Simple Harmonic Motion
The differential equation this whole course has been building toward: \(\dfrac{d^2x}{dt^2} = -\omega^2 x\), whose solution — verified by substitution, the same technique Unit 102 used for the drag equation — is \(x(t)=A\cos(\omega t+\phi)\). Everything about SHM (period, energy conservation, the spring's \(\omega=\sqrt{k/m}\)) follows from this one equation and its solution, rather than a list of separate formulas.
What you'll learn
- Verify that x(t)=A cos(ωt+φ) satisfies the SHM differential equation d²x/dt²=−ω²x by direct substitution
- Connect a physical system's restoring force (e.g., a spring, F=−kx) to the differential equation it obeys, and identify ω from it
- Differentiate x(t) to find velocity and acceleration at any point in the cycle
- Show, using calculus, that total mechanical energy in SHM is constant
1. The SHM Differential Equation and Its Solution
A system is in simple harmonic motion whenever its position obeys \(\dfrac{d^2x}{dt^2}=-\omega^2x\) — acceleration always proportional to position, always pointing back toward \(x=0\). In a first course, the standard approach isn't to derive the solution from scratch, but to verify that a proposed solution, \(x(t)=A\cos(\omega t+\phi)\), actually satisfies the equation — the same substitution-and-check technique Unit 102 used for the drag equation.
2. Connecting a Physical System to Its ω
For a mass on a spring, Newton's second law gives \(m\dfrac{d^2x}{dt^2}=-kx\), i.e., \(\dfrac{d^2x}{dt^2}=-\dfrac{k}{m}x\). Comparing this to the standard form \(d^2x/dt^2=-\omega^2x\) identifies \(\omega=\sqrt{k/m}\) directly — not a separate formula, just matching coefficients between the physical equation and the general one.
3. Energy Conservation in SHM, Shown by Calculus
Total mechanical energy, \(E=\tfrac{1}{2}mv^2+\tfrac{1}{2}kx^2\), can be shown to be constant directly: substitute \(x(t)=A\cos(\omega t)\) and \(v(t)=dx/dt\), use \(k=m\omega^2\), and differentiate \(E\) with respect to time. Every term cancels, giving \(dE/dt=0\) — energy conservation isn't assumed here, it's a consequence of the equation of motion.
Key equations
- d²x/dt² = −ω²x — The defining differential equation of simple harmonic motion — any system obeying this equation oscillates sinusoidally.
- x(t) = A cos(ωt + φ) — The general solution to the SHM differential equation — verified by substitution, not derived from scratch in a first course.
- ω = √(k/m) — For a mass on a spring, substituting F=−kx into F=ma=m d²x/dt² gives this specific ω that makes the differential equation match.
- E = ½mv² + ½kx² = ½kA² (constant) — Total mechanical energy in SHM — shown via calculus to be constant (dE/dt=0) by substituting x(t) and v(t)=dx/dt and differentiating.