05 / CHAPTER 02

Phase space and energy

See oscillation as a closed orbit in position–velocity space.

Oscillations & Waves: original course diagram — A visual model for this course. Read it as a prompt for the assumptions behind the chapter.

ANCHOR EQUATION

Oscillator energy

E = \frac{1}{2} m ẋ^{2} + \frac{1}{2} k x^{2}

In symbols: E = ½mẋ² + ½kx²

Kinetic and elastic potential energy trade while total energy stays fixed in the ideal model.

Phase space displays energy exchange

An oscillator’s state is the pair (x,v). Plotting v against x shows whether motion is periodic, damped, or driven; the curve is a geometric record of energy.

Rewrite energy as a curve

For an ideal oscillator E = ½mv² + ½kx². Rearranging gives v²/(2E/m) + x²/(2E/k) = 1, an ellipse in x–v space. At x = ±A, v = 0; at x = 0, speed is maximum. The motion moves around the ellipse as time advances.

For m = 1 kg, k = 4 N/m, and A = 0.5 m, E = ½(4)(0.25) = 0.50 J. At x = 0.3 m, v = √[(2E − kx²)/m] = √(1 − 0.36) = 0.80 m/s.

Damping spirals inward

If energy decreases, successive phase-space loops shrink toward the equilibrium point. A straight line or noisy cloud indicates that the harmonic model or measurement is inadequate.

Work it: Sketch the phase-space ellipse for E = 2 J, m = 0.5 kg, k = 8 N/m. Mark turning points and maximum-speed points with coordinates.