its own length. This pull is tension. A string can only pull, never push. In a light (massless) string, the tension is the same all along it, and a smooth light pulley changes its direction but not its size.

The spring force

Stretch or squash a spring by $x$ and it pulls or pushes back with

The spring constant $k$, in $\text{N/m}$, says how stiff the spring is. The force always tries to bring the spring back to its own length. Try it.

You should find that doubling the load doubles the stretch, and a spring twice as stiff stretches half as much.

A worked example

A pot plant of mass $1.2\ \text{kg}$ hangs from a spring with $k = 150\ \text{N/m}$. Then a child pulls the empty hook of a second spring, the same kind, with $8.4\ \text{N}$.

1. The plant. At rest, $kx = mg$: $x = \dfrac{1.2\ \text{kg} \times 9.8\ \text{m/s}^2}{150\ \text{N/m}} = 0.0784\ \text{m} = 7.84\ \text{cm}$.
2. The child. $x = \dfrac{8.4\ \text{N}}{150\ \text{N/m}} = 0.056\ \text{m} = 5.6\ \text{cm}$.

Think first. A spring does not care what pulls it, a weight or a hand: only the size of the force matters.

Practise with new numbers

Try it in Python

The program checks a table of loads and stretches, to see whether a spring obeys Hooke's law.

Remember these

• Weight $W = mg$, down. Mass stays the same everywhere; weight does not.
• The normal reaction pushes at right angles to a surface and is only as big as it needs to be.
• Tension pulls along a string, the same all along a light string.
• Hooke's law: $F = kx$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A $70\ \text{kg}$ astronaut is on Mars, where $g = 3.7\ \text{m/s}^2$. What are his mass and weight?

2. You press down on a $2.0\ \text{kg}$ book on a table with $15\ \text{N}$. What is the normal reaction? ($g = 9.8\ \text{m/s}^2$.)

3. A spring of $k = 80\ \text{N/m}$ is squashed by $5.0\ \text{cm}$. How hard does it push back?

4. Can a rope push a box? Why not?