Simple Harmonic Motion – Springs, Pendulums and Oscillations

Introduction

Simple harmonic motion (SHM) is a special kind of oscillation that appears in springs, pendulums, molecules and even electrical circuits. Understanding SHM gives you insight into waves, quantum mechanics and countless laboratory instruments.

Background / Prerequisites

• Newton’s laws of motion.
• Basic idea of restoring force.
• Simple trigonometric functions.

Core Concepts

• SHM means acceleration is proportional to displacement and directed toward equilibrium.
• The general solution is

\[ x(t) = A \cos(\omega t + \phi). \]

Detailed Explanation

Mass-spring system

• Hooke’s law: \(F = -k x\).
• Equation of motion: \(m \ddot{x} = -k x\) leading to \(\ddot{x} + (k/m) x = 0\).
• Angular frequency \(\omega = \sqrt{k/m}\).
• Time period \(T = 2\pi \sqrt{m/k}\).

Simple pendulum (small-angle)

• Restoring torque is proportional to displacement angle.
• For small angles: \(T = 2\pi \sqrt{l/g}\).

Energy view

• Total energy remains constant: \(E = \tfrac{1}{2} k A^2\).
• Energy oscillates between kinetic and potential forms.

Examples / Applications

• Spring on a horizontal surface acting as a vibration absorber.
• Pendulum clocks keeping time through SHM.
• Lattice vibrations in solids and phonons modeled as coupled oscillators.

Common Mistakes & Tips

• Using SHM formulas for large pendulum angles (>15 degrees).
• Forgetting that restoring force must be proportional to displacement.
• Confusing angular frequency \(\omega\) with frequency \(f\).

Summary / Key Takeaways

• SHM models many physical systems accurately near equilibrium.
• Time period depends on system parameters like mass, spring constant or length.
• Energy shuttles smoothly between kinetic and potential forms.

Further Reading / Related Topics

• Energy in SHM.
• Damped and driven oscillations.
• Waves and resonance.