Experimental Methods
Physics is an empirical science. Every theory must survive confrontation with experiment, and every experiment must be designed, conducted, and analyzed with rigorous methodology. This skill covers the practical foundations of experimental physics: measurement, uncertainty, dimensional analysis, experimental design, data analysis, and the landmark experiments that shaped our understanding. It also covers Fermi estimation — the physicist's art of getting useful answers from minimal information.
Agent affinity: faraday (pedagogy, Sonnet), curie (department chair, Opus)
Concept IDs: phys-motion-kinematics, phys-wave-properties (measurement context)
Experimental Methods at a Glance
| # | Topic | Key idea | |---|---|---| | 1 | Scientific method | Hypothesis -> prediction -> experiment -> analysis -> revision | | 2 | SI units | Seven base units; all physics quantities derived from them | | 3 | Significant figures | Report precision honestly | | 4 | Uncertainty & error analysis | Quantify how much you trust your result | | 5 | Dimensional analysis | Units constrain possible equations | | 6 | Experimental design | Controls, variables, reproducibility | | 7 | Data analysis | Regression, chi-squared, curve fitting | | 8 | Graphical analysis | Linearize to extract parameters | | 9 | Landmark experiments | The experiments that built physics | | 10 | Fermi estimation | Order-of-magnitude reasoning | | 11 | Lab safety | Practical hazard awareness |
Topic 1 — The Scientific Method in Physics
The cycle:
- Observe a phenomenon.
- Hypothesize a mechanism or relationship.
- Predict quantitative consequences.
- Experiment to test the predictions.
- Analyze data; compare with predictions.
- Revise the hypothesis if predictions fail; confirm (tentatively) if they succeed.
Key discipline. A theory is not proved by experiment — it is merely not yet disproved. A single reproducible disagreement between theory and experiment can overthrow a theory that has survived millions of confirmations. This asymmetry between confirmation and falsification (Popper, 1934) is the logical structure of empirical science.
Reproducibility. An experiment that cannot be independently reproduced is not evidence. The design, procedure, and raw data must be documented in sufficient detail for others to replicate the result.
Topic 2 — The SI Unit System
Seven base units:
| Quantity | Unit | Symbol | |---|---|---| | Length | meter | m | | Mass | kilogram | kg | | Time | second | s | | Electric current | ampere | A | | Temperature | kelvin | K | | Amount of substance | mole | mol | | Luminous intensity | candela | cd |
All other physical quantities are derived from these. Examples: force (N = kg m/s^2), energy (J = kg m^2/s^2), pressure (Pa = kg/(m s^2)), voltage (V = kg m^2/(A s^3)).
SI prefixes: From yocto (10^-24) to yotta (10^24). Common in physics: nano (10^-9), micro (10^-6), milli (10^-3), kilo (10^3), mega (10^6), giga (10^9).
Unit consistency. Every term in a physics equation must have the same dimensions. If your answer comes out in "meters per kilogram," something is wrong. Always check units before and after calculation.
Topic 3 — Significant Figures
Rules:
- All nonzero digits are significant. (4.53 has 3 sig figs.)
- Zeros between nonzero digits are significant. (1.003 has 4 sig figs.)
- Leading zeros are not significant. (0.0045 has 2 sig figs.)
- Trailing zeros after a decimal point are significant. (2.300 has 4 sig figs.)
- Trailing zeros with no decimal point are ambiguous. (1500 could be 2, 3, or 4 sig figs — use scientific notation to be clear: 1.50 * 10^3 has 3 sig figs.)
Arithmetic rules:
- Multiplication/division: Result has the same number of sig figs as the input with the fewest.
- Addition/subtraction: Result has the same number of decimal places as the input with the fewest.
Important caveat. Significant figures are a rough method for tracking precision. For serious experimental work, use explicit uncertainty analysis (Topic 4) instead.
