Tried-and-tested sequences for physics teaching · Lothar Bodingbauer
These sequences have proven themselves in physics lessons, above all at the Abendgymnasium Wien, an evening upper-secondary school for adults. They are not ready-made recipes but frameworks: an opening that sparks curiosity, a phase of working on one’s own, and a conclusion in which one’s own findings meet what is already known.
Everyone gets a pendulum: a string with some mass at its lower end. Swing along with it! Sing “bim – bam” as you do! Get used to the rhythm and find out how fast the pendulum swings.
Step one: the qualitative measurement
A table of guesses: what might the period depend on? The position of the sun, the mass, the length – what else? For each parameter, tick: strongly, weakly, not at all, cannot be tested. Tick only what has been observed, not what is suspected.
No stopwatch, no tape measure: the ear is a very good measuring instrument. When you sing along, differences in the period are easier to hear than to see.
If you change the length and the mass at the same time, you don’t know which change had the effect. So always change only one parameter.
Step two: the quantitative measurement
Usually it turns out that the period depends only on the length. That is a hypothesis. New question: how exactly does the period depend on the length?
Measure ten swings and divide by ten – a single swing is too short.
Measure three times for each length and take the average.
Change the length deliberately; put the values into a table and a graph.
Step three: prediction
From our own measurements we develop the formula for the period. The values show a square-root relationship: a pendulum four times as long swings twice as slowly. With the formula we predict the period for a new length that has not yet been measured.
Write down the prediction, then measure.
How good was the prediction? The deviation is given as a percentage.
Comparison with the textbook formula: T = 2π · √(l / g).
Step four: other kinds of pendulum
The spring pendulum oscillates in the same way, except that here the period depends on the mass and the stiffness of the spring. And circular motion: if you move a ball in a circle and look at its shadow from the side, you see an oscillation. That is where the π in the formula comes from.
Conclusion
Deviations come from friction, draughts, the shape of the weight – the setup is not ideal. This is a good opportunity to discuss what a model is. Going further: how do you swing properly on a swing? Resonance – and why it can destroy a bridge.
A single radioactive decay cannot be predicted, but a large number of atoms can be predicted very well. How do these fit together? What is the difference between objective and subjective chance?
The coin simulation
Many coins stand for many atomic nuclei. All the coins are tossed; those showing “heads” have decayed and are put aside. The rest are tossed again, round after round.
After each round, count and enter the result in a table.
The values go into a graph: number of remaining coins against the number of tosses.
Which individual coin “decays” is completely random. Roughly how many will, can be predicted well.
From the result to the formula
In equal time intervals, the same proportion always disappears, not the same amount. This gives the law of radioactive decay.
What is the lambda responsible for?
Why is there a minus sign in the exponent?
What is the difference between a linear function and an exponential function?
Going further
Half-lives range from archaeological finds (carbon-14 dating) through short-lived isotopes in medicine to long-lived nuclear waste. Is a short or a long half-life more dangerous?
Sisyphus, king of Corinth, deceived the gods and even outwitted death. For this he is punished: every day he must roll a heavy stone up a mountain, and every night it rolls back down. He has to roll it up again, over and over.
Do it now
Why is this a punishment and not a reward? Write down several reasons.
From force to work
Starting from the force Sisyphus has to exert along the way, we develop the concept of work: work is force times distance. The heavier the stone and the longer the way up, the more work.
Work and energy
The work is not gone. It is stored in the stone as long as it lies at the top. Energy is the ability to do work.
Potential and kinetic energy
At the top the stone has energy of position, potential energy: mass · gravitational acceleration · height. At night it rolls down, and the energy of position turns into energy of motion, kinetic energy: ½ · mass · speed².
The golden rule of mechanics
What you save in force, you must add in distance. A long, gentle path needs less force, a short, steep one more – the work stays the same. The direct path is the shortest, but also the steepest. Would Sisyphus have it easier if he chose a different path?
Going further: how high can you get on one biscuit?
A biscuit has about 500 kJ. The body needs about two thirds of that for the heart, breathing and body temperature; one third is left for climbing. Weighing 80 kg, that gets you about 200 metres up.
Guiding questions
What links motion and height – kinetic and potential energy?
If energy is conserved, what does “using up energy” mean in everyday life?
Balls are interesting in themselves. But stacking several of them is no easy task at all. First of all, they roll away.
Working phase
A thousand small balls, about 10 mm in size. The task: stack pyramids.
How do you stop the balls from rolling away? What kind of barriers are needed?
Is there only one way to stack them? Which pyramids are possible – with a triangular base, with a square base?
Rolling friction is learned along the way.
Going further
Older students take a closer look at the concept of the densest sphere packing – exciting not only for orange sellers. Works from primary school all the way to the school-leaving exam.
We hear opinions, convictions and claims. Some sound convincing, some don’t. How can you actually tell what science is?
Ways of knowing something
Self-perception (“I am tired”), everyday knowledge (“fire is hot”), information from others, and finally scientific knowledge. Which of these can be checked from the outside?
The four stars of science
Measurability
Repeatability
Predictability
Freedom from contradiction
Can you prove it?
“All sheep are white.” No pile of white sheep, however large, proves the statement; a single black sheep refutes it. That is why in science we prefer to say “show” rather than “prove”.
