Introduction to Physics

Wind, rainbows, the warmth of rubbed hands, the Sun’s light, the beating of a heart. Nature looks endlessly varied. Physics is the bet that underneath it all lie a few simple rules.

Looking for the rules#

Physics is the study of nature and its laws. Its aim is to find the basic rules from which the events around us follow. The payoff is economy: Newton’s law of gravitation together with his laws of motion explains why an apple falls, why the Moon circles the Earth and why the sea rises highest on full-moon nights. Because the rules are basic, they apply everywhere, so physics feeds into engineering, chemistry, biology and medicine.

Nature’s chess game#

Richard Feynman offered a lovely picture. Imagine watching chess games without being told the rules. Watch long enough and you start to guess them: bishops move diagonally, pawns move forward. Knowing the rules isn’t the same as understanding every game, but it is the minimum needed to make any sense of it.

Now suppose a player castles for the first time. Your guessed rules said the king moves one square, yet here it jumps two. You must revise your rules. Physics works exactly like this. Nature plays the game; we watch its moves (experiments and observations) and guess the rules (laws). When a new observation refuses to fit, the old law is declared incomplete or wrong and a better one is found.

Physics and mathematics#

Saying “the gravitational force between two masses is proportional to the product of the masses and inversely proportional to the square of their separation” is clumsy. Writing $F \propto m_1m_2/r^2$ is not. Mathematics states laws compactly, and its tools (algebra, trigonometry, calculus) let us work out consequences. From that one rule, calculus tells us the pull of a whole rod or a whole planet.

So mathematics is the language of physics. But a language is a means, not the message. Fluent grammar helps you express an idea; the idea is what matters. Keep this in mind throughout the course: an equation is a compact way of saying something about nature, and the goal is to understand what it says.

What the world is made of#

Ancient thinkers proposed that matter is built from tiny indivisible particles; the Indian philosopher Kanada (whose name comes from kana, a tiny particle) is one of them. Experiments eventually showed that everyday matter is made of atoms and molecules, and each atom is a small, dense nucleus of protons and neutrons with electrons arranged around it. The numbers of these three particles decide how a material behaves.

The laws we develop for ordinary-sized objects make up classical physics. They don’t always hold for single atoms, nuclei or elementary particles, which need quantum physics. In classical physics a “particle” just means an object small compared with everything else in the problem; a speck of dust containing billions of atoms still counts. Twentieth-century experiments revealed hundreds of elementary particles that can transform into one another: nature turns out to be deeply interconnected.

“The station is 5 away.” Five what? Steps, kilometres, minutes? A number alone says nothing. Physics is quantitative, so every measurement needs an agreed unit.

Number times unit#

To measure a quantity we compare it with a standard amount of the same kind, the unit. If a unit of mass is agreed, and an elephant is 2000 units while a goat is 40, the elephant is 50 times heavier. A measured value therefore always has two parts:

$\text{value} = (\text{how many units}) \times (\text{unit})$, for example $2\ \text{h} = 2 \times \text{hour}$.

Who decides the units?

A unit is useless unless everyone uses the same one. By international treaty, the General Conference on Weights and Measures (CGPM, from its French name) decides the standard units and publishes any changes.

Fundamental and derived quantities#

Not every quantity needs its own independent unit. Once you have a unit of length, a unit of area follows: the area of a square one unit on each side. A unit of length and a unit of time give a unit of speed. So we choose a small set of fundamental (base) quantities that are

• independent of one another, and
• enough to express every other quantity by multiplying and dividing.

Everything else is a derived quantity, with a derived unit. The choice of base quantities is a convention (one could take speed and time as basic, making length derived), and different systems have made different choices. Seven turn out to be enough for all of physics.

The International System (SI)#

Adopted by the CGPM in 1971, the Système International d’Unités (SI) is used across the world. Its seven base quantities and units:

Two more named units cover angles: the radian (rad) for plane angle and the steradian (sr) for solid angle.

Prefixes

Nature spans huge ranges: the separation of two protons in a nucleus is about 10⁻¹⁵ m, the distance to a quasar about 10²⁶ m. Standard prefixes stand for powers of ten:

What makes a good standard?#

The length from elbow to fingertip (the old cubit) fails the first test: it differs from person to person. For over a century the kilogram was a platinum–iridium cylinder kept near Paris; tiny drifts in its mass compared with its copies exposed the weakness of a physical object.

Forget the numbers for a moment and ask only what kind of thing a quantity is. A speed is a length per time. A force is a mass times a length per time squared. That skeleton is its dimensions.

Building quantities from base quantities#

Every physical quantity can be built from the base quantities by multiplying and dividing. Follow force back to its roots:

The power to which each base quantity appears is called the dimension of the quantity in that base. Force has dimension 1 in mass, 1 in length and −2 in time, and zero in every other base quantity.

Only the kind of quantity matters, never its size. Initial velocity, final velocity, change in velocity and average velocity are all simply length/time.

Dimensional formulas#

Use one-letter symbols: M for mass, L for length, T for time, I for electric current; temperature, amount of substance and luminous intensity use K, mol and cd. Square brackets mean “the dimensions of”:

Dimensions look like bookkeeping, but they are a pocket tool: they catch wrong formulas in seconds, convert units between systems, and can even guess the form of a law you haven’t derived.

1. Checking equations#

You can add apples to apples but not apples to kilometres. In physics, you can only add, subtract or equate quantities of the same kind:

Check $v^2 = u^2 + 2ax$, for a particle accelerating uniformly from speed u through distance x:

2. Converting units#

Change the units of mass, length and time, and every derived unit changes with them. If a quantity has dimensions $\mathrm{M^aL^bT^c}$, then

3. Guessing the form of a law#

If you know which quantities something depends on, and suspect the dependence is a product of powers, dimensions can find the powers. What sets the speed v of a wave on a string? Plausibly the tension F and the mass per unit length μ. Try

where k is a dimensionless number. Then

Match the powers: M: a + b = 0; L: a − b = 1; T: −2a = −1. So a = ½, b = −½, and

Limits of the method

• You must already know which quantities are involved.
• It only works for products of powers. It cannot produce $x = ut + \tfrac12at^2$, which is a sum of two terms.
• It never gives dimensionless constants such as 2π or ½.
• You get one equation per base quantity. In mechanics that is three (M, L, T), so at most three unknown powers can be found.

A mountain and a pinhead, a galaxy and a hydrogen atom, the age of the universe and an electron’s orbit. The numbers in physics are so wildly different that the most useful first question is often: roughly how big, to the nearest power of ten?

Powers of ten#

Write any number as $a\times10^b$ with $1 \le a < 10$ and b a whole number. The Sun’s diameter is $1.39\times10^9$ m; a hydrogen atom’s is $1.06\times10^{-10}$ m.

So the Sun’s diameter is of order 10⁹ m and the atom’s of order 10⁻¹⁰ m. The Sun is 9 − (−10) = 19 orders of magnitude larger: ten million million million times.

Order-of-magnitude estimates#

To get a quick, roughly right answer, replace every number by its power of ten and do the arithmetic with exponents. You won’t get the exact value, but you will rarely be off by more than one order of magnitude, which is often all you need to decide whether something is plausible.