Introduction physics laboratory
Module 1: Cover sheet
Name and matriculation number for the report
Both appear on the PDF report and in the verification view behind the QR code. Enter them before you start with the first module.
One cover sheet, one report. If the name or the matriculation number is changed later, this experiment starts over: the progress of every module is deleted.
Module 2: Record keeping
Structure and content of the documentation in physics laboratory
Script: Ch. 3.1–3.3, pp. 4–6 (Open PDF)
Step 1: Why an elaboration?
Meaning and purpose
The experiment documentation consists of two parts: the Development and that Laboratory report (which is attached as an attachment).
The Development The exercise has to document the entire experiment from the question to the implementation up to the analysis. It must be able to be read and understood by a stranger who is familiar with the matter and, in principle, enable them to repeat the experiment at any time.
The Laboratory report will be against it during experiments handwritten and contains the original measured values and observations.
TipGood laboratory reports ...
- ... make this possible complete reproducibility of the experiment
- ... also document unexpected observations and problems
- ... contain one critical discussion of the results
- ... contain the handwritten one Laboratory report as an attachment
Exercise: purpose of elaboration
What is it? the Requirement for an elaboration?
Step 2: Building an elaboration
The 8 components
A complete elaboration contains the following sections in this order:
- Subject of the experiment: What is the goal? What is being measured? (max. 1 page)
- Theoretical basics: Relevant physical relationships and formulas
- experimental procedure: Description with schematic sketches
- Experimental conditions: Temperature, sources of error, etc.
- Measurement results: Tabulated, numbered, with signatures
- Error calculation: Measurement uncertainty and error estimation
- Result: Commentary and comparison with literature values
- Appendix: laboratory report as an attachment
Exercise: Assign sections of the elaboration
Match each description to the correct section of the paper:
1. "The aim of experiment is to experimentally determine the acceleration of gravity \(g\) using a string pendulum." — Which section?
2. "The experiment was carried out at a room temperature of 22 °C and an air pressure of 1013 hPa." — Which section?
3. "The specific value \(g = 9{,}78\,\text{m/s}^2\) deviates from the literature value by 0.3% and is therefore within the expected range." — Which section?
4. "Using Gaussian error propagation, the overall uncertainty \(\Delta g\) is calculated from the individual uncertainties of the pendulum length and the oscillation period." — Which section?
Step 3: Preparation and laboratory report
Before experiment
Good preparation is crucial to a successful experiment:
- experimentRead the description carefully and acquire the basic physics
- Basics and formulas summarize by hand (max. 2 pages)
- Answer additional questions in writing and submit them at the beginning
- Prepare tables and charts at home
The laboratory notes (appendix)
The laboratory report is during experiments handwritten and is an independent document that is attached to the draft as an appendix. It contains:
- All Measured values in the original — no subsequent changes
- Observations and abnormalities during the experiment
Important: The laboratory notes are kept with a permanent, document-proof writing instrument (e.g. a ballpoint pen) — never in pencil. Measured values must never be erased or overwritten. Incorrect values are crossed out and the correct value is noted next to them.
Exercise: What belongs in the laboratory notes?
Which of the following statements about Laboratory report (which is carried out during experiment) is correct?
Module 3: Measurement errors and their classification
Systematic vs. random errors, absolute and relative error information
Script: Ch. 4.1, pp. 7–9 (Open PDF)
Step 1: Why do measurements have errors?
Basic principle
There is a true value for every measurable quantity. However, it is fundamentally impossible to determine this exactly with measurements. Every measurement is subject to a measurement error. The goal of a measurement is therefore to provide an estimated value together with a measurement uncertainty that defines a range of values for the true value.
Important: More precise is not always better!
Example elementary charge (literature value: \(e = 1{,}602 \cdot 10^{-19}\,\text{C}\)):
- measurement 1: \(e = (1{.}7 \pm 0{.}1) \cdot 10^{-19}\,\text{C}\) → consistent with literature value
- measurement 2: \(e = (1{.}62 \pm 0{.}01) \cdot 10^{-19}\,\text{C}\) → more precise, but not consistent!
