Small oscillations and the period of the pendulum

The pendulum equation $\ddot\theta = -\dfrac{g}{\ell}\sin\theta$ becomes manageable if the angle is small. For $\theta \ll 1$ rad (roughly $10^\circ$ or less) the approximation $\sin\theta \approx \theta$ holds, and the equation becomes linear:

$\ddot\theta = -\frac{g}{\ell}\,\theta$

This is now exactly the form of simple harmonic motion $\ddot\theta = -\omega^2\theta$. Comparing coefficients we read off the angular frequency and the period:

$\ev{\omega = \sqrt{\frac{g}{\ell}},\qquad T = 2\pi\sqrt{\frac{\ell}{g}}}$

Principle — Period of the simple pendulum

The period of a simple pendulum of length $\ell$ does not depend on the mass, nor on the amplitude (for small oscillations), but only on the length of the string and the local gravity (French 1971). See Riferimenti bibliografici.

Galileo in Pisa

Two pendulums with wooden and lead balls, of equal length, swing in sync: the mass is irrelevant, it is $\sqrt{\ell/g}$ that governs. It is the same isochronism that makes the pendulum an excellent regulator for clocks.

The fact that $T$ depends on $g$ has a notable practical consequence: the pendulum is a gravimeter. For a given length $\ell$, measuring the period lets us work back to the local value of $g$ — on Earth, on the Moon, or on another planet.

Collegamenti

Argomenti: Oscillations and harmonic motion Concetti: Simple pendulum · Period and frequency Competenze: Small-oscillation approximation Metodi: Linear approximation of small oscillations Oggetti: Simple pendulum

Esercizi collegati: Problem — Length of a pendulum that beats seconds · Problem — Amplitude and period of a swing · Problem — Ranking pendulums