Is an elegant approximation always accurate?
Is an elegant approximation always accurate?
- Convert 10° to radians before estimating error.
- Compare 4° and 14° release angles.
- Estimate whether a 0.1% timing target is plausible.
What to measure
Units and parameter definitions are shown beside every laboratory control and live reading.
The idea underneath
The small-angle pendulum replaces sinθ with θ. The first neglected term is −θ³/6, so relative force error grows roughly as θ²/6. Even when an approximation looks visually convincing, that error may matter for precision timing. This lab deliberately stays below 15° and reports the small-angle period. It does not pretend to show nonlinear large-angle motion. Compare the predicted correction with the accuracy required by a proposed measurement.
sinθ=θ−θ³/6+… · T/T₀≈1+θ₀²/16Units and parameter definitions are shown beside every laboratory control and live reading.
Build the argument
- Write a prediction before running the experiment.
- Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
- Compare the observations to an analytic limit and document the remaining discrepancy.
Worked example 1
At 10°, θ=0.1745 rad and θ²/16≈0.19%.
Worked example 2
At 4°, the leading period correction is about 0.030%.
Applications
A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.
Common misconception
More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.
A further investigation
The argument of sin in a power series must be in…
Check your understanding
The argument of sin in a power series must be in…
- Radians
- Degrees
- Metres
Explanation
The series coefficients assume a dimensionless radian argument.
Model limits
A force changes velocity, not just speed. A massless rod or ideal spring is a model assumption, and small-angle pendulum formulas have a limited range.