Classical Dynamics of Particles and Systems · Problem 3-12

Nonlinear Oscillations and Chaos — Marion & Thornton

Physics — Mechanics · simple-pendulum

Problem

A simple pendulum consists of a mass mm suspended from a fixed point by a weightless, extensionless rod of length ll. Obtain the equation of motion and, in the approximation that sin⁡θ≅θ\sin\theta \cong \theta, show that the natural frequency is ω0=g/l\omega_0 = \sqrt{g/l}, where gg is the gravitational field strength. Discuss the motion in the event that the motion takes place in a viscous medium with retarding force 2mgl θ˙2m\sqrt{gl}\,\dot{\theta}.
Intuitive explanation

The key is recognizing that a pendulum's motion is governed by the restoring torque from gravity—which depends on the angle θ\theta, not the arc length. For small angles, sin⁡θ≈θ\sin\theta \approx \theta linearizes the equation into simple harmonic motion, revealing that the natural frequency depends only on the ratio g/lg/l. When damping enters through a viscous force proportional to angular velocity, you're looking at a driven harmonic oscillator problem where the damping coefficient determines whether the system oscillates, critically damps, or overdamps.

Intuitive explanation

The key is recognizing that a pendulum's motion is governed by the restoring torque from gravity—which depends on the angle θ\theta, not the arc length. For small angles, sin⁡θ≈θ\sin\theta \approx \theta linearizes the equation into simple harmonic motion, revealing that the natural frequency depends only on the ratio g/lg/l. When damping enters through a viscous force proportional to angular velocity, you're looking at a driven harmonic oscillator problem where the damping coefficient determines whether the system oscillates, critically damps, or overdamps.

Related theory

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