Experiment Design & Estimation (IPhO Experimental)
IPhO experimental lesson on planning: choosing what to measure, setting ranges, controlling variables, and using estimates to design efficient procedures.
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In IPhO experiments, you are not just “collecting data”: you are choosing what to measure so the parameter appears cleanly (often as a slope), and you are choosing ranges and methods so uncertainty stays small. Estimation is the hidden tool that makes a plan work on the first try.
1. Definitions (Must Know)
- Design variable: what you can choose (range, number of points, instrument, method).
- Signal: the change in measured quantity caused by changing the variable of interest.
- Noise: random scatter or measurement uncertainty.
- Sensitivity: how strongly the measured quantity changes with the parameter (often the slope).
- Dynamic range: the useful range of a sensor or method (before saturation/nonlinearity).
- Pilot measurement: one quick trial to estimate scales (then refine the plan).
2. Key Ideas (What Earns Marks)
- State the model and show the linear form you want.
- List what you will measure and with what instrument (including uncertainties).
- Choose a range that makes the change in the measured variable much larger than noise.
- Say what you will keep constant and how.
- Include repeats where scatter matters, and one “sanity check” measurement.
3. Detailed Explanations
A. The “slope-first” design pattern
Whenever possible, design the experiment so the desired parameter is obtained from a gradient. Gradients are robust to offsets and are easy to defend with uncertainty estimates.
Workflow:
- write the model with parameters
- rearrange to Y = mX + c
- choose X you can control over a wide range
- measure Y with good resolution
B. Use estimation to pick ranges and instruments
Before collecting lots of points, do a rough estimate:
- expected magnitude of Y
- expected change Δ Y over your X range
- expected uncertainty in Y
If Δ Y is comparable to uncertainty, expand the range, change the method, or choose a different linearisation.
C. Choosing number of points and repeats
Rule of thumb:
- many points reduce slope uncertainty if scatter is random
- repeats reduce random uncertainty in each point
If setup time is expensive, fewer points with repeats may beat many points with no repeats.
4. Common Mistakes
- Choosing a tiny range so the slope is dominated by noise.
- Collecting lots of points with a bad transform (wrong Y vs X choice).
- Not stating control variables, then blaming “human error”.
- Using an instrument at its limit (resolution too coarse, or saturating).
- Making a plan that needs precision alignment but providing no alignment method.
5. Exam Tips
- Start your method section with: “We aim to plot Y against X so that Y = mX + c.”
- Write one sentence for each: measure, control, repeat, calculate, plot, conclude.
- Include a quick “pilot test” note if the plan depends on an unknown scale.
6. Worked Examples (use Toggle)
1) Design: measure g using a pendulum, choose L to improve precision
For a simple pendulum, T = 2π square root of (L/g) so T² = (4π²/g)L.
Design choices:
- pick several lengths L over a wide range (e.g. 0.3 m to 1.0 m) to make T² change significantly
- time many oscillations (e.g. 20 periods) to reduce relative timing uncertainty
Why long lengths help: if your timing uncertainty is roughly a fixed reaction time per timing event, larger T means smaller percentage uncertainty in T.
Mark-winning sentence: “We time 20 oscillations and divide by 20 to reduce the fractional timing uncertainty.”
2) Design: measure resistivity using a wire, avoid a hidden offset
Model for a uniform wire: R = ρ L/A.
If contact resistance adds a constant offset R₀, then
Design: measure R for several lengths L (same wire, same diameter), plot R against L.
- slope m = ρ/A gives ρ = mA
- intercept R₀ accounts for contact resistance automatically
This is a classic “slope-first” design: it converts a systematic offset into a harmless intercept.
7. Mind Stretchers (use Toggle)
1) Tradeoff: more points or more repeats?
You can take 12 minutes of data. Each “new X value” takes 2 minutes to set up, but you can repeat measurements at the same X quickly.
Strategy A: 6 distinct X values, 1 measurement each.
Strategy B: 3 distinct X values, 3 repeats each.
Which strategy gives a smaller slope uncertainty if scatter dominates? Which is better if drift (slow systematic change) dominates?
8. Practice
- You want to determine the spring constant k by measuring extension x under load F. What plot gives k as a slope? What systematic error becomes an intercept?
- You suspect I ∝ 1/r². Propose an X range for r that makes the change in I large compared to a 2 percent sensor uncertainty.
- You want to measure a time constant τ in an exponential decay. What linearisation would you use?
Back to IPhO Experimental Skills Hub
Answer key and feedback
- Plot F against x so the slope is k. A constant zero-force or zero-extension offset appears as a nonzero intercept.
- For q = q₀e^(-t/τ), plot ln q against t; the slope is -1/τ and the intercept is ln q₀.
- Widely separated X values reduce slope uncertainty when random scatter dominates. Interleaving or reversing the measurement order is safer when slow drift dominates.
Syllabus and review details
No official syllabus alignment is listed for this lesson.