The Forces of Nature

Push a door, hang a bag on a hook, feel a gust of wind on your face. Every one of these is a force, and every one of them involves two objects, not one.

Pushes and pulls#

In everyday language a force is a push or a pull. Push a trolley and the force points away from you; pull a drawer and it points toward you. Hold a heavy suitcase and you exert a large force; hold an empty cup and the force is small.

You do not need muscles to exert a force. A strong wind leans on you. A rope holds a hanging lantern exactly as a hand could. A plastic ruler rubbed on dry hair lifts scraps of paper off a desk. A beach ball pushed under water shoots back up: the water is pushing it. Non-living things exert forces all the time.

A book lying on a table, for instance, feels an upward force exerted by the table on the book. Saying just “there is a force on the book” leaves out half the story.

Measuring force: the newton#

The SI unit of force is the newton (N). To get a feel for it: holding a 102 g mass (a small apple) steady near the Earth’s surface takes about 1 N. A precise definition comes from Newton’s second law in the next chapter: 1 N is the force that gives 1 kg an acceleration of 1 m/s².

Force has both size and direction and adds by the triangle rule, so it is a vector. When several forces act on a particle we combine them with vector addition into a single resultant force, exactly as in the chapter on vectors.

Forces come in pairs#

Look again at the examples above. The table pushes the book up, and the book presses down on the table. The rope holds the lantern up, and the lantern stretches the rope downward. Whenever body A exerts a force on body B, body B exerts a force on body A too. This observation is Newton’s third law:

The two forces form an action–reaction pair. Which one you call “action” is arbitrary. The crucial point, explored below, is that they act on different bodies.

Only four kinds of force#

Friction, tension, the push of a spring, the lift on a wing, the force that holds an atom together: the variety looks endless. Yet physicists have found that every force in nature is one of just four fundamental interactions:

1. Gravitational: every mass attracts every other mass.
2. Electromagnetic: forces between electric charges, which secretly account for almost every everyday push and pull.
3. Nuclear (the strong force): holds protons and neutrons together inside nuclei.
4. Weak: responsible for certain radioactive decays.

The rest of this chapter visits each of them in turn.

The force that drops an apple also steers the Moon. It acts between you and the person sitting next to you, too, just far too weakly to notice.

The law of universal gravitation#

Any two bodies attract each other simply because they have mass. For two point masses $m_1$ and $m_2$ a distance $r$ apart, the attraction has magnitude

$G$ is a universal constant, the same for apples, planets and galaxies. Two features matter most: the force grows with each mass, and it falls off as the inverse square of distance. Double the separation and the pull drops to a quarter.

Big bodies: treat spheres as points#

Real objects are not points. To find the pull of one extended body on another, you would have to add up the force of every particle of the first body on every particle of the second, as vectors. Even two bodies of three particles each produce nine forces to add; real bodies contain trillions of trillions.

Treating the bodies as continuous turns that sum into an integral, and for one important case the answer is beautifully simple (we prove it in a later chapter):

Gravity at the Earth’s surface#

The Earth’s pull on objects is called gravity. Model the Earth as a uniform sphere of mass $M \approx 6\times10^{24}$ kg and radius $R \approx 6400$ km. A small body of mass $m$ near the surface feels a force $GMm/R^2$ toward the Earth’s centre, the direction we call “vertically downward.” Group the constants:

The quantity $g$ has the dimensions of acceleration, and in the next chapter we will see that it is exactly the acceleration of any body falling freely near the surface. It is called the acceleration due to gravity. We often round it to 10 m/s² to keep arithmetic simple. So the Earth pulls on a small body of mass $m$ with a force $mg$, straight down. That force is the body’s weight.

When your hand presses a wall, no atom of your hand ever touches an atom of the wall. Electric forces between their charges push back first. Almost every force you feel is electromagnetism in disguise.

Coulomb’s law#

On top of gravity, particles that carry electric charge exert electromagnetic forces on one another. For two charges $q_1$ and $q_2$ at rest relative to the observer, a distance $r$ apart:

Charges of the same sign repel; opposite signs attract. This is the force a rubbed plastic ruler uses to lift paper. Moving charges also feel magnetic forces, which makes the full electromagnetic force richer than Coulomb’s law; we meet that in electromagnetism.

