Unit 7 · 7.4 — Energy of Simple Harmonic Oscillators

Everything to learn here

• Concept: In an undamped ideal oscillator, mechanical energy is perfectly conserved, continuously transferring back and forth between maximum kinetic energy (at equilibrium) and maximum potential energy (at the amplitude).
• Concept: Because total energy depends on the square of the amplitude, doubling the amplitude of a spring oscillator quadruples the total mechanical energy of the system.
• Equation: $$E_{t}otal=\frac{1}{2}k{A}^2\text{(Total Energy constraint of a horizontal spring)}$$
• Equation: $$\frac{1}{2}k{A}^2=\frac{1}{2}m{v}^2+\frac{1}{2}k{x}^2\text{(Energy conservation expanded at any intermediate position where E = K + U)}$$
• Vocabulary: Energy "Seesaw" — The conceptual framework of energy trading 100% back and forth between maximum K (at equilibrium) and maximum U (at the extremes)
• Skill: Mathematical Routines — Substituting the energy constraint equation to calculate maximum speed (v_max = A*sqrt(k/m)) using algebraic substitution rather than calculus derivatives
• Skill: Creating Representations — Draw precise energy bar charts for a block-spring system at x = A, x = 0, and x = -A to visually demonstrate the constant conservation of total mechanical energy across all positions
• Equation: $$Determinesystemmassbyrelatingtheoscillationperiodtotheratioofmaximumforceanddisplacementusingm=(t0/\pi {)}^2\cdot \text{(Fmax/x0)}$$
• Concept: Understand that total energy E = 1/2*k*A^2 is conserved, shifting between maximum potential energy at endpoints and maximum kinetic at equilibrium.

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