Welcome to the Foundation of Physics!

Ever wondered how scientists can confidently say a planet is exactly \(149.6 \text{ million km}\) away, or why your phone knows its battery is at exactly \(14\%\)? It all starts here. Physics is the study of the universe, but without measurement, it’s just storytelling. In this chapter, we will learn the universal "language" of units and how to handle the "fuzziness" (uncertainties) that comes with every single measurement we take.

1. The SI System: Our Universal Language

To make sure a scientist in Tokyo understands a scientist in Rio de Janeiro, we use the SI system (Système International). Think of these as the "alphabet" of physics.

The Base Units

According to the IB syllabus, you must be familiar with these fundamental building blocks:

  • Mass: kilogram (\(kg\))
  • Length: metre (\(m\))
  • Time: second (\(s\))
  • Electric Current: ampere (\(A\))
  • Temperature: kelvin (\(K\))
  • Amount of substance: mole (\(mol\))

Derived Units and Index Notation

Most other units (like Newtons for force or Joules for energy) are "derived" from these base units. In the IB, we use index-form unit strings. Instead of writing \(m/s\), we write \(m s^{-1}\). Instead of \(kg \cdot m/s^2\), we write \(kg m s^{-2}\).
Quick Tip: If a unit is in the denominator (bottom), just give it a negative power!

Non-SI Units You Need to Know

Sometimes, SI units are too big or too small for specific contexts. You are expected to recognize and use these specific non-SI units:

  • Energy: electronvolt (\(eV\))
  • Mass: unified atomic mass unit (\(u\)) or \(eV c^{-2}\)
  • Distance: astronomical unit (\(AU\)), light year (\(ly\)), parsec (\(pc\))
  • Time: hour, day, year
  • Astronomy: solar luminosity

Key Takeaway: Always check your units! If you calculate a speed and get \(kg\), something went wrong. Using units to check your work is called dimensional analysis.

2. Scientific Notation and Estimation

Physics deals with the massive (the mass of the Sun) and the tiny (the mass of an electron). To stay sane, we use Scientific Notation.

Instead of writing \(150,000,000,000 m\), we write \(1.5 \times 10^{11} m\).
Instead of \(0.0000000005 m\), we write \(5 \times 10^{-10} m\).

Orders of Magnitude

An order of magnitude is just the power of ten. If a car's mass is \(10^3 kg\) and a person's mass is \(10^2 kg\), the car is "one order of magnitude" heavier. This is great for estimation—a skill where you make a sensible guess when exact data isn't needed.

Did you know? Estimating to the nearest power of 10 is often called a "Fermi Problem." It’s a great way to check if your final answer on an exam actually makes sense!

3. Significant Figures (SF)

Significant figures tell us how precise a measurement is. If you measure a table with a ruler marked in millimeters, saying the table is \(1.2345678 m\) long is dishonest—you can't actually be that sure!

The Golden Rules:
  • When multiplying or dividing, your answer should have the same number of SF as the value with the least SF used in the calculation.
  • When recording data, keep the number of decimal places consistent with the precision of your instrument.

4. Uncertainties: Managing the "Fuzziness"

In Physics, no measurement is perfect. There is always an uncertainty. We record this as: \(\text{value} \pm \text{uncertainty}\).

Types of Uncertainty

  • Absolute Uncertainty: Has the same units as the measurement (e.g., \(5.0 \pm 0.1 m\)).
  • Fractional Uncertainty: \(\frac{\text{absolute uncertainty}}{\text{measured value}}\).
  • Percentage Uncertainty: \((\text{fractional uncertainty}) \times 100\%\).

Propagating Uncertainties (The Math Part)

When you use uncertain measurements in a formula, the uncertainty "spreads" or propagates. Here are the rules you must master:

1. Addition and Subtraction

When you add or subtract values, add the absolute uncertainties.
Example: If \(A = 10 \pm 1\) and \(B = 5 \pm 1\), then \(A - B = 5 \pm 2\).

2. Multiplication and Division

When you multiply or divide values, add the percentage uncertainties.
Example: If you calculate Area (\(Length \times Width\)), and Length has \(2\%\) uncertainty while Width has \(3\%\), the Area has \(5\%\) uncertainty.

3. Powers

If a value is raised to a power \(n\), multiply the percentage uncertainty by \(n\).
Example: If the radius of a sphere has a \(1\%\) uncertainty, the Volume (\(V \propto r^3\)) will have a \(3\%\) uncertainty (\(1\% \times 3\)).

Common Mistake: Students often try to subtract uncertainties when subtracting values. Never subtract uncertainties! Doing more math always makes you less certain, not more.

5. Accuracy vs. Precision

These two terms are often confused in daily life, but they are very different in the lab:

  • Precision: How close the measurements are to each other (consistency). This is affected by random errors.
  • Accuracy: How close the measurements are to the true value. This is affected by systematic errors (like a scale that isn't zeroed properly).

Analogy: Imagine a dartboard. If all your darts hit the very top-left corner in a tiny cluster, you are precise but inaccurate. If they are scattered all around the bullseye, you are accurate (on average) but imprecise.

Quick Review Box

  • Base Units: \(m, kg, s, A, K, mol\).
  • Index Notation: Use \(m s^{-1}\) instead of \(m/s\).
  • Add/Sub: Add absolute uncertainties.
  • Mult/Div: Add percentage uncertainties.
  • Powers: Multiply percentage uncertainty by the power.

Note: For details on how to show these uncertainties on a graph, see the next chapter: "Graphing and data analysis".