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Dynamics

Frictionless harmonic oscillator

Observe the spring’s restoring force, the motion and energy conversion together. In the frictionless model, total mechanical energy is conserved.

Which energy is greatest when the mass passes through equilibrium?

Spring–mass motion

Ready

02 / LIVE MEASUREMENTS

Position x—m
Velocity v—m/s
Acceleration a—m/s²
Net force—N
Damping coefficient b—kg/s
Damping regime

Position versus time

Change the parameters

Calculated period:

Total physical time:

One cycle is one full period: . Changing mass or stiffness changes both period and total duration. Playback speed only slows the display.

Kinetic, potential and total energy

Kinetic energy KPotential energy UMechanical energy E
Kinetic energy K—J
Potential energy U—J
Mechanical energy E—J
Transferred energy E₀ − E—J

K: solid blue line; U: dashed orange line; E = K + U: dotted lavender line. Graphs are drawn up to the selected instant. E₀ − E is the energy transferred to the surroundings by the resistive force.

Grid spacing: 0.25 m. The position scale is fixed; equilibrium is at x = 0.

The F arrow shows the direction of the net force; its length is not a quantitative force scale.

Equation of motion and its solution

Equation of motion and its solution

For an ideal horizontal spring, the restoring force is Fₛ = −k x. The position x is measured relative to equilibrium. The initial conditions are x(0) = x₀ and v(0) = v₀. The natural angular frequency is ω₀ = √(k/m), and the period is T₀ = 2π/ω₀.

Dots denote derivatives with respect to time. In the expressions below, t = 0 is the selected initial instant.

Kinetic, potential and total energy

K = m v²/2, U = k x²/2 and E = K + U. Since the power delivered by the resistive force is −b v², mechanical energy cannot increase. Energy is not destroyed; it is transferred from the mechanical system described by the model to the surroundings.

Virial theorem: time averages

For stable, bound, conservative motion, an appropriate time average gives 2⟨K⟩ = ⟨x dU/dx⟩. For a harmonic potential, averaging over a full period gives ⟨K⟩ = ⟨U⟩ = E/2. This does not mean the energies are equal at every instant; we do not apply the same equality directly to damped transient motion.

Model and assumptions

One-dimensional horizontal motion, constant mass, a massless ideal Hookean spring and constant coefficients. Weight and the normal force balance vertically. There is no external driving force or measurement noise. The damped page uses only resistance proportional to velocity. Changing a parameter starts the experiment with new initial conditions; the transition during the parameter change is not modelled.

Damped spring–mass system

Linear viscous friction: three damping regimes, an analytical solution and mechanical energy loss.

Open the experiment

Symbols and derivatives: how to read them

NotationMeaning and unit
Time; s denotes seconds. The light clock uses ns = 10⁻⁹ s.
Position relative to equilibrium (m); x₀ is initial position. Its sign indicates the side of equilibrium.
First time derivative of position: velocity (m/s). One dot means one derivative with respect to time.
Second time derivative of position: acceleration (m/s²), the rate of change of velocity. Two dots mean two time derivatives.
Mass m (kg), spring stiffness k (N/m), viscous damping coefficient b (kg/s). Drag is −bv and spring force is −kx.
ω₀ is natural angular frequency (rad/s); β = b/(2m) is decay rate (s⁻¹); ζ = β/ω₀ is dimensionless damping ratio. This β is not the light clock’s v/c.
Damped angular frequency (rad/s) in the underdamped regime; used as an oscillation frequency only for ζ < 1.
v₀ is initial velocity (m/s). A, B and C± are solution coefficients (m) fixed by initial position and velocity, not independent parameters.
Two exponential decay rates (s⁻¹) for overdamping. C₊ + C₋ = x₀ and r₊C₊ + r₋C₋ = v₀ enforce the initial conditions.
K is kinetic, U is spring potential and E is total mechanical energy (J). With damping, lost mechanical energy is transferred to the environment as thermal energy.
Angle brackets indicate time averages, here over a complete period or an integer number of periods of undamped periodic motion.

References and further reading

The books below offer further study of the relevant concepts. The simulation’s assumptions are stated separately above.

  1. Stephen T. Thornton, Jerry B. Marion. Classical Dynamics of Particles and Systems. Edition 5 · Chapter(s) 3. Cengage. View source ↗
  2. David Halliday, Robert Resnick, Jearl Walker. Fundamentals of Physics. Edition 12 · Chapter(s) 15. Wiley. View source ↗
  3. Hugh D. Young, Roger A. Freedman. University Physics with Modern Physics. Edition 15 · Chapter(s) 14. Pearson. View source ↗