Energy Interchange and Phase in Oscillations

In simple harmonic motion the total mechanical energy E = ½kA² is constant, and it is continuously exchanged between kinetic energy ½mv² (maximum at equilibrium) and potential energy ½kx² (maximum at the extremes); phase describes the position within the cycle, and the velocity leads the displacement by a quarter of a cycle (π/2). In brief: at any displacement x, E_k = ½k(A² − x²) and E_p = ½kx². Both oscillate at twice the frequency of the displacement. The speed at displacement x is v = ω√(A² − x²). Doubling the amplitude quadruples the energy.

An oscillating system trades energy between two forms, like a pendulum between height and speed. This guide shows the energy of a spring-mass system and a pendulum, explains phase and phase difference, compares the energy at different points in a table, works through numerical examples and corrects common mistakes.

How is energy exchanged in SHM?

In a frictionless mass-spring system, the only forces doing work are the conservative spring force, so the total mechanical energy is conserved. The potential energy stored in the spring is E_p = ½kx², and the kinetic energy of the mass is E_k = ½mv². At the extremes (x = ±A), v = 0, so E = ½kA² is all potential. At the equilibrium position (x = 0), the potential energy is zero and all of the energy is kinetic: ½mv_max² = ½kA², which gives v_max = A√(k ÷ m) = Aω. At a general position, E_k = E − E_p = ½k(A² − x²), so v = ±ω√(A² − x²). For a pendulum, the exchange is between gravitational potential energy (maximum at the highest points) and kinetic energy (maximum at the lowest point). The total energy is proportional to the square of the amplitude, and is also E = ½mω²A².

What do the energy graphs show?

Against displacement, E_p is a parabola opening upward, E_k is an inverted parabola, and their sum is a horizontal line at the total energy E. The particle is confined to the region where E_k ≥ 0, which is |x| ≤ A (the turning points are where the potential curve meets the total energy line). Against time, E_k = E sin²ωt and E_p = E cos²ωt (starting from the extreme position): both are positive, oscillate at twice the frequency 2f of the motion (they reach their maxima twice per cycle) and are in antiphase. The time-average of each is E/2.

What is phase?

The phase of an oscillation is the argument (ωt + φ) of the sinusoidal function, which specifies the stage of the cycle. The phase difference between two oscillations of the same frequency is the difference of their phases, expressed in radians (or degrees, or as a fraction of a cycle). Two oscillations are in phase if the difference is 0 (or a multiple of 2π), and in antiphase if it is π. In SHM:

• the velocity leads the displacement by π/2 (a quarter of a period): when the displacement is at its maximum the velocity is zero, and when the displacement passes through zero the speed is at maximum;
• the acceleration is in antiphase with the displacement (phase difference π);
• two pendulums released together from the same position are in phase; if one is released a quarter of a period later, they have a phase difference of π/2. A phase difference of Δφ corresponds to a time delay Δt = (Δφ ÷ 2π) T.

Energy at different points

Worked example: energy of a spring oscillator

A 0.50 kg mass on a spring (k = 200 N/m) oscillates with amplitude 5.0 cm.

1. Total energy: E = ½kA² = ½ × 200 × 0.050² = 0.25 J.
2. Maximum speed: ½mv_max² = 0.25, so v_max = √(2 × 0.25 ÷ 0.50) = 1.0 m/s. Check: ω = √(200 ÷ 0.50) = 20 rad/s, so Aω = 0.050 × 20 = 1.0 m/s ✓.
3. At x = 3.0 cm: E_p = ½ × 200 × 0.030² = 0.090 J. E_k = 0.25 − 0.090 = 0.16 J. Speed: v = √(2 × 0.16 ÷ 0.50) = 0.80 m/s (check: ω√(A² − x²) = 20 × 0.040 = 0.80 m/s ✓).

4. At x = A/2 = 2.5 cm: E_p = ½ × 200 × 0.025² = 0.0625 J = E/4, and E_k = 0.1875 J = 3E/4.
5. Doubling the amplitude: to 10 cm, the energy is 4 × 0.25 = 1.0 J and v_max = 2.0 m/s.
6. Phase and time: T = 2π ÷ 20 = 0.314 s. A phase difference of π/2 corresponds to a time difference of T/4 = 0.0785 s. The velocity is maximum 0.0785 s after the mass is at the extreme.

Why is this important?

The exchange between potential and kinetic energy is used to analyze oscillating systems from molecules to bridges and electrical circuits (in an LC circuit, energy oscillates between the capacitor and the inductor). Quantum mechanics describes the vibrations of molecules with discrete energy levels of a harmonic oscillator, E = (n + ½)hf, the zero-point energy ½hf being nonzero even at absolute zero. Max Planck’s 1900 treatment of oscillators in a black body began the quantum theory.

Which mistakes are common?

Mistake 1: Saying that energy is lost at the turning points

It is all potential there. The total is unchanged.

Mistake 2: Thinking that the kinetic energy has the same frequency as the displacement

The energies oscillate at twice the frequency.

Mistake 3: Mixing up phase leads and lags

The velocity leads the displacement by π/2; the displacement lags the velocity.

Mistake 4: Forgetting damping in real systems

Real oscillations lose energy to friction. SHM with constant amplitude is an idealization.