A measurement result is only meaningful if the specified uncertainty is realistic. An uncertainty that is too small gives the impression of an accuracy that was not achieved at all.
Step 2: Distinguish types of errors
In order to specifically reduce measurement errors, you have to know their cause. There are two fundamentally different types of errors that are treated differently:
Systematic errors
They arise from the measurement system or instrument. Measured values usually deviate in the same direction from the true value and are reproducible.
Examples: shifted thermometer scale, incorrect clock movement, zero point error on a scale
Reduction: through calibration, better measuring devices or correction calculations.
Random (statistical) errors
Unavoidable, partly caused by human inadequacies. Lead to Scattering in both directions around the mean.
Examples: reading accuracy of a scale, reaction time when stopping
Reduction: by repeating the measurement and statistical analysis (see Module 5).
Exercise: Classify errors
Select an example, then its category. You can also drag the example. To correct an assignment, select the assigned example again.
Systematic errors
Random errors
Step 3: Quantify errors
Once you have identified the nature of the error, you must also be able to quantify it. There are three forms of representation that are useful in different ways depending on the context:
Error representations
- Absolute error: has the same unit as the measured quantity, e.g. \(\Delta l = 1\,\text{mm}\).
Tells directly how much the measured value can deviate from the true value. - Relative error: dimensionless, e.g. \(\Delta l / l = 0{,}05\).
Allows you to compare the measurement accuracy of different quantities - regardless of the unit. - Percentage error: relative error in percent, e.g. \(5\,\%\).
Most clear representation, especially when communicating results.
Exercise: Calculate relative error
A length measurement gives \(l = 20\,\text{mm}\) with an absolute error of \(\Delta l = 1\,\text{mm}\). What is the relative error \(\Delta l / l\)?
Module 4: Calculation accuracy
Why intermediate results cannot be rounded
Calculate with machine precision - only round at the end
A common mistake when evaluating laboratory experiments: intermediate results — a mean value, say, or a partial result of a formula — are rounded to a few decimal places before they feed into the next calculation. Every rounding step discards information, and the rounding errors can add up — the end result then deviates significantly from the correct value.
Rule
In every intermediate step, always calculate with the full precision of your calculator or computer (machine precision). Only round the Final result to a sensible number of digits.
TipWhen intermediate rounding does the most damage
The critical case is the difference of two numbers of similar size: the leading digits cancel out, and what remains in the result are precisely the trailing digits — the ones lost in rounding. Rounding the intermediate results by a few hundredths can thus produce an error of 20 % or more in the final result.
In practice this means: keep intermediate results stored in the calculator (memory or ANS key), or write them down with at least two more digits than the final result is meant to have.
Exercise: Effects of intermediate rounding
The difference in area \(\Delta A = \bar{a}^2 - \bar{b}^2\) of two square plates with edge lengths \(a\) and \(b\) is to be determined. Both edge lengths were measured three times with a vernier calliper:
| Edge length | Measurement 1 | Measurement 2 | Measurement 3 |
|---|---|---|---|
| \(a\) / cm | 4,73 | 4,74 | 4,70 |
| \(b\) / cm | 4,47 | 4,46 | 4,50 |
The calculation has two stages: First of all form the mean values \(\bar{a}\) and \(\bar{b}\) from the three measurements each — these are the intermediate results —, then calculate \(\Delta A\) from them. You take this route twice: in step 1 at machine precision, in step 2 with rounded intermediate results. Comparing the two results shows what the rounding does.
Step 1: Calculate with machine precision
Form the mean values \(\bar{a}\) and \(\bar{b}\) and insert them into \(\Delta A = \bar{a}^2 - \bar{b}^2\) — with every digit your calculator shows. Round only the final result to 2 decimal places.
Step 2: Calculate with rounded intermediate results
Now round both mean values to one decimal place — that is \(\bar{a} \approx 4{,}7\,\text{cm}\) and \(\bar{b} \approx 4{,}5\,\text{cm}\) — and insert these rounded values into \(\Delta A = \bar{a}^2 - \bar{b}^2\). Round the result to 2 decimal places.
