Physics

SHM and resonance

Springs and pendulums, energy swap, damping, driving, and resonance.

Basics

Simple harmonic motion

If the restoring force is proportional to displacement and opposite (Hooke), the motion is sine or cosine. A spring has ω = √(k/m); a small-angle pendulum √(g/ℓ). Period T=2π/ω does not depend on amplitude in the ideal case. Phase says when the peak occurs. Real springs go nonlinear at large amplitude. Clocks and the basics of sound sit on this motion.

Energy exchange

Potential (1/2)kx² and kinetic (1/2)mv² trade while their sum stays (1/2)kA² if there is no friction. Speed is greatest in the middle and zero at the ends. Phase space traces an ellipse. Damping makes a spiral into the origin. Measured amplitude scales as the square root of energy.

Damping

A drag proportional to speed makes amplitude decay exponentially. Underdamping oscillates as it dies; overdamping returns without oscillating; critical damping settles fastest (a door closer). High Q rings longer and has a sharp resonance. Car suspensions are tuned near critical.

Driving and resonance

A periodic drive eventually makes the system follow that driving frequency. Near the natural frequency the amplitude grows — resonance. Tacoma Narrows and shattering glass are the folklore. Damping lowers and widens the peak. Radio tuning and NMR in MRI pick a resonance too. Large amplitude leaves the linear model.

Formulas

Spring angular frequency

ω = √(k / m), T = 2π / ω

Heavier mass or weaker spring → slower.

Symbols

k spring constant
m mass

Simple pendulum (small angle)

T ≈ 2π √(ℓ / g)

θ≪1 rad. Mass drops out.

Symbols

ℓ length
g gravitational acceleration

Displacement

x = A cos(ω t + φ)

A amplitude, φ initial phase.

Symbols

φ phase constant

SHM energy

E = ½ k A² = ½ m v_max²

Conserved if there is no friction.

Symbols

A amplitude

Key table

1 Hz	one cycle per second. ω = 2π f
Resonance hazard	bridges, buildings, shafts keep natural frequencies off the working ones
Q	roughly energy / energy lost per cycle; high Q is a sharp peak