SCIENCE · INVESTIGATE AND EXPLAIN

Challenge a lens fit with distance-reference errors

One object-image pair can produce a focal-length estimate, but it cannot strongly test the thin-lens model or the distance references. Use several real-image observations with object and image distances measured from the same lens reference plane. Rearranging gives 1/i=1/f−1/o, so plotting 1/i vertically against 1/o horizontally predicts slope −1 and intercept 1/f.

Learning outcome: Compare offsets 0, +1 and −1 cm at f=15 cm. Retain inferred focal length, slope and residuals. Explain a direct lens-plane calibration or independent geometry check that could challenge the offset hypothesis.

Starting recap

Define normal angles and signed object/image distances. Distinguish actual ray intersections from backward extensions, and keep the focal/critical boundaries explicit rather than forcing a finite answer.

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Try

Without added errors, f=15 cm and o=30,40,50,60 cm give i=30,24,21.4286,20 cm. Their reciprocal plot has slope −1 and intercept 1/15 cm⁻¹. Adding 1 cm to every measured i changes reciprocal values unequally, so it is not merely a vertical shift of that ideal line.

Open the experiment

Predict

Will a common +1 cm image-distance offset necessarily leave the fitted reciprocal slope unchanged?

Your prediction and reason

Experiment

Compare offsets 0, +1 and −1 cm at f=15 cm. Retain inferred focal length, slope and residuals. Explain a direct lens-plane calibration or independent geometry check that could challenge the offset hypothesis.

Challenge focal-length inference with multiple observations

Worked teaching model · synthetic values · Original synthetic object distances 30,40,50,60 cm. True image distance fo/(o−f), with fixed errors [0.10,−0.10,0.10,−0.10] cm plus common image-distance offset. Fit 1/di versus 1/do with intercept; predicted ideal slope −1. Positive measured image distances throughout. A position offset is not a constant reciprocal-axis offset.

True focal length: 15 cm Image-distance zero offset: 1 cm

Reset keeps your prediction, captures and practice history.

Axes rescale with settings; compare numerical values, not screen angles or apparent lengths. Solid/points show the selected model relationship or observations; the dashed comparison is explained below.

Calculated evidence ledger — synthetic teaching model
Quantity	Value	Unit
Fitted reciprocal-axis slope	-0.93266	ratio
Fitted reciprocal intercept	0.063288	cm⁻¹
Focal length inferred from intercept	15.801	cm
Largest reciprocal residual	0.0002469	cm⁻¹

Accessible plot coordinates

Primary plotted points
1 / measured object distance (cm⁻¹)	1 / measured image distance (cm⁻¹)
0.033333	0.032154
0.025	0.040161
0.02	0.044388
0.016667	0.047847

Dashed comparison coordinates
1 / measured object distance (cm⁻¹)	1 / measured image distance (cm⁻¹)
0.033333	0.032199
0.025	0.039972
0.02	0.044635
0.016667	0.047744

A thin-lens line should have slope −1 and intercept 1/f; a small residual alone does not certify distance references.

All image distances are measured from the assumed lens midline. A common image-position offset transforms nonlinearly after reciprocals, so it can bias both slope and intercept. Original fixed reading errors are separate from the selected offset. The source focal length is available for model testing, not evidence a real lens was measured.

A thin-lens line should have slope −1 and intercept 1/f; a small residual alone does not certify distance references.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Establish the model and evidence

The supplied original packet has four object distances and computed real images. Small fixed reading errors and a controllable common image-position offset are then added before taking reciprocals. A constant length offset does not become a constant offset on the reciprocal plot. It can bias slope, intercept and inferred f in different ways, even when a fitted line looks convincing.

A line-fitting routine will return a numerical slope and intercept even when systematic errors undermine the model. Compare the slope with −1, inspect residuals and independently verify the lens-plane reference. Real finite-thickness lenses can shift principal planes, and misidentifying the sharpest screen position adds another uncertainty. The synthetic packet does not assert any of those physical checks occurred.

Keep inference distinct from known model parameters. The true focal-length control is exposed so you can test your procedure; it would be unknown in a genuine experiment. A useful report preserves raw distances, transformations, fitted parameters, units and a comparison capable of separating focal error from a position zero error. No sunlight focusing or laser measurement is requested.

Worked evidence

Predict and investigate

Will a common +1 cm image-distance offset necessarily leave the fitted reciprocal slope unchanged?

Transfer to changed conditions

A fitted reciprocal line has slope −1 and intercept 0.040 cm⁻¹. Infer f and predict image distance for o=40 cm under the ideal model.

Compare the worked transfer

f=25 cm. Then 1/i=0.040−0.025=0.015 cm⁻¹, giving i≈66.67 cm and M≈−1.667. Agreement of one predicted point does not eliminate common calibration errors.

Explain the boundary

Why does a small residual not prove that the measured coordinate origin is correct?

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. Original text, rays and computed/synthetic records, not completed optical measurements. All investigations are virtual: no lasers, intense-light viewing, sunlight focusing, flames or eye tests are requested. Any real practical work needs qualified supervision and independent safety review.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Why does a small residual not prove that the measured coordinate origin is correct?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Locate thin-lens images with rays and signed distances Next: Explain focusing changes without a clinical-eye claim

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