Harmonic Oscillator & Phase Space Explorer

Drag Mass or Phasor Vector directly to perturb state

Physical Parameters Active

Mass (m) 1.00 kg

Spring Constant (k) 4.00 N/m

Damping Ratio (γ) 0.00 s⁻¹

Derived State & Kinematics

Angular Frequency (ω) 2.00 rad/s

Natural Period (T) 3.14 s

Position x(t) 1.06 m

Velocity v(t) -2.12 m/s

Trajectory Classification closed_ellipse

Mechanical Energy Partition

Kinetic (E_k)

0.00 J

Potential (E_p)

0.00 J

Total (E_tot)

4.50 J

Demonstrates continuous phase shift energy balance: Total energy $E = \frac{1}{2}kA^2$ transfers losslessly between elastic potential and kinetic states under zero damping ($\gamma = 0$).

Trace position, velocity and the damping envelope

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The simple harmonic preset has mass one and stiffness four. Angular frequency is square root of stiffness divided by mass: two radians per second. Its period is two pi divided by two, or pi seconds. With amplitude one point five and initial phase pi over four, initial position is about one point zero six one and initial velocity minus two point one two one. The source uses these analytic cosine and sine formulas. The source damped preset uses amplitude one point eight, angular frequency two and damping parameter zero point four. Its envelope is amplitude times exponential minus zero point four times time. At time zero it is one point eight; at one second about one point two zero seven; at two seconds about zero point eight zero nine. The damped oscillation angular rate is square root of four minus zero point one six, about one point nine six. Envelope and instantaneous position are different quantities. For simple harmonic mass one, stiffness four and amplitude one point five, total energy is one half times stiffness times amplitude squared, or four point five joules. At maximum displacement potential energy is four point five and kinetic zero. At the center speed is amplitude times angular rate, three, making kinetic energy one half times nine, also four point five. The original analytic CSV and phase-space display describe this educational oscillator; drag actions alter its initial state.