V05 - spring-mass oscillator (static)
First the fundamentals, then the digital experiment — you answer the guiding questions at the end, with your own measured values at hand. A ready-made solution is deliberately not provided here.
Cover sheet
Name and matriculation number for this experiment’s report
Both appear on the PDF report and in the verification view behind the QR code. Enter them before you start with the fundamentals.
One cover sheet, one report. If the name or the matriculation number is changed later, this experiment starts over: measured values, answers and assignments are deleted and the apparatus is set up anew.
Chapter 1: The spring and Hooke's law
Why a spring can be a measuring instrument at all
Pull on a helical spring and it pulls back — the further it is stretched, the more strongly. In 1676 Robert Hooke put this into a sentence that still holds today: The force is proportional to the deflection.
The proportionality factor C means spring constant and is measured in N/m. It is the whole story of a spring: a stiff spring has a large C, a soft one a small one. Determining it is the goal of this part of the experiment.
Figure: the same spring without a load, with the mass M and with 2M. The drawing is to scale — twice the load stretches it exactly twice as far.
This does not hold without limit: overload a spring and it stays deformed, and the relation is gone. That is why the experiment instructions state for every spring a maximum load — it is not a recommendation.
Two meanings for “m” — a trap
In this module M is the attached mass. The lower-case m is reserved for the slope of the regression line — that is its traditional name and the one the tool uses. The measurement column of the attached mass is nevertheless labelled “m / kg” at the test rig, for historical reasons.
Two springs instead of one
In the on-site laboratory the springs are also measured in parallel and in series. In parallel the two share the load — the arrangement becomes stiffer. In series the two stretch one after the other — it becomes softer.
At the digital test rig you measure a single spring; the combinations remain reserved for the laboratory.
Chapter 2: Static equilibrium
What happens when the mass simply hangs there
When a mass hangs quietly on the spring, seemingly nothing happens. That is exactly why measurements here are so clean: two forces balance each other precisely.
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Two forces on the mass. Downwards the weight pulls with M·g, upwards the stretched spring pulls with C·s.
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At rest they cancel out. The sum of all forces on the mass is zero — otherwise it would move.
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The equilibrium. Every attached mass belongs to exactly one deflection; the spring constant C links the two.
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More mass, more deflection — in the same ratio. A single weighing yields C already; a series of Measurement series averages out the reading errors and at the same time shows whether the spring stayed in the elastic range.
Chapter 3: Why linearise at all?
Why a straight line is worth more than a curve
Each measurement scatters. One Straight line can be determined robustly and transparently from scattered data points, whereas fitting a curved relationship is more difficult. A nonlinear relationship is therefore linearised by choosing suitable variables for the axes. The quantity being determined is then contained in the Increase or in Axle section the straight line.
Here you have it easy: the equilibrium from Chapter 2 is already the equation of a straight line. Nothing has to be rearranged — the only question is which measured quantity belongs on which axis, and what the intercept then means.
Chapter 4: What does linear regression achieve?
Slope and intercept — and when they mean something
Linear regression determines that straight line y = m·x + cthat best fits your measurement points. Slope m and intercept c only carry a physical meaningif you derive the axes from the basic equation of the experiment. That choice is yours to make.
A word on naming: what the supplementary questions and the lab instructions call y-intercept is called here and in the evaluation tool c — both mean the value of the line at the position x = 0.
Watch the intercept: here it should be close to zero. If it is not, either the scale has a zero offset or the rest position was not read without a load — in this part of the experiment it is a check value, not a measurement result.
Part 2 of the experiment determines the same spring constant along a completely different route: by way of time instead of distance. Its principles are on the page of the second part.
The experiment
Record and evaluate a series of measurements — on the test-rig page
Attach the intended masses one after another and read off the deflection — you must not exceed the maximum load of the spring.
A button then takes you straight on to station 7, the guiding questions.
Your evaluation
Regression parameters
Questions
Answer them with your own measured values — the answers go into the report
The spring, once more
You know the drawing from Chapters 1 and 2. Now the labels are missing: drag them to the right place — with the mouse, by tapping, or by first tapping a building block and then a field.
Task: Name the four quantities on the loaded spring.
Questions on the static determination
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Which quantities do you have to plot against each other in a diagram in order to perform a linear regression?
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What is the relation between the spring constant you are looking for and the regression parameters to be determined, that is, slope and intercept?
Your answers are stored locally and are transferred into your Preparation report taken over. There is no automatic evaluation — discuss your thoughts in the lab and in the colloquium.
Calculating the quantities you are after
Finally, complete the evaluation: for each quantity you are after, first state the relationship symbolically and then evaluate it with your own regression parameters.
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Spring constant C
N/m
There is no automatic assessment here either. Both entries go into your preparation report — so that in the end it contains not only the approach but also the result.
Preparation report
Series of measurements, diagram and your answers as a PDF
The PDF contains your series of measurements, the diagram with the regression line, the calculated parameters and your answers to the guiding questions — plus a QR code for verifying the report.
Checking the measurement status …
Further reading
- Walcher, W.: Praktikum der Physik 8th edition, Teubner, Wiesbaden 2004
- Schenk, W.; Kremer, F. (eds.): Physikalisches Praktikum 14th edition, Springer Spektrum, Wiesbaden 2014
- Demtröder, W.: Experimentalphysik 1: Mechanik und Wärme 9th edition, Springer Spektrum, Berlin/Heidelberg 2021
The titles lead to the publisher's page; the full text is available via the university network.