An Experiment That Could Change Our Minds

Back to the cart from chapter seven. Timur and Dana are not pulling the string yet: they are drawing up an experiment plan. On the sheet are the same numbers — mass 19.0 kg, acceleration 1.5 m/s², and the assumed dynamometer reading of 30 N.

Mass times acceleration is 28.5 N. The gap to the instrument’s reading is one and a half newtons.

“I would still start with the dynamometer,” says Timur.

“And if it is fine?” asks Dana. “Maybe we missed resistance. Or wrote down the mass wrong.”

Before reading on, decide: can these three numbers pick one explanation?

Two explanations of one reading

First recall what exactly we compare. A dynamometer on a horizontal string measures its tension. Mass times acceleration gives the resultant. These forces need not match: something may resist motion. The bare difference of numbers proves no instrument fault.

Consider two possible explanations. For now they are guesses fitted to a conditional example, not measurement results. In both calculations we assume the force and acceleration readings carry no substantial systematic shift.

First: the mass is right, but there is resistance. The cart with the load weighs 19.0 kg. Suppose a 1.5 N force opposes its motion. Then from the 30 N tension, subtracting resistance leaves 28.5 N — exactly what mass times acceleration gives.

For further calculation suppose resistance stays roughly constant over the picked stretch and in the regime at hand. That assumption remains to be checked.

Second: the mass is written down wrong, and resistance may be neglected. Suppose the cart with all its fastened equipment weighs 20.0 kg. Then at 1.5 m/s² acceleration 30 N are needed. Perhaps some part never made it into the mass entry.

Both explanations fit the starting numbers. But these are no two independent confirmations: in each case we picked the unknown so the calculation would give the wanted result. Now check what these explanations predict beyond the starting case.

Change the acceleration

“And if we pull harder?” Timur suggests.

Plan motion at about 3.0 m/s² acceleration — twice as much, with the same cart and load. What tension does each explanation predict?

In the first case mass times acceleration makes 57 N. Add the assumed 1.5 N resistance: the expected tension is 58.5 N.

In the second case the mass is 20.0 kg, and resistance we neglect. The expected tension is 60 N.

In the first row the predictions match. In the second they differ by 1.5 N. Now we have a chance to tell the explanations apart, if measurement precision suffices and the accepted conditions hold.

Repeating motion at the old acceleration is useful too: the readings’ spread can be estimated and the result checked for reproducibility. But if we get about 30 N again, preferring one of these two explanations over the other on such a match will not do. Repetition answers its own question.

There is a more direct move — weighing the cart with all its equipment anew, checking the weighing method and the result’s entry. For choosing between 19 and 20 kg masses that may suffice. Raising acceleration after that is optional. It is needed if we also want to check how tension depends on acceleration.

Which conditions we check with the explanation

For the calculation we accepted that the cart moves horizontally, the string pulls along motion, mass stays unchanged through the experiment, and acceleration is measured against the laboratory. As in chapter seven, we count its frame of reference as inertial with precision enough for this experiment.

Force and acceleration must be compared over one and the same stretch of motion. A reading at the first jerk cannot be set against an acceleration measured later. Nor may the number 3.0 be counted exact just because we wrote it in the plan: the calculation takes the actually measured acceleration.

A separate question is whether resistance will hold. A stronger pull may change the motion’s conditions. So the string’s direction, the track’s state and the speed range we compare results in must be watched. The same instruments and the same cart by themselves do not secure the same conditions.

We need not change exactly one quantity in every experiment. But we must understand which changes the calculation allows for and which may spoil the comparison. If resistance shifts markedly, the 58.5 N prediction no longer follows from the first explanation as stated.

Will precision suffice

One and a half newtons between predictions looks convincing on paper. In the laboratory the whole comparison’s uncertainty must be estimated.

It depends on measuring tension and on the calculation’s input quantities: mass, acceleration, assumed resistance. If the calculations use shared readings or corrections, that too must be allowed for. From one quivering needle no one can decide whether the predictions can be told apart.

For the example suppose such an estimate is done. A rule picked in advance: a result counts as compatible with a prediction if their difference in magnitude is under 0.5 N. Here 0.5 N is a conditional threshold for the whole comparison, grounded in the uncertainty estimate. It is not the dynamometer’s division value, nor a promise that error will never exceed half a newton.

At 3.0 m/s² acceleration a 58.6 N result satisfies this rule for the first prediction only. A 59.9 N result — for the second only.

And 59.25 N fits neither: it differs from both predictions by 0.75 N. It cannot be entered for a favoured explanation just because it landed somewhere in the middle.

Now picture a less precise comparison, with a 1 N threshold set by the same principle. The same 59.25 N is compatible with both predictions. The number stayed put, but the chance to conclude changed.

So we provide for three cases in advance:

• Result compatible with one prediction only. Grounds to prefer this explanation over the second under the check’s accepted conditions.
• Result compatible with both. This check failed to pick. Uncertainty may need shrinking, or predictions spreading wider.
• Result compatible with neither. Measurements, their uncertainty estimate and the explanations themselves need checking. One of the two need not turn out right.

Even the first outcome will not prove the only possible explanation found. A constant overread of the dynamometer, say, may look in calculation like constant resistance. Telling even these options apart takes a separate check of the instrument.

Write down before you know

Timur reaches for the string, but Dana turns the sheet to him.

“First let us write down what we will count as a result.”

The sheet must keep both explanations, their assumptions, expected values, the measurement method and the comparison rule. The number of repeats and how to process their results must also be decided in advance, so the handiest reading is not picked afterwards.

A plan does not make a researcher infallible. It shows which decisions were taken before new data arrived, and which appeared after.

If a fault or an unallowed-for condition shows, the plan may change. But then the old entry must be kept, the edit explained, and the refined explanation checked apart. Otherwise an assumption invented after the experiment is easily passed off as a prediction the experiment confirmed.

Here the interlude’s dates matter again: what information was available when we drew the conclusion? For an experiment one particular question joins it: was the comparison criterion picked before the new result or fitted to it?

What could change our minds

Timur and Dana have no new reading yet. But now they hold a plan by which it can move their opinion. They know what they compare, which differences they expect, and under which conditions the comparison makes sense.

The answer “I am ready to change my mind” grew more definite. Name a result weakening the first explanation, a result weakening the second, and a result after which both must be revised. The chance that precision will fall short of a pick can be admitted in advance too.

Such an experiment promises no certain end to the dispute. It lets the next conclusion rest on new grounds and show how they differ from the old ones.

In the last chapter we move to music. There a check establishes the rule kept, while the question of whether it turned out well stays open all the same.