Module 5: Statistical key figures from measurement series
Calculate mean, standard deviation and mean error interactively
Script: Ch. 4.2, p. 10 (Open PDF)
Why statistical analysis?
If you repeat a measurement several times, you will get slightly different values due to random errors. A single measured value is therefore not very meaningful. The statistical analysis provides two crucial pieces of information: the best estimate for the quantity being determined and a Information about uncertainty this estimate.
The four key figures
i) Average — the best estimate of the true value. The more measurements, the more reliable:
$$ \bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i $$
ii) Deviations — show how far each measured value is from the mean. The sum of all deviations must be exactly zero (important calculation check!):
$$ \delta_i = x_i - \bar{x} \qquad \text{(Control: } \sum \delta_i = 0\text{)} $$
iii) Standard deviation — a measure of the spread of the individual measurements. It describes how much the measured values typically deviate from the mean. The factor \(\frac{1}{n-1}\) (instead of \(\frac{1}{n}\)) corrects a systematic underestimation in small specimens:
$$ s_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} $$
iv) Mean error of mean — the actual measurement uncertainty. It indicates how accurately the mean estimates the true value. The more measurements (\(n\)) are performed, the smaller the mean error becomes — the uncertainty decreases in proportion to \(\frac{1}{\sqrt{n}}\):
$$ \Delta x = \frac{s_x}{\sqrt{n}} $$
Result: The measurement result is always given as an average with uncertainty:
$$ x = \bar{x} \pm \Delta x $$
Exercise: Evaluate measurement series
The height h of a wooden block was measured 12 times (values in cm). Enter the measured values and calculate the statistical key figures step by step.
Step 1: Calculate average
The mean is the best estimate of the true value of the measured quantity. Add all 12 measured values and divide by the number.
Calculate the average (round to 3 decimal places):
Step 2: Deviations \(\delta_i = x_i - \bar{x}\)
The deviations show how much each individual measured value deviates from the mean. Positive values are above the mean, negative values are below. Important control: The sum of all deviations must be exactly zero - otherwise there is a calculation error. The squared deviations \(\delta_i^2\) are needed in the next step for the standard deviation.
Step 3: Calculate standard deviation (to 3 decimal places)
The standard deviation summarizes the spread of all measured values into a single number. It describes how far away a typical measured value is from the mean. Use the sum of squared deviations \(\sum \delta_i^2\) from the table above.
$$ s_x = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2} $$
Step 4: Calculate mean error (to 3 decimal places)
The mean error indicates the uncertainty of the mean — how accurately our mean estimates the true value. It is always smaller than the standard deviation because averaging over many measurements makes the result more accurate. For \(n = 12\) measurements the average error is smaller than \(s_x\) by a factor of \(\sqrt{12} \approx 3{,}5\).
$$ \Delta x = \frac{s_x}{\sqrt{n}} $$
Step 5: Specify result
Enter the final result in the form \(\bar{x} \pm \Delta x\) (each to 3 decimal places):
Module 6: Error propagation
Apply the Gaussian error propagation law
Script: Ch. 4.3, pp. 10–15 (Open PDF)
Why error propagation?
In Module 5 you learned how to determine the uncertainty of a directly measured quantity. But often one is interested calculated quantity, which depends on several measured quantities — for example an area from length and height, or a speed from distance and time. Each of these quantities carries its own uncertainty. Uncertainty propagation answers the central question: How great is the uncertainty of the final result if the input variables are error-prone?
Procedure in three steps
For a composite quantity \(X = f(X_1, X_2, \ldots, X_k)\) the uncertainty is determined in three steps. Requirement: Each input variable has been measured and statistically evaluated several times (Module 5) - so you already know the mean value \(\bar{x}_j\) and the uncertainty \(\Delta x_j\) of each individual measured quantity.