Exercise: false conclusions
Ten short stories: Lisa wears her favourite cap, and the sun shines. Tim drinks a green smoothie before the exam. What is the false conclusion in each case? Not everything that occurs together is connected.
How many piano tuners are there in Vienna? You want to know something but have almost no data. Still, you can estimate a good number. This way of calculating goes back to the physicist Enrico Fermi, who was famous for his accurate estimates.
Where the name comes from
At the first atomic bomb test on 16 July 1945, Fermi dropped small scraps of paper as the shock wave arrived. From the distance they were blown away, he estimated the explosive yield at about 10,000 tonnes of TNT. The instruments later showed around 20,000 tonnes – astonishingly close for a few scraps of paper. At the University of Chicago, Fermi asked his students questions like these, such as how many piano tuners there are in Chicago. Today, estimates of this kind are named after him: Fermi problems.
Working phase
The big problem is broken down into small sub-questions. How many people live in Vienna? How many households have a piano? How often is a piano tuned? How many pianos can a tuner manage in a day? Simple assumptions, step by step, until an order of magnitude emerges.
Another example: the tyre train
How long is the freight train that carries all the new car tyres needed in Austria in one year?
How many cars are there in Austria? (about 5 million)
How often does a car need new tyres? (a set of four about every 4 to 5 years – so roughly 1 tyre per car per year)
How much space does a tyre take up? (about 65 cm in diameter, 20 cm wide, so roughly 0.1 cubic metres)
How much fits into a freight wagon? (about 20 m long, 3 m wide, 3 m high, so about 150 cubic metres – roughly 1,500 tyres)
How many wagons, how long is the train?
One possible estimate: 5 million tyres divided by 1,500 per wagon gives a good 3,000 wagons, times 20 metres – a train of about 60 kilometres, roughly from Vienna to St. Pölten. Whether it is 30 or 100 kilometres depends on the assumptions. That is exactly what is worth arguing about.
More questions
How many leaves are there on a tree – and how much do they weigh?
What is the difference between an observation and an explanation?
How far away is America from here? A guess is quick: any number, from the gut. Estimating is something else.
Lower limit
America is certainly further away than 75 kilometres, because that is where St. Pölten is. And St. Pölten is not America.
Upper limit
America cannot be more than 40,000 kilometres away, because that is the circumference of the Earth. Actually only 20,000 kilometres: we cannot get any further away from here than that – beyond it, we come closer again on the other side of the Earth.
Closing the pincers
With further arguments, the limits move closer together: how long is the flight to New York? How fast does a plane fly? How far is it to the Atlantic coast?
Every limit needs an argument that can be checked.
We learn to pin down the “right value” with arguments.
The difference: a guess is a number. An estimate is a number with reasons and limits.
Going further
This is the basis of Fermi estimates: breaking big questions down into small parts that can be justified.
Thomas Mann describes the Buddenbrooks’ house and the autumn garden, “whose beds were protected against the frost with straw mats”. A small remark that leads straight to a question of physics.
Does straw really warm?
No. Straw does not produce heat; it slows down the loss of heat. The actual insulation is done by the still air between the stalks. The heat itself comes from the ground, which stores solar energy during the day.
The three ways heat is transferred
Conduction: air is a poor conductor of heat and therefore a good insulator.
Convection: the mat slows the wind, so warm air is not blown away.
Thermal radiation: radiation to the cold night sky is reduced.
The same principle everywhere
Down jackets, vacuum flasks, double glazing, sleeping bags, igloos and blankets of snow all slow down heat loss with a layer of still air.
Tasks
What is the difference between “producing heat” and “preventing heat loss”?
Why does snow often protect plants from hard frost?
Why does metal feel colder than wood, even though both are at the same temperature?
Phases of the moon · for German as a second language
Vocabulary
This sequence was originally designed for learners of German. Nouns, verbs and adjectives about the moon: die Mondphase (phase of the moon), der Vollmond (full moon), der Neumond (new moon), die Mondsichel (crescent), der Halbmond (half moon), der Horizont (horizon); aufgehen (to rise), untergehen (to set), zunehmen (to wax), abnehmen (to wane); hell (bright), dunkel (dark). Practising separable verbs: „Der Mond geht im Osten auf.“ (The moon rises in the east.)
Prior knowledge check
Ten statements, true or false: the moon makes its own light. We almost always see the same side of the moon. The moon rises about 50 minutes later each day. It is no problem to be unsure – we discuss it together.
Observation task over several days
Note at the same time every day: date, time, compass direction, height above the horizon, appearance (a small sketch). Rule of thumb: a fist at arm’s length is about 10 degrees.
Comparison with another place
Ask a person in another place, if possible at a different latitude: what does the moon look like there? Is the phase of the moon the same? How do you explain this?
Content: Lothar Bodingbauer, from weekly plans, worksheets and model lessons taught at the Abendgymnasium Wien and from the former phyx.at (in German). Selection, compilation, structuring into phases, abridgements and connecting sentences, the description of the procedure in the coin simulation, the paragraph on the origin of the name in the Fermi estimates and the figures of the tyre-train example: Claude, an AI model by Anthropic, on behalf of Lothar Bodingbauer. Contact: lothar@sprechkontakt.at
