Periodic Motion & Simple Harmonic Oscillations Explained | Chapter 14 of University Physics

Chapter 14 delves into periodic motion and simple harmonic motion (SHM), covering how restoring forces produce oscillations, the mathematical models for springs and pendulums, energy exchange in oscillators, and the effects of damping and driving forces.

Watch the full video summary here for step-by-step derivations and examples.

Describing Periodic Motion

Periodic motion repeats in cycles, characterized by:

• Amplitude (A): Maximum displacement from equilibrium
• Period (T): Time for one complete cycle
• Frequency (f): Cycles per second (Hz), f = 1/T
• Angular frequency (ω): 2πf

Simple Harmonic Motion

SHM arises when a restoring force is proportional to displacement, as in Hooke’s Law:

F = –k x

The solutions for motion are sinusoidal:

• x(t) = A cos(ωt + φ)
• v(t) = –ω A sin(ωt + φ)
• a(t) = –ω² x(t)

Where ω = √(k/m), giving the period and frequency independent of amplitude:

T = 2π √(m/k), f = 1/T

Energy in SHM

Total mechanical energy in SHM oscillates between kinetic and potential:

E = ½ k A² = ½ m v² + ½ k x²

At displacement x, the speed is:

v = ±√(k/m) √(A² – x²)

Pendulums & Angular SHM

Small-angle pendulums approximate SHM:

• Simple pendulum: T = 2π √(L/g)
• Physical pendulum: T = 2π √(I/(m g d))

Angular SHM also applies where torque is proportional to angular displacement:

τ = –κ θ, ω = √(κ/I)

Damped & Forced Oscillations

Damping introduces an opposing force F = –b v, producing three regimes:

• Underdamped: Oscillations decay exponentially
• Critically damped: Returns to equilibrium fastest without oscillating
• Overdamped: Slow return without oscillations

Forced oscillations with driving frequency ω_d yield:

A = F_max / √[(k – m ω_d²)² + (b ω_d)²]

Resonance occurs when ω_d ≈ ω, producing large amplitudes in lightly damped systems.

Conclusion

Chapter 14 provides the framework for understanding oscillatory systems—from springs and pendulums to circuits and seismic waves. Master these concepts to analyze real-world periodic phenomena and design systems that exploit or mitigate resonance.

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