Test where the small-angle approximation is useful

Ask how accurate the approximation must be

The small-angle period formula predicts the same period for every release angle at a fixed length. The full sine restoring law does not. At L = 1 m, compare the 10° and 60° settings and decide whether that difference matters for a task requiring period prediction within 1%.

The simple estimate stays at 2.006067 s. The nonlinear reference is approximately 2.009893 s at 10° and 2.152875 s at 60°. The larger-angle motion takes longer. The approximation underestimates period because replacing sin(theta) with theta overstates restoring acceleration away from equilibrium.

Use radians in the restoring law

Ten degrees is about 0.174533 radians; sixty degrees is about 1.047198 radians. At 60°, sin(theta) ≈ 0.866025, noticeably smaller than theta itself. In the full model, release angular acceleration is −(g/L) sin(theta); the linearized model uses −(g/L)theta. Compare those two entries in the ledger.

A force discrepancy at the release point is not itself the period discrepancy: the period depends on the whole motion. The nonlinear reference accounts for changing restoring force across the swing. It does not merely adjust the starting acceleration and assume it remains constant.

State the denominator and tolerance

The lab reports 100 × (Treference − Tsmall)/Treference, the percentage by which the approximation is below the reference. With 10°, this is about 0.190%; with 30°, about 1.711%; with 60°, about 6.819%. A calculation dividing by Tsmall would produce a different percentage, so name the denominator.

For a teaching target of at most 1% period error, explore the offered 5° increments. The largest offered angle meeting that criterion is 20°, with about 0.761% discrepancy. At 25° it exceeds 1%. This is a result for the stated model and discrete control choices, not a universal definition of “small angle.” A timing task with a different tolerance would choose a different range.

Now change length while keeping angle fixed. Both periods scale with √L, so the percentage discrepancy stays the same. Absolute error in seconds grows with length. Decide whether a specification is relative or absolute before choosing an approximation.

What the reference calculation does

The reference keeps the point-bob, fixed-pivot, no-drag assumptions but removes the small-angle replacement. Its period follows from the full sine restoring law and conservation of energy. The browser evaluates a smooth transformed integral using 256 fixed Simpson intervals, a bounded numerical sum.

Optional: inspect the nonlinear period integral

Let theta0 be the release angle in radians and phi an integration variable from 0 to π/2. The period is 4√(L/g) times the integral of 1/√(1 − sin²(theta0/2) sin²(phi)) over that interval. This is a complete elliptic integral. As theta0 tends to zero, the integrand tends to 1, recovering 2π√(L/g). The transformed form avoids an endpoint singularity in a direct angle-time integral.

The NIST pendulum reference gives the elliptic-integral period relation. Numerical verification compares the implementation with an independent arithmetic-geometric-mean relation across the supported settings. You do not need elliptic integrals to compare the two period predictions.

A numerical reference is not a measurement and its printed decimal is not exact. Integration error, small-angle approximation error, and physical-model error are different. Real length uncertainty, bob size, pivot friction and air resistance are absent from both model predictions. More numerical precision cannot repair an incorrect physical model.

Transfer to a measurement decision

A timing instrument resolves 0.02 s. At a 1 m length and 10° release, the model difference between the two periods is about 0.003826 s. Can one rounded timing reading reliably distinguish the predictions? Would this observation prove that the small-angle model is exactly correct?

Check the approximation transfer

The difference is smaller than the instrument’s stated resolution, so one rounded reading does not reliably distinguish the predictions. Agreement at that resolution does not prove exact equality. Timing many cycles, defining consistent start/end events and assessing random and systematic uncertainty could improve the comparison, but repeated readings do not automatically remove systematic errors.