Damped Spring-Mass Oscillator | Interactive Hooke's Law Simulation

Physical Oscillator

↕ Click & Drag mass to displace

Velocity Vector (v) Spring Force (Fs) Damping Force (Fd) Scale Ruler

E = Ek + Ep

Kinetic Energy (0.5·m·v²) 0.00 J

Potential Energy (0.5·k·x²) 33.75 J

Total Mechanical Energy 33.75 J

Space = Pause

Playback Speed:

m Mass 2.0 kg

k Spring Constant 30 N/m

c Damping Coefficient 1.5 Ns/m

x₀ Initial Displacement +1.50 m

Underdamped Δ = -237.75

Oscillatory decay with exponential envelope (ζ < 1, Δ < 0). System oscillates across equilibrium with decaying amplitude.

Damping Ratio ζ

0.19

Natural Frequency ω₀

3.87 rad/s

Displacement x

+1.50 m

Velocity v

0.00 m/s

Acceleration a

-22.50 m/s²

Frequency f₀

0.62 Hz

▾ Details

The equation of motion for a damped spring-mass oscillator is given by Newton's Second Law combined with Hooke's Law and viscous fluid resistance:

m·x'' + c·x' + k·x = 0

Dividing by mass gives the standard canonical form:

x'' + 2ζω₀·x' + ω₀²·x = 0

Where: