Small-amplitude pendulum measurements give these (recorded length L in m, period squared T² in s²) pairs: (0.25, 1.036), (0.5, 2.016), (0.75, 2.996), (1, 3.976). Treat the listed central values as exactly collinear. The model is T² = (4π²/g)L_true, but L_true may equal L + δ because of a constant length offset. (a) Determine the gradient, g and δ. (b) Explain how timing 20 oscillations reduces fractional timing uncertainty and identify the correct endpoints for measuring pendulum length.

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Gradient 3.92 s² m⁻¹; g = 10.07 m s⁻²; δ = +0.01429 m. Measure pivot to bob centre; time repeated oscillations. The straight-line gradient is ΔT²/ΔL = (3.976 − 1.036)/(1 − 0.25) = 3.92 s² m⁻¹. Use separated points rather than dividing one T² by one L. Comparing with T² = (4π²/g)L + (4π²/g)δ gives g = 4π²/gradient = 10.07 m s⁻². The intercept is 0.056 s². Since intercept = gradient × δ, δ = 0.01429 m. Its positive sign means the recorded length is smaller than the effective pivot-to-centre length. For roughly fixed start/stop reaction-time uncertainty, timing 20 cycles makes the measured duration about twenty times longer. Dividing that duration by 20 therefore reduces the fractional uncertainty of the period. Measure from the pivot to the bob's centre, keep the release angle small, and repeat timings. A nonzero intercept is consistent with a length offset but does not, by itself, prove that this is the only systematic error.

Determine gravity from pendulum data · set 8

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Pendulum observations

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Pendulum observations — supplied values
Recorded length L / m	Period squared T² / s²
0.25	1.036
0.5	2.016
0.75	2.996
1	3.976