- Best estimate: Insert the average values of the individual variables into the formula - this gives the best estimate of the result
- Partial derivatives: Derive the formula for each variable in error (treating all others as constants). These derivations show which measured quantity has the greatest influence on the overall error.
- Calculate uncertainty: Insert the partial derivatives (evaluated at the mean values) and the individual uncertainties into the error propagation law
Gaussian error propagation law
\(\bar{s} = 2{,}00\,\text{m}\), \(\Delta s = 0{,}02\,\text{m}\), \(\bar{t} = 0{,}50\,\text{s}\), \(\Delta t = 0{,}01\,\text{s}\).
Step 1 — Best estimate: Each input variable was measured several times. From the measurement series you know the mean values \(\bar{x}_1, \bar{x}_2, \ldots\) and the uncertainties \(\Delta x_1, \Delta x_2, \ldots\) (from module 3). You can get the best estimate of the result by substituting the average values into the formula:
$$ \bar{X} = f(\bar{x}_1, \bar{x}_2, \ldots, \bar{x}_k) $$
Step 2 — Partial Derivatives: Derive the formula for each input quantity. The partial derivatives \(\frac{\partial f}{\partial x_j}\) act as Weighting factors — they indicate how sensitive the result reacts to a change in the respective input variable:
$$ \frac{\partial f}{\partial x_1}, \quad \frac{\partial f}{\partial x_2}, \quad \ldots, \quad \frac{\partial f}{\partial x_k} $$
\(\dfrac{\partial v}{\partial t} = -\dfrac{s}{t^2} = -\dfrac{2{.}00\,\mathrm{m}}{(0{.}50\,\mathrm{s})^2} = -8{.}00\,\dfrac{\mathrm{m}}{\mathrm{s}^2}\)
Step 3 — Calculate Uncertainty: Depending on whether the individual errors are random or systematic, there are two variants:
Statistical error — the errors are random and independent, so the contributions add up squarely. The overall error is smaller than the sum of the individual contributions because random deviations partially compensate for each other:
$$ \Delta x = \sqrt{\sum_{j=1}^{k} \left(\frac{\partial f}{\partial x_j} \cdot \Delta x_j\right)^2} $$\(\phantom{\Delta v} = \sqrt{0{,}0016\,\frac{\text{m}^2}{\text{s}^2} + 0{,}0064\,\frac{\text{m}^2}{\text{s}^2}} = 0{,}09\,\frac{\text{m}}{\text{s}}\)
Systematic error — the errors work in the same direction (worst case), the amounts add up directly. This gives the maximum possible deviation:
$$ \Delta x = \sum_{j=1}^{k} \left|\frac{\partial f}{\partial x_j}\right| \cdot \Delta x_j $$\(\phantom{\Delta v} = 0{,}04\,\frac{\text{m}}{\text{s}} + 0{,}08\,\frac{\text{m}}{\text{s}} = 0{,}12\,\frac{\text{m}}{\text{s}}\)
Exercise: Side surface of a wooden cuboid
The side area A of a cuboid is calculated from the length l and height h: \(A = l \cdot h\). Both variables were measured several times and statistically evaluated (as in Module 5). Now determine the best estimate for the area and its uncertainty using error propagation.
Given (Results of the statistical analysis from the measurement series of \(l\) and \(h\)):
\(\bar{l} = 9.73\,\text{cm}\), \(\Delta l = 0.03\,\text{cm}\) (Mean value and uncertainty of length measurements)
\(\bar{h} = 3.9\,\text{cm}\), \(\Delta h = 0.05\,\text{cm}\) (Mean value and uncertainty of height measurements)
Step 1: Best estimate of the side area (to 2 decimal places)
You can get the best estimate of the area by substituting the averages of \(l\) and \(h\) into the formula. Since \(A = l \cdot h\), calculate \(\bar{A} = \bar{l} \cdot \bar{h}\).
Step 2: Calculate partial derivatives
The partial derivatives determine how strongly an error in \(l\) or \(h\) affects the area \(A\). For \(A = l \cdot h\), you derive from \(l\) (treating \(h\) as a constant) and vice versa. Enter the result as a variable (e.g. "h" or "l").