Topic 4 — Uncertainty and Error Analysis
Types of Error
Systematic errors shift all measurements in the same direction. Examples: a miscalibrated instrument, a consistently misread scale, air resistance neglected in a free-fall experiment. Systematic errors are not reduced by repeating measurements — they must be identified and corrected.
Random errors cause measurements to scatter unpredictably around the true value. Examples: timing variations in manual stopwatch use, electronic noise. Random errors are reduced by averaging multiple measurements.
Accuracy vs. precision. Accuracy is how close the measurement is to the true value (affected by systematic errors). Precision is how reproducible the measurement is (affected by random errors). A measurement can be precise but inaccurate (tight cluster far from center), accurate but imprecise (scattered around center), or both or neither.
Reporting Uncertainty
Standard form: x = x_best +/- delta_x (units). Example: L = 1.53 +/- 0.02 m.
The uncertainty delta_x is typically the standard deviation of the mean (for repeated measurements) or the instrument resolution (for single measurements): delta_x = sigma / sqrt(N) for N repeated measurements.
Propagation of Uncertainty
When a result depends on several measured quantities, how does the uncertainty propagate?
For f = f(x, y, z):
delta_f = sqrt((df/dx)^2 (delta_x)^2 + (df/dy)^2 (delta_y)^2 + (df/dz)^2 (delta_z)^2)
Common special cases:
- Addition/subtraction: f = x + y. delta_f = sqrt(delta_x^2 + delta_y^2).
- Multiplication/division: f = xy. delta_f/f = sqrt((delta_x/x)^2 + (delta_y/y)^2).
- Power: f = x^n. delta_f/f = |n| delta_x/x.
Worked example. You measure the period of a pendulum as T = 2.05 +/- 0.03 s and the length as L = 1.04 +/- 0.01 m. Calculate g from T = 2 pi sqrt(L/g) and its uncertainty.
Solution. g = 4 pi^2 L / T^2 = 4 9.87 1.04 / 4.2025 = 9.77 m/s^2.
Fractional uncertainties: delta_L/L = 0.01/1.04 = 0.0096. delta_T/T = 0.03/2.05 = 0.0146. Since g proportional to L / T^2: delta_g/g = sqrt((delta_L/L)^2 + (2 delta_T/T)^2) = sqrt(0.0096^2 + 0.0293^2) = sqrt(0.000092 + 0.000858) = sqrt(0.000950) = 0.0308.
delta_g = 0.0308 * 9.77 = 0.30 m/s^2. Result: g = 9.8 +/- 0.3 m/s^2.
Topic 5 — Dimensional Analysis
Principle. Every valid physics equation must be dimensionally consistent. This provides a powerful constraint on possible relationships between variables.
Uses:
- Checking equations. If the dimensions do not match, the equation is wrong.
- Deriving functional forms. If you know which variables a quantity depends on, dimensional analysis can determine the functional form up to a dimensionless constant.
Buckingham Pi theorem. If a physical quantity depends on n variables involving k independent dimensions, then the relationship can be expressed in terms of (n - k) independent dimensionless groups (Pi groups).
Worked example. Find the period of a simple pendulum using dimensional analysis. The period T might depend on length L, mass m, and gravitational acceleration g.
Solution. Assume T = C L^a m^b g^c, where C is dimensionless.
Dimensions: [T] = [L]^a [M]^b [L/T^2]^c = L^(a+c) M^b T^(-2c).
Matching: For T: 1 = -2c, so c = -1/2. For L: 0 = a + c = a - 1/2, so a = 1/2. For M: 0 = b.
Therefore T = C sqrt(L/g). The mass does not appear. Dimensional analysis cannot determine C, but the full derivation gives C = 2 pi.
When dimensional analysis fails. It cannot determine dimensionless constants, and it cannot discover relationships between quantities with the same dimensions (e.g., distinguishing kinetic and potential energy).
Topic 6 — Experimental Design
Independent variable: The quantity you deliberately vary. Dependent variable: The quantity you measure