Forces and the equation of the pendulum

A simple pendulum is a point mass $m$ hanging from an inextensible string of length $\ell$, free to swing in a vertical plane. To write its equation of motion we draw the force diagram on the mass and resolve the weight along two convenient directions: the one along the string (balanced by the tension $\vv T$) and the one tangent to the trajectory (responsible for the motion).

The weight $m\vv g$ resolved along the string and along the tangent. Only the tangential component $-mg\sin\theta$ moves the mass; the tension $\vv T$ and the radial component of the weight balance each other (except for the centripetal force).

The tangential component of the weight is $F_t = -m g\sin\theta$ (negative because it pulls back towards equilibrium). Calling $s = \ell\theta$ the arc length travelled, Newton’s second law along the trajectory is $m\ddot s = -mg\sin\theta$, i.e. $m\ell\ddot\theta = -mg\sin\theta$. The mass cancels out and we are left with

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

This equation is not yet that of a harmonic oscillator: the term $\sin\theta$ makes it non-linear. The motion is nonetheless periodic, but to obtain harmonic motion one more step is needed: linearisation for small oscillations.

Collegamenti

Argomenti: Oscillations and harmonic motion Concetti: Simple pendulum · Weight force Competenze: Free-body diagram · Vector resolution Oggetti: Simple pendulum

Esercizi collegati: Problem — Length of a pendulum that beats seconds · Problem — Lead and cork pendulums · Problem — Pendulum clock and temperature