Applied mathematics and modelling · Maths EE idea · Solid

How small is a small angle?

A research question to start from

How does the period of a pendulum depend on amplitude, and at what amplitude does the small-angle approximation become unacceptable?

A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.

Why it works as a maths EE

An exact integral against an approximation, with a clear threshold to decide.

Mathematics you would need

Second-order ODEs
Energy methods
Elliptic integral (numerically)
Series approximation

Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.

One possible line of attack

Derive the small-angle period.
Derive the exact period integral.
Compute and compare, adding the first series correction.

Scope and difficulty

Solid. Solid; the experiment is optional.

Pitfalls

Physics lab report.
Exact integral quoted without derivation.

Where to start reading

Search a library catalogue or a university's open lecture notes for: pendulum period amplitude exact integral; small angle approximation error. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).

Make it your EE

Plan this EEOpens the free EE planner with this question as a starting point. Sharpen the questionSix quick checks for a focused research question. Self-check a draftCheck a section against each criterion, free.