Q&A — Frequently asked questions for Chapter 1

Here are questions that often come up while learning this chapter — with short, clear answers. If you have a question that's not listed, tell us and we'll add it.

How to use: questions are ordered from simple to advanced. If you're interested in a specific topic, jump ahead using the section links.

Foundations — why measure?

1. Why does SI have seven base units, not six or eight?

The choice of seven is pragmatic, not necessary. With effort you could get by with fewer (natural units set all constants to 1, effectively zero units). With seven, each common physical phenomenon (length, mass, time, current, temperature, amount of substance, luminous intensity) gets its own dedicated unit — a good balance between simplicity and applicability.

Historically: three (length, mass, time) initially, then current (electrical applications), temperature (thermodynamics), mole (chemistry), and candela (photometry) were added. They were added because they were independent — they can't be built precisely from just the first three.

2. Why was the kilogram redefined?

Before 2019, the kilogram was defined as the mass of a specific platinum-iridium cylinder (the IPK) kept in Paris. The problem: this cylinder's mass changed over time (surface contamination, wear). This meant the kilogram was officially "shrinking" over time — unacceptable for a base unit.

The solution: redefine the kilogram via Planck's constant h (2019). Now h is exactly 6.62607015 × 10^{-34} J·s, and the kilogram is computed from it (via the Kibble balance as seen in §1.5). Result: the kilogram never changes.

3. If the speed of light c is exactly 299,792,458 m/s, how can it be measured precisely?

It can't be! Since 1983, c is no longer measured — the meter is defined from c. That is:

\[ 1\ \mathrm{m} = \text{the distance light travels in }\frac{1}{299{,}792{,}458}\ \mathrm{s} \]

If tomorrow someone "measures" and claims c = 299,792,459 m/s, that means they measured the meter incorrectly — not that c has changed. This is the key metrology trick: pin down the fundamental constants exactly, then build the units from them.

Significant figures and precision

4. What is the difference between accuracy and precision in one sentence?

Visual example: four darts on a bullseye.

In our pendulum experiment, RULER_BIAS is systematic error (reduces accuracy), and reading noise is random error (reduces precision).

5. Are trailing zeros always significant?

Depends on the decimal point:

Golden rule: if in doubt, use scientific notation — it's always unambiguous.

6. Why is 20°C = 68°F but Δ20°C = Δ36°F?

Because Fahrenheit has both an offset (shift from zero) and a different scale factor:

\[ T_F = \tfrac{9}{5} T_C + 32 \]

For absolute value, both terms apply: 20°C → (9/5)(20) + 32 = 36 + 32 = 68°F

For difference, the offset cancels (it subtracts out): Δ20°C → (9/5)(20) = 36°F

This means: in thermodynamics formulas with ΔT, it doesn't matter whether you use C or F (just the scale factor). But for absolute value (e.g. pV = nRT), you must use Kelvin (which has zero at absolute zero).

Dimensional analysis

7. If the dimensions are right, does that guarantee the formula is right?

No! Dimensional analysis is a necessary condition, not sufficient. Example:

\[ T = 2\pi\sqrt{\frac{\ell}{g}} \quad \text{(correct)} \]

\[ T = 5\pi\sqrt{\frac{\ell}{g}} \quad \text{(dimensionally right, physically wrong)} \]

Both give [T] = s. Dimensions can't distinguish dimensionless factors. To find the coefficient, you must either experiment or solve the full equations of motion (which gives 2π for a pendulum).

Rule: if the dimensions are wrong, the formula is definitely wrong. If the dimensions are right, the formula might be right — but you still have to check.

8. How does a Fermi estimate work?

The method: break the problem into small pieces, estimate each within ± an order of magnitude, then combine.

Classic example: "How many piano tuners in Chicago?"

Actual answer: about 40-50. Off by less than 2×!

The secret: positive and negative errors tend to cancel. With 5–6 steps each ±30% off, the final result is roughly 2× accurate — often enough for physics estimation.

9. When does Kelvin differ from Celsius in physics?

Whenever you use an absolute temperature value, you must use Kelvin, not C. Examples:

Anywhere you have ΔT (temperature difference), you can use C or K — the offset cancels.

Application and choosing units

10. Why does particle physics use eV instead of Joule?

Because at atomic/subatomic scales, energy values in joules are very small:

eV is defined as the energy of an electron in a 1 V potential: 1 eV = 1.602 × 10^{-19} J. For protons, mesons, quarks, ... all values sit at a few GeV or TeV — only eV lets you say this cleanly.

Rule of thumb: in astronomy, use parsecs/light-years. In biochemistry, use kcal/mol. In particle physics, use GeV. Let the scale of the problem drive the unit choice.

11. In Python, is import numpy as np enough, or should I use a units library?

Depends:

Example with pint:

from pint import UnitRegistry
u = UnitRegistry()

v = 72 * u.km / u.hour
v_si = v.to(u.meter / u.second)
print(v_si)  # → 20.0 meter/second

If you make a mistake (v = 72 * u.km), pint throws an error. The cost of this safety: slightly slower (2-3×) and heavier syntax.

12. What's the difference between atm, bar, mmHg, torr, psi?

All are pressures, but from different historical sources:

Unit SI equivalent Context
Pa 1 SI base
atm 101,325 Pa Mean sea-level atmospheric pressure (historical definition)
bar 10^5 Pa Rounded atmosphere (engineering, meteorology)
mmHg (torr) 133.322 Pa Height of a mercury column at 0°C (medicine, vacuum)
psi 6894.76 Pa pound/inch² (US engineering, tire pressure)

In physics calculations, convert everything to Pa and work from there. Only in the final report translate to the desired unit.

Advanced questions

13. Is natural units (ħ = c = 1) really a unit system, or a theoretical trick?

Both. In particle physics, when people write E = m (instead of E = mc²), they mean both are measured in units where c = 1. Similar to measuring distance in light-seconds — the speed of light is no longer multiplied.

Advantage: simpler equations. Maxwell equations in Heaviside-Lorentz are much simpler than in SI.

Disadvantage: you lose dimensional awareness. If you say "this particle has energy 3", it's meaningless until you know the reference unit (GeV? Planck energy?).

In final reports, always convert back to SI (or at least eV).

14. How does one detect systematic error in a measurement?

Three ways:

  1. Repeat with a different instrument. If two different chronometers give different values, one (or both) has systematic error.
  2. Compare with an independent calculation. If you measured g by pendulum (g₁) and free-fall (g₂) and they differ a lot, one has a systematic problem.
  3. Check the shape of the residuals in a fit. If the residuals have a pattern (e.g., all positive in one region), the model or data has systematic bias. If residuals are randomly scattered, only random noise remains.

Method 3 is the most useful in real experiments. In our Hooke experiment (§1.11 problem 15), a non-zero intercept b in the linear fit is a signature of ruler zero-error.

15. If I want to improve measurement precision by 10×, how much does the cost grow?

Usually more than 10×! This is an empirical rule in metrology.

Example:

The cost includes production, maintenance, and calibration control. This is why any atomic physics lab is expensive — and why anyone unsatisfied with millisecond precision has to pay a thousand-fold more.

Practical lesson: before spending on precision, ask: is this precision really needed for the problem? If you have a 5% systematic error, going from 1% to 0.01% precision will make no impact.

Question not listed here?

Previous section: 1.12 Further reading Next chapter: 2.1 Motion Along a Straight Line — 1D kinematics (coming soon)

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