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Waves & Optics · Physics

Simple harmonic motion is the template for oscillations.

When the restoring force is proportional to displacement, motion becomes sinusoidal and energy swaps cleanly between kinetic and potential forms.

This topic

Simple Harmonic Motion

Build the core model: restoring force, equations of motion, period and angular frequency, and energy in SHM.

Model
Restoring force and equilibrium position
SHM begins with equilibrium: the net force is zero there. Near equilibrium, many forces behave approximately like linear restoring forces.
  • Equilibrium: where net force vanishes
  • Displacement measured from equilibrium
  • Restoring force points back toward equilibrium
  • “Small oscillations” idea
Start nano-lesson Define equilibrium before writing equations.
Forces
Hooke’s law and linear restoring forces
Hooke’s law is the simplest linear restoring force. Many systems reduce to an effective Hooke-like force near equilibrium.
  • Hooke’s law form and meaning
  • Spring constant as stiffness
  • Linear approximation for small displacement
  • When the approximation breaks
Start nano-lesson Linear force → sinusoidal motion.
Dynamics
SHM equations of motion
The defining differential equation has solutions that are sinusoidal. The key is not memorizing forms, but recognizing the SHM structure.
  • Second-order equation viewpoint
  • Position, velocity, acceleration relationships
  • Phase and amplitude interpretation
  • Initial conditions set the phase
Start nano-lesson Recognize the SHM pattern quickly.
Timing
Period and angular frequency
Period T, frequency f, and angular frequency ω describe how quickly oscillation repeats. ω is the natural constant that appears in the equations.
  • Relationships among T, f, and ω
  • How stiffness and mass affect ω
  • Interpret ω as “radians per second”
  • Common unit checks
Start nano-lesson ω is the most natural parameter.
Energy
Energy in SHM (kinetic and potential)
SHM is an energy exchange: kinetic energy peaks at equilibrium, and potential energy peaks at turning points. Total energy is constant (without damping).
  • Turning points: v = 0
  • Equilibrium: maximum speed
  • Energy swapping over a cycle
  • How amplitude sets total energy
Start nano-lesson Energy view reduces memorization.
Practice
Practice & Exercises
Practice identifying SHM structure, interpreting graphs, and connecting parameters to physical meaning.
  • Recognize SHM from force–displacement info
  • Compute T, f, ω from system parameters
  • Energy and amplitude questions
  • Graph interpretation: x(t), v(t), a(t)
  • Exam-style SHM mixed sets
Open practice Start with a diagram and definitions.