Formulating falsifiable physical questions
Turn a force-law question into contrasting quantitative predictions while making calibration, background and source intervention explicit.
On this page
The question for today
What you’ll understand
- Construct a statistic that distinguishes two specified force exponents.
- Eliminate constant nuisance parameters without assuming away spatial backgrounds.
- Separate a prediction, an observation and a conditional rejection.
A physical question becomes testable when competing explanations predict different outcomes for a specified measurement. Asking whether a force is unusual leaves its observable and alternatives undefined. Asking whether a source-dependent radial force follows an inverse square or inverse cube over a stated distance interval permits a calculation. The experiment must still connect an ideal force to instrument readings. Unknown gain, background, source geometry and distance offsets can imitate changes in the proposed law. Designing a useful question includes identifying which of these ambiguities the available measurements can actually remove.
Build a contrast before collecting the data
Consider a deliberately simplified point-source experiment. Distances are known center-to-center separations, the source size is negligible, and force magnitudes remain positive over the interval. Write the source contribution as a positive amplitude times a power of a dimensionless distance ratio. A distance-independent additive background is allowed. The exponent is the target; the amplitude and background are nuisance parameters. Comparing candidate exponents requires giving each candidate the same opportunity to fit those nuisance parameters.
Two readings can determine an amplitude and a constant background for either exponent, so a perfect two-point fit cannot distinguish them. Three geometrically spaced distances produce a stronger contrast. Subtract neighboring readings to eliminate the common offset, then divide the differences to eliminate the amplitude. The resulting ratio is a power of two. This elimination is exact within the toy measurement model, rather than a proof that every real background is constant.
A measured voltage is not yet a source force
Suppose an instrument multiplies every force by an unknown constant gain and adds one constant electronic offset. Both cancel from the same difference ratio. A distance-dependent background does not generally cancel. Neither does a distance calibration error that changes the true separation from the recorded value. Near a small denominator, even modest noise can make the ratio unstable; fitting the original readings with their covariance may then be more informative than quoting an apparent exponent.
Changing the source state supplies another observable. If switching the source leaves the background unchanged, subtracting source-off from source-on readings isolates the source-dependent part at each distance. The assumption must be examined physically: switching might also change temperature, charging or mechanical alignment. A differential signal identifies the effect of the intervention under those controls. It does not automatically identify gravity, a new interaction, or a unique microscopic mechanism.
State what an unfavorable outcome rejects
Before running an experiment, specify the distance range, candidate laws, observable, treatment of missing runs and a statistical comparison. A discrepancy rejects the combined model of interaction and measurement at a stated evidential level. Locating its physical cause requires further checks. Conversely, agreement over a finite range constrains alternatives at the experiment’s sensitivity; it does not establish an exact law at every distance. A small residual should be compared with predicted corrections and uncertainty, rather than described as decisive by appearance.
Real inverse-square tests integrate over extended source and detector geometries and investigate systematic effects. Kapner and collaborators provide a primary experimental example of such a specified search; the numbers below are invented for transparent reasoning and are not their measurements. The central research skill is to produce an observable that would differ between viable explanations, then preserve the assumptions needed to interpret that difference. A proposed improvement should name which ambiguity or sensitivity limitation it addresses.
Work it through
Separate exponent from constant offset
Toy force readings at r0, 2r0 and 4r0 are 17, 5 and 2 nN. Treat the readings as exact for this algebra exercise. Compare p=2 with p=3 under the stated constant-background model, and determine A and B for the compatible exponent.
The neighboring differences are 12 nN and 3 nN. Their ratio is D=4, which is dimensionless because the force units cancel. The inverse-square candidate predicts four; the inverse-cube candidate predicts eight.
For p=2, the first difference equals A(1−1/4)=3A/4. Hence A=16 nN. Substitution into the reading at r0 gives B=17−16=1 nN.
Check the unused combination explicitly: A/16+B=16/16+1=2 nN at 4r0. Thus one shared amplitude and offset reproduce all three readings. No noise-based confidence statement follows from this exact-data exercise.
The data select p=2 within the two proposed ideal models, with A=16 nN and B=1 nN. This conclusion depends on the spatially constant background and correctly known separations.
The living notebook
Explore the model
Interactive lab · function
Use reciprocal distance as the horizontal variable
Set x=r0/r and interpret y=ax² as a normalized inverse-square source contribution. Reset and set both controls to Coefficient a=1 and Position x=1; predict y=1. For a fresh comparison, Reset and set both controls to a=1 and x=0.5; predict y=0.25, corresponding to doubling r.
- y at chosen x
- 0.608
- Tangent slope · 2ax
- 1.35
- Signed area · ax³/3
- 0.182
This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.
Look forThe lab checks the proposed inverse-square reference in a transformed coordinate. It has no inverse-cube alternative, background, noisy force data, source geometry or statistical model selection.
Pause & reason
Check your understanding
Your turn
Try it yourself
At r0, 2r0 and 4r0, source-off readings are 9, 2 and 1.125 nN; paired source-on readings are 25, 6 and 2.125 nN. The background is unchanged by switching but varies with distance. Find the source-dependent force ratios and exponent. Explain why the raw on-readings should not be analyzed using a constant-background assumption.
Take a moment to make a prediction or work it out on paper. Open a hint when you need one.
Give me a hint
Subtract each off-reading from its paired on-reading before comparing distances.
Compare with the solution
The paired differences are 16, 4 and 1 nN. Each doubling of distance reduces the source-dependent signal by four, so p=2 and its reference amplitude is 16 nN. In contrast, raw on-reading differences are 19 and 3.875 nN, whose ratio is about 4.903 rather than four. Those readings include a spatially changing background and violate the earlier constant-offset model. Pairing removes that background only under the stated unchanged-background assumption; the exercise therefore tests a source intervention as well as a scaling relation.
Keep the idea
Take it with you
- A falsifiable question names contrasting quantitative predictions and an observable.
- Algebraic nuisance cancellation has explicit assumptions and can become ill-conditioned.
- Differential measurements constrain source-dependent effects without uniquely naming their mechanism.
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