Step 3: Calculate statistical error (to 2 decimal places)
If the measurement errors of \(l\) and \(h\) random and independent are different from each other, the deviations partially compensate for each other. That's why the posts combined in quadrature (Gaussian error propagation). The result is smaller than the simple sum of the individual contributions:
$$ \Delta A_{\text{stat}} = \sqrt{\left(\frac{\partial A}{\partial l} \cdot \Delta l\right)^2 + \left(\frac{\partial A}{\partial h} \cdot \Delta h\right)^2} = \sqrt{(\bar{h} \cdot \Delta l)^2 + (\bar{l} \cdot \Delta h)^2} = \;? $$
Step 4: Calculate systematic error (to 2 decimal places)
If the errors systematic (e.g. an incorrectly calibrated measuring device), in the worst case they can all act in the same direction. Then the amounts add directly - without quadratic attenuation. This gives the maximum possible error:
$$ \Delta A_{\mathrm{sys}} = \left|\frac{\partial A}{\partial l}\right| \cdot \Delta l + \left|\frac{\partial A}{\partial h}\right| \cdot \Delta h = \bar{h} \cdot \Delta l + \bar{l} \cdot \Delta h = \;? $$
Step 5: Result and comparison
Compare both errors. The systematic error is always greater than or equal to the statistical one because during quadratic addition the contributions partially compensate for each other. In practice, both uncertainties are often stated separately because they have different causes and can be reduced differently: statistical errors through more measurements, systematic errors through better calibration.
Enter the final result with the statistical error (each to 2 decimal places):
Module 7: Linear Regression
Experience the least squares method interactively
Script: Ch. 4.4, pp. 16–18 (Open PDF)
Principle
There is often a linear relationship between two measurements: \(y = m \cdot x + b\). Due to measurement errors, the data points are not exactly on a straight line. Linear regression determines the straight line that minimizes the sum of the squared deviations of all data points.
Goodness of fit
The regression coefficient \(R\) is in the range \(-1 \le R \le 1\). The closer \(R\) is to 1 or \(-1\), the better a straight line describes the measurement data. \(R^2\) close to 1 means a very good linear fit.
Regression parameters
Increase — indicates how much \(y\) changes per unit of \(x\):
$$ m = \frac{n \sum x_i y_i \;-\; \sum x_i \cdot \sum y_i}{n \sum x_i^2 \;-\; \left(\sum x_i\right)^2} $$Axle section — the \(y\) value at \(x = 0\):
$$ b = \frac{\sum x_i^2 \cdot \sum y_i \;-\; \sum x_i \cdot \sum x_i y_i}{n \sum x_i^2 \;-\; \left(\sum x_i\right)^2} $$Specific measure — describes how well the straight line represents the measurement data. \(R^2 = 1\) means perfect linear agreement:
$$ R^2 = \frac{\left(n \sum x_i y_i - \sum x_i \cdot \sum y_i\right)^2}{\left(n \sum x_i^2 - \left(\sum x_i\right)^2\right) \cdot \left(n \sum y_i^2 - \left(\sum y_i\right)^2\right)} $$Uncertainties of the regression parameters
The following applies to the associated errors of the regression parameters:
Uncertainty of the slope \(\Delta m\):
$$ (\Delta m)^2 = \frac{n}{n-2} \cdot \frac{\sum y_i^2 - m \cdot \sum x_i y_i - b \cdot \sum y_i}{n \sum x_i^2 - \left(\sum x_i\right)^2} $$Uncertainty of the intercept \(\Delta b\):
$$ (\Delta b)^2 = \frac{\sum x_i^2}{n} \cdot (\Delta m)^2 $$Exercise: Calculate linear regression by hand
For the following data set (\(n = 5\)), calculate the slope \(m\), the intercept \(b\), their uncertainties \(\Delta m\) and \(\Delta b\) and the coefficient of determination \(R^2\) by hand.
| No. | \(x_i\) | \(y_i\) |
|---|---|---|
| 1 | 1 | 1,9 |
| 2 | 2 | 3,8 |
| 3 | 3 | 5,6 |
| 4 | 4 | 7,8 |
| 5 | 5 | 9,9 |
Step 1: Calculate subtotals
Add the values column by column. These sums are required for all further calculations.