Ordinary matter is built from electrons (charge −1.6 × 10⁻¹⁹ C each), protons (+1.6 × 10⁻¹⁹ C) and uncharged neutrons. Electromagnetic forces bind electrons to nuclei in atoms, bind atoms into molecules, and lie behind almost all of chemistry. Compared with gravity they are unimaginably strong:

Electromagnetism in everyday forces#

(a) Contact forces between surfaces

Put a book on a table. The atoms at the bottom of the book come very close to those at the top of the table, and their charged constituents push on each other strongly. The net result is a measurable force: the table pushes the book up, and (third law) the book pushes the table down.

Contact forces usually act along the common normal (perpendicular) to the touching surfaces, as a push. But they can also have a component along the surfaces. That component is friction, the subject of a later chapter. An idealised surface that can only push perpendicular to itself is called smooth or frictionless.

(b) Tension in a string or rope

In a tug-of-war the rope is pulled from both ends. It stretches a tiny, usually invisible amount and is said to be under tension. A rope holding a hanging bucket is in tension too: the atoms at its lower end pull on the atoms of the bucket’s handle, and the resultant of all those electromagnetic forces holds the bucket up. A taut string always pulls on whatever is tied to its ends.

(c) The spring force

Coil a metal wire and you have a spring. The straight-line distance between its ends is its length; with no forces on it, that length is its natural length $x_0$. Stretch it and it pulls on whatever is attached; compress it and it pushes. As long as the change in length is not too large, the size of the force is proportional to that change:

Why gravity wins at large distances#

If electric forces are so much stronger, why does gravity rule the motion of planets and falling apples? Because large objects are almost perfectly neutral. The attractions and repulsions between the charges of one body and those of another very nearly cancel. Gravitational forces never cancel: every particle attracts every other, so they all add up. A 1 kg block 100 m above the ground feels about 9.8 N of gravity from the Earth and essentially zero electric force, even though both are packed with charged particles.

Inside every atom sits a nucleus that, by the rules of electricity alone, should fly apart. Something stronger holds it together.

A puzzle in the nucleus#

An atom’s nucleus is astonishingly small: about 10⁻⁴⁵ m³ against roughly 10⁻³⁰ m³ for the whole atom, a mere 10⁻¹⁵ of its volume. Yet it holds about 99.98% of the atom’s mass. And the nuclei of ordinary (non-radioactive) elements are stable.

Take the nucleus of helium, called an alpha particle: two protons and two neutrons. The protons repel each other through the Coulomb force, which at such tiny separations is fierce. The neutrons carry no charge, so they are mere spectators in the electrical tug-of-war. Gravity between the four particles is far, far too weak to help. So why doesn’t the alpha particle burst apart?

The nuclear (strong) force#

A third kind of force is at work, separate from gravity and electromagnetism: the nuclear force. It acts between protons and neutrons (in any combination), is mostly attractive, and has a peculiar character:

• At separations of about 10⁻¹⁵ m it is much stronger than the Coulomb repulsion, which is why nuclei hold together.
• Beyond a few times 10⁻¹⁵ m it fades to almost nothing: it is short-ranged.

Because of that short range, nuclear forces only matter when something changes inside a nucleus. Bare nuclei are rare in daily life, so you never feel them directly. But radioactivity, nuclear power (fission) and the energy of the Sun (fusion) all come from them.

The weak force#

A fourth interaction shows up when particles transform into one another. A free neutron can turn into a proton while emitting an electron and an almost massless particle called an antineutrino. This is beta decay. A proton inside a nucleus can likewise become a neutron, emitting a positron and a neutrino (beta-plus decay).

Where classical physics stops#

A remarkably small set of laws (Newton’s laws of motion, his law of gravitation, Maxwell’s electromagnetism, the Lorentz force and the laws of thermodynamics) describes the behaviour of everything from raindrops to planets. Physics built on them is called classical physics. It has two borders:

For most of this course we work inside the classical region: bodies larger than about 10⁻⁹ m moving well below the speed of light. There, nuclear and weak forces have no noticeable effect, and only gravity and electromagnetism matter. Occasionally we will apply classical ideas to subatomic particles; the answers are then approximations, but often illuminating ones. We also assume Newton’s third law holds for every force we use. Near the end of the course we take a first look at quantum physics.