2.1. Simple Harmonic Motion (C1 - AHL)

SHM describes any system where a restoring force is proportional to displacement and directed toward equilibrium: F=−kx

This simple condition occurs in many real systems:

• Mass-spring systems

• Pendulums (small angles)

• Vibrating molecules

• Electrical circuits (LC oscillators)

• Even quantum harmonic oscillators!

Because many systems approximate this behavior when displacements are small, SHM serves as a universal model for analyzing oscillations in physics, engineering, and beyond.

SHM is defined by the condition that acceleration is directly proportional to negative displacement.

This arises from Newton's Second Law and Hooke's Law. 

This second-order differential equation defines SHM, leading to solutions like: x(t)=Acos⁡(ωt+ϕ)

The form ensures the motion is sinusoidal, with constant frequency and a restoring force pulling the system back toward equilibrium.

• Total mechanical energy in SHM is constant: E=Ek+Ep

• Energy oscillates between kinetic and potential, but their sum stays constant.

✅ Graphically:

• Displacement-time, velocity-time, and acceleration-time graphs are sinusoidal.

  • Displacement: x(t)=Acos⁡(ωt)

  • Velocity: v(t)=−Aωsin⁡(ωt)

  • Acceleration: a(t)=−Aω^2cos⁡(ωt)

• Energy-time graphs show:

  • Ek and Ep​ varying sinusoidally, out of phase.

  • Total energy as a horizontal line (constant).