Step 2: Calculate the slope and intercept (to 2 decimal places)
Insert the subtotals into the formulas. The denominator is the same for both: \(N = n \sum x_i^2 - \left(\sum x_i\right)^2\).
Step 3: Calculate uncertainties (to 2 decimal places)
Calculate the uncertainties \(\Delta m\) and \(\Delta b\) using the above formulas. To do this, enter the subtotals and the calculated values for \(m\) and \(b\).
Step 4: Calculate the coefficient of determination (to 3 decimal places)
The coefficient of determination \(R^2\) indicates how well the regression line describes the data.
Interactive regression
Enter your own measured values or customize the sample data. The regression line is calculated and displayed live.
| No. | x | y |
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| 1 | ||
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TipRegression results
Rise: \(m\) = ±
Axle section: \(b\) = ±
Regression coefficient: \(R^2\) =
Module 8: Linearize non-linear relationships
Variable substitution for linear regression
Script: Ch. 4.4.1, p. 19 (Open PDF)
Step 1: Why linearize?
The problem
The linear regression from Module 7 only works for relationships of the form \(y = m \cdot x + b\). But many physical laws are nonlinear — they contain squares, roots, exponential functions or powers. If you were to plot the measurement data directly, you would not be able to calculate a regression line.
The solution: Through clever Variable substitution brings the connection into a linear form. Regression can then be used and the desired physical quantities can be calculated from the slope and intercept.
The three most important types
| Type | Reference | Substance | Linear shape |
|---|---|---|---|
| Square | \(y = a \cdot x^2 + b\) | \(u = x^2\) | \(y = a \cdot u + b\) |
| Exponential | \(y = a \cdot e^{b x}\) | \(\ln\) both sides | \(\ln(y) = bx + \ln(a)\) |
| Potency law | \(y = a \cdot x^b\) | \(\ln\) both sides | \(\ln(y) = b\cdot\ln(x) + \ln(a)\) |
TipProcedure for linearization
Once you are asked to use a nonlinear formula for regression, do this:
- What is the theoretical connection?
- What type does it correspond to (quadratic, exponential, power)?
- Which quantity belongs on the x-axis, and which on the y-axis?
- What do slope and intercept mean physically?
Step 2: Example — Free Fall
We work through linearization step by step using a concrete example.
Initial position
In free fall, the fall distance \(s\) is as a function of the fall time \(t\):
$$ s = \frac{1}{2}\,g\,t^2 $$
Goal: The acceleration due to gravity \(g\) is to be determined using linear regression.
Step by step
1. Recognize type: \(s\) depends quadratically on \(t\) → quadratic type.
2. Substitution: We set \(u = t^2\). This will:
$$ s = \frac{g}{2} \cdot u $$
This has the form \(y = m \cdot x\) with \(y = s\), \(x = u = t^2\) and slope \(m = g/2\).
3. Application: \(s\) on the y-axis, \(t^2\) on the x-axis.
4. Physical quantity: From the slope \(m_{\text{reg}}\) of the regression line it follows:
$$ g = 2 \cdot m_{\mathrm{reg}} $$
Understanding
The regression results in a slope of \(m_{\text{reg}} = 4{,}89\,\text{m/s}^2\). What is the acceleration due to gravity \(g\)?
Module 9: Creating a report
Check your progress and export your PDF log
Script: Ch. 5, pp. 20–24 (Open PDF)
Module overview
Here you can see the current status of all modules. Complete all modules to get the full log.
| Module | Status |
|---|---|
| 2. Record keeping | — |
| 3. Measurement error | — |
| 4. Calculation accuracy | — |
| 5. Statistical key figures | — |
| 6. Error propagation | — |
| 7. Linear regression | — |
| 8. Linearization | — |
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