Testing a pendulum with a length offset

Part (a) marking points

1. Calculate T by dividing each time by 20, then calculate T²: about 1.288, 1.689, 2.089, 2.497 and 2.894 s². (1 independent mark)
2. Plot T² / s² vertically against L / m horizontally, with labelled linear axes and a scale using most of the grid. (1 independent mark)
3. Plot the data accurately and draw a straight best-fit line without forcing it through the origin. (1 independent mark)

Credit equivalent correct explanations. Award each distinct point once; a contradiction invalidates the affected point.

Acceptable alternatives: Plot L against T² with the axes and later gradient calculation changed consistently.

Part (b) marking points

Final answer: 9.83 m s⁻². Numerical checks assess the final value only; they do not check working or the number of digits typed.

1. Use two well-separated points on the best-fit line; gradient about 4.02 s² m⁻¹. (1 method mark)
2. Use g = 4π²/gradient. (1 method mark)
3. Obtain a value in the accepted range 9.5–10.2 m s⁻². (1 accuracy mark)

Allow a defensible best-fit gradient within the stated range. After a plotting/arithmetic error, credit the correct relationship to the student’s own gradient; no full accuracy credit outside the range. Do not credit g = 4π² × gradient for these axes.

Acceptable alternatives: For L vertically against T², g = 4π² × gradient.

Part (c) marking points

Final answer: 0.0805 s². Numerical checks assess the final value only; they do not check working or the number of digits typed.

1. The effective length is L + 0.020 m. (1 method mark)
2. Use intercept = 4π² × 0.020 / 9.81 = 0.0805 s². (1 accuracy mark)

No earlier numerical result is required. Credit a valid alternative method. Accuracy credit requires a physically valid method.

Part (d) marking points

Assess the response as a whole. Indicative content:

• Measure pivot to bob centre; repeat with both length conventions and predict the intercept difference using the bob radius.
• Use a small measured angular amplitude and keep it similar; release without pushing.
• Time many complete cycles with a fixed fiducial point, repeat timings, and calculate a mean period with a repeat-based uncertainty.
• Use a wide range of measured lengths; plot T² against effective length and determine g from a best-fit gradient.
• Compare the fitted intercept with its uncertainty; use extreme acceptable slopes or regression uncertainty to assess uncertainty in g.
• Check pivot position, bob radius, timing/counting errors and amplitude dependence; repeating timings cannot remove a shared length bias.

Level 1 (1–2): Some relevant apparatus or timing steps, with little connection to analysis or the intercept.

Level 2 (3–4): A workable measurement and graph method, with an explained improvement or length-offset test. Analysis or uncertainty is incomplete.

Level 3 (5–6): A coherent, reproducible method links centre-of-bob length, timing and graph analysis to g and the intercept test, and evaluates the main uncertainties with justified improvements.

Judge the response as a whole against the levels. The indicative content is not a checklist of six marks. Choose the lower mark in a band for a borderline response and the upper mark for a response that consistently meets the descriptor. No relevant correct physics: 0 marks.