Oscillations · Beginner · ⏱ 25 min

Spring-Mass System

Attach a mass to a spring, release it, and verify Hooke's law F = –kx. Measure the period, visualise phase space, and add damping.

Step 01

Add a Spring and Mass

Drag a Circle body onto the canvas. In the Forces panel, add a Spring force and attach the other end to a fixed anchor point directly above the mass. Set spring constant k = 50 N/m, natural length = 2 m, and mass m = 1 kg.

Note: The spring appears as a coiled line. You can drag the anchor end to reposition it. Make sure the anchor is set to Fixed in its properties so it doesn't move.
Step 02

Verify Hooke's Law

Displace the mass 0.5 m from equilibrium and pause the simulation. Check the force readout in the body panel — it should read F = 50 × 0.5 = 25 N directed toward the anchor. Try three different displacements and confirm F ∝ x.

Step 03

Measure the Period

Release the mass from 1 m displacement. Enable the graph panel (G key) and plot position-y vs time. Measure the time between two peaks — this is the period T. Compare to the formula T = 2π√(m/k). For k=50, m=1, expect T ≈ 0.89 s.

Note: Period is independent of amplitude in ideal SHM — verify this by repeating with 0.5 m and 2 m displacements. The period should be the same.
Step 04

Plot Phase Space

Switch the graph panel to position-y vs velocity-y (phase space). An undamped oscillator traces a perfect ellipse. Note how the ellipse's size corresponds to the total energy of the system — more displacement means a larger ellipse.

Step 05

Add Damping

In the body's force list, add a Linear Drag force with coefficient b = 2 N·s/m. Re-run the simulation and observe the graph: the ellipse spirals inward toward the origin as the system loses energy. This is underdamping. Increase b to 20 to reach critical damping — the mass returns to equilibrium without oscillating.

Note: Critical damping occurs at b = 2√(km). For k=50, m=1, this gives b ≈ 14.1 N·s/m. Over-damped: b > 14.1. Under-damped: b < 14.1.
Key formula:
T = 2π√(m/k) | F = –kx

Discussion

Questions, corrections, and insights welcome.

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