DYNAMICS · CLASSICAL OSCILLATORS
Oscillations: Spring and Pendulum
Explore harmonic motion, damping and energy transformations. Compare a linear spring with a nonlinear pendulum, distinguishing exact laws from small-angle approximations.
Physical Parameters
Simulation and Force Diagram
The pendulum has the same scale on both axes: angles are drawn geometrically without distortion. Torque arrows are schematic.
Synchronized Graphs
Click a graph or enter an exact time to update the diagram, values and worked solution. The dashed pendulum comparison uses a linearized model and may be inaccurate at large amplitudes.
Position / Angle vs Time
Velocity vs Time
Acceleration vs Time
Kinetic, Potential and Mechanical Energies
Mechanical Energy, Dissipation and Balance
Values Table
| t (s) | Type | q (m or rad) | v (m/s or rad/s) | a (m/s² or rad/s²) | K (J) | U (J) | E (J) | Q (J) | Residual (J) | Force (N or N·m) |
|---|
Comma-separated CSV with decimal points and SI units.
Model Assumptions
Spring-Mass: a point mass on a horizontal guide and an ideal Hookean spring with constant k; linear damping Fᵥ = −bv. No dry friction, impacts or external driving forces. An exact analytical solution is used for all three damping regimes.
Pendulum: a point mass attached to a rigid, inextensible, massless rod; frictionless pivot when c = 0 and uniform gravity. Viscous torque is τᵥ = −cω. The nonlinear equation with sin θ is integrated numerically; complete rotations are also possible.
The pendulum period T₀ = 2π√(L/g) is a small-angle approximation, not the exact period of large-amplitude oscillations. With damping, the motion is not strictly periodic.
Energies are K and U, with U = 0 J at equilibrium. Cumulative dissipation Q is nonnegative, and the energy balance is E(t) + Q(t) = E(0). Numerical integration error is reported as a residual.
FROM THE MODEL TO THE RESULTS
Step-by-Step Calculations
General equations, SI substitutions and intermediate steps.
One left-aligned equation per line. MathML fallback is provided when MathJax is unavailable. Rounding affects display only.
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