O-Level Physics Graphing & Linearisation Clinic

> **TL;DR**\
> Paper 3 expects clean tables, full-page graphs, and linearised relationships that feed ACE analysis (Section 4, [SEAB 2026 syllabus](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)).\
> Make every plot a straight line by choosing the right axes (reciprocals, logs, gradients/intercepts tied to physical constants).\
> Always include raw points, best-fit line, large gradient triangle, and uncertainty commentary.

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## 1 | Graphing checklist before you touch the ruler

- Use at least half of the grid in both x and y directions.\\
- Mark raw points with small crosses; no thick dots.\\
- Label axes with quantity + unit (e.g. `1/u / cm⁻¹`).\\
- Choose a sensible scale (multiples of 1, 2, 5 × powers of ten).\\
- Draw a thin best-fit line; no connecting-the-dots.\\
- Sketch a large triangle covering ≥2/3 of the line for gradient.

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## 2 | Common Paper 3 relationships and how to linearise them

| **Experiment**   | **Original relationship**                                   | **Linear form**                         | **Gradient meaning**                   | **Intercept meaning**                        |
| ---------------- | ----------------------------------------------------------- | --------------------------------------- | -------------------------------------- | -------------------------------------------- |
| Lens equation    | ( \frac{1}{f} = \frac{1}{u} + \frac{1}{v} )                 | Plot (1/v) vs (1/u)                     | Gradient = −1                          | Intercept on (1/v) axis = (1/f)              |
| Spring constant  | \\( F = kx \\)                                              | Plot F vs x                             | Gradient = k                           | Passes through origin if no systematic error |
| Resistivity      | (R = \rho L / A)                                            | Plot R vs L                             | Gradient = (\rho / A)                  | Intercept ≈ contact resistance               |
| Cooling (Newton) | (T = T\_\text{ambient} + (T_0 - T\_\text{ambient}) e^{-kt}) | Plot (\ln (T - T\_\text{ambient})) vs t | Gradient = −k                          | Intercept = (\ln(T_0 - T\_\text{ambient}))   |
| Wave speed       | (v = f \lambda)                                             | Plot v vs f (if λ fixed) or λ vs (1/f)  | Gradient = λ (first case)              | Intercept ideally 0                          |
| Inverse square   | (I \propto 1/d^2)                                           | Plot I vs (1/d^2)                       | Gradient ∝ luminous intensity or power | Intercept = background light                 |

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## 3 | Worked example — converging lens

**Scenario:** You record object distance u (cm) and image distance v (cm) for a thin lens.\
**Steps:**

1. Compute 1/u and 1/v (cm⁻¹) with appropriate significant figures.\\
2. Plot 1/v (y-axis) against 1/u (x-axis), scale 0.00–0.12 cm⁻¹.\\
3. Draw the best-fit line; add a large triangle to compute gradient \\( ≈ −1 \\).\\
4. The y-intercept gives 1/f. If intercept = 0.050 cm⁻¹, focal length f = 20.0 cm.\\
5. ACE commentary: discuss scatter, lens alignment, and reading precision; suggest improvements (longer bench, blackout curtaining).

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## 4 | Worked example — cooling curve linearisation

**Scenario:** Hot water cools from 70 °C to ambient 26 °C.\
**Steps:**

1. Subtract ambient temperature from each reading (T − T_a).\\
2. Take natural log values using a calculator or spreadsheet.\\
3. Plot ln(T − T_a) vs time (minutes).\\
4. Gradient magnitude gives k (cooling constant); intercept shows ln(T_0 − T_a).\\
5. ACE commentary: note if later points deviate (heat loss to surroundings); propose insulation improvements.

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## 5 | Gradient and intercept uncertainty

- Choose two points on the best-fit line as far apart as possible.\\
- Calculate gradient using triangle sides (Δy/Δx).\\
- For uncertainty, use max/min lines method (if taught) or quote percentage spread from repeated runs.\\
- Intercept uncertainty can be estimated by propagating gradient and point coordinate uncertainties.\\
- Always state units: gradient of I vs 1/d² has units W·m² if current converted to luminous power.

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## 6 | ACE-ready commentary phrases

- “Gradient magnitude of 0.050 ± 0.003 cm indicates f = 20.0 ± 1.2 cm, matching manufacturer spec within 6 %.”\\
- “Negative intercept suggests 0.12 A of background current; subtracting this improves linear fit.”\\
- “Log plot deviated after 12 min, implying convective drafts; repeating with a lid should tighten the line.”

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## 7 | Drills to try with past papers

1. Re-plot any curved data question using the correct linear axes; annotate gradient/ intercept meaning.\\
2. Take an old Paper 3 graph that used centimetre paper; recreate it digitally, then compare gradients.\\
3. Swap datasets with a friend and critique each other’s plots for label/scale errors.\\
4. Build a gradient/ACE sentence bank: one for every major practical topic.

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## Sources

1. [SEAB, GCE O-Level Physics (6091) Syllabus 2026](https://www.seab.gov.sg/files/O%20Lvl%20Syllabus%20Sch%20Cddts/2026/6091_y26_sy.pdf)