Experimental Physics, Error Analysis and Estimation — NSEP and INPhO

Weightage: Physics olympiads test whether you can reason about the real world, not only solve ideal problems. Estimation and dimensional analysis appear in theory papers, and the national physics programme includes an experimental component at its later stages, so check the current HBCSE notice for the format. Error analysis is the common skill under both.

1. Dimensional analysis

Every physical equation must be dimensionally consistent, and that fact can derive relationships up to a constant.

Worked example. Guess the period $T$ of a pendulum from $L$ (length), $m$ (mass) and $g$. Write $T=k{L}^a{m}^b{g}^c$. Matching dimensions ($T$: $\text{s}$; $L$: $\text{m}$; $g$: ${\text{m s}}^{-2}$) forces $b=0$ because only $m$ has mass dimensions, then $a+c=0$ and $-2c=1$. So $c=-\frac{1}{2}$, $a=\frac{1}{2}$, giving $T=k\sqrt{L/g}$. The constant $k=2\pi$ needs the full theory, but the dependence did not.

Also use dimensions to check a result: if an answer for speed is in ${\text{m}}^2/\text{s}$, it is wrong. Dimensionless ratios, such as the Reynolds number, show when different situations behave alike.

2. Estimation

An order-of-magnitude estimate needs a model, numbers you know and arithmetic in powers of ten.

Worked example. The mass of the atmosphere. The pressure at the surface, $10^5$ Pa, equals the weight of the air column per unit area, so the total weight is $P\times 4\pi R^2$ with $R=6.4\times 10^6$ m. Then

$M=\frac{4\pi R^{2}P}{g}=\frac{4\pi (6.4\times 10^6{)}^2(10^5)}{9.8}\approx 5\times 10^{18}\text{kg}.$

This matches the accepted value to within a few percent.

Skills: know constants to one significant figure (Earth radius $6\times 10^6$ m, $c=3\times 10^8$ m/s, $G=6.7\times 10^{-11}$ in SI), split a problem into factors you can estimate, and state the assumptions so the estimate can be judged. Check the answer against a limit or a known number.

3. Significant figures and units

Report a result to the precision justified by its uncertainty, with the unit. In multiplication and division, the result has as many significant figures as the least precise factor. Convert to SI before calculating, and never round until the end.

4. Errors and their propagation

Every measurement has an uncertainty. Random errors scatter and can be reduced by repeating and averaging, while systematic errors shift all readings the same way and need a better method or a calibration.

For a set of repeated readings, the mean is the best estimate, and the standard error of the mean falls as $1/\sqrt{N}$.

Propagation. For independent uncertainties:

• Sum or difference $Q=A\pm B$: add absolute errors in quadrature, $\delta Q=\sqrt{(\delta A{)}^2+(\delta B{)}^2}$.
• Product or quotient $Q=A^p{B}^q$: add relative errors in quadrature, $\frac{\delta Q}{Q}=\sqrt{p^2{\left(\frac{\delta A}{A}\right)}^2+q^2{\left(\frac{\delta B}{B}\right)}^2}$.

A simpler, conservative rule adds the errors linearly instead of in quadrature, which is how many school problems are marked.

Worked example. Measure $g$ from $T=2\pi \sqrt{L/g}$, so $g=4{\pi }^{2}L/T^2$. With $L=1.00\pm 0.01$ m and $T=2.01\pm 0.02$ s, $g=39.48/4.040=9.77$ m/s². The relative errors are $0.010$ for $L$ and $2\times 0.00995$ for $T^2$, so $\frac{\delta g}{g}=\sqrt{0.010^2+0.0199^2}=0.022$, and $g=9.77\pm 0.22$ m/s². The error in $T$ dominates because it is squared.

The lesson: invest your effort in reducing the error of the quantity with the largest contribution, such as timing many oscillations rather than one.

5. Graphs and straight-line fits

Turn a relation into a straight line so that it can be tested and its constant found. For $T^2=\frac{4{\pi }^2}{g}L$, plot $T^2$ against $L$, and the gradient is $4{\pi }^2/g$. For an exponential decay $N=N_0{e}^{-\lambda t}$, plot $\ln N$ against $t$, and the gradient is $-\lambda$. For a power law $y=k{x}^n$, plot $\log y$ against $\log x$, and the gradient is $n$.

Good practice:

• Label axes with quantity and unit, and choose a scale that uses most of the sheet.
• Plot error bars if the uncertainties are known.
• Draw the best-fit line through the points, balanced on both sides, and also the steepest and shallowest lines consistent with the error bars. The spread of their gradients gives the uncertainty of the gradient.
• Read the gradient with a large triangle, using points on the line and not data points.
• Check whether the intercept should be zero, since a non-zero intercept can reveal a systematic error.

6. Instruments and common experiments

• Vernier calliper: least count $=1$ main scale division $-$ $1$ vernier division, and check for zero error.
• Screw gauge: least count $=$ pitch divided by the number of circular divisions. Correct for zero error and avoid backlash.
• Simple pendulum: time at least 20 oscillations, use small angles, and measure length to the centre of the bob.
• Resistance by a metre bridge or an ammeter-voltmeter method, with attention to contact and meter resistances.
• Focal length of a lens by the displacement or $u$-$v$ method.
• Sonometer, Young's modulus (Searle's apparatus) and surface tension experiments for mechanics.

For each experiment, be ready to say what is measured, what is plotted, which error dominates and what could make the result systematically wrong.

Common traps

• Quoting a result to more digits than the uncertainty supports.
• Adding relative errors for a sum, where absolute errors apply.
• Ignoring the zero error of an instrument.
• Drawing a line through the origin without checking whether it should pass through it.
• Taking an estimate to be exact, with no stated assumptions.

Memory aids

• "Relative errors for products, absolute for sums."
• "Linearise, plot, read the gradient."
• "Model, numbers, powers of ten": estimation.

Summary

Dimensional analysis gives the form of a relation and checks any result, and estimation turns known constants into plausible numbers. Error analysis separates random from systematic errors and shows which measurement dominates the final uncertainty.

Graphs linearise a law so that the gradient gives the constant, and experiments are judged by what is measured, plotted and controlled.

Exam protocol

• Check dimensions on every answer.
• State assumptions in an estimate.
• Propagate errors and identify the dominant one.
• Plot a straight line, with units, error bars and a large gradient triangle.