Seismograph
The ground shakes, the frame moves with it, and the suspended mass stays still. A pen records the difference — the seismogram. Adjust damping and natural frequency to see how the instrument shapes what it records.
A mass that stays still while the world shakes
A seismograph exploits Newton’s first law: a mass at rest tends to stay at rest. The instrument consists of a heavy mass suspended by a spring from a frame that is bolted to the ground. When the ground shakes, the frame moves with it, but the mass — being loosely coupled through a soft spring — resists the motion due to its inertia. A pen attached to the mass writes on a paper drum attached to the frame, recording the relative displacement between mass and ground.
This is the fundamental insight: the mass provides an approximate inertial reference frame against which ground motion can be measured. The quality of this approximation depends on the natural frequency of the mass-spring system relative to the frequency of the ground motion. For ground frequencies well above the natural frequency, the mass barely moves and the seismograph faithfully records ground displacement.
How the instrument shapes what it records
The seismograph is a damped harmonic oscillator driven by ground motion. The equation of motion for the relative displacement z = xₚₐₛₛ − xₙ₣₠₱ₙₖ is: z'' + 2ζω₀z' + ω₀²z = −x''ₙ₣₠₱ₙₖ, where ω₀ = 2πf₀ is the natural angular frequency and ζ is the damping ratio.
The frequency response of the instrument — the ratio of recorded amplitude to true ground amplitude as a function of frequency — is H(f) = (f/f₀)² / √[(1 − (f/f₀)²)² + (2ζf/f₀)²]. At frequencies far above f₀, H → 1 and the instrument faithfully records ground displacement. Near f₀, the response peaks (resonance). Below f₀, the response falls off as (f/f₀)².
With ζ ≈ 0.7 (critical damping), the response is flat above f₀ with no resonance peak — the ideal broadband seismometer. Underdamped instruments (ζ < 1) ring at their natural frequency; overdamped ones (ζ > 1) respond sluggishly to all frequencies.
Logarithmic scales for a logarithmic world
In 1935, Charles Richter developed a magnitude scale to classify earthquake size. The original Richter magnitude Mₗ is defined as the logarithm (base 10) of the maximum trace amplitude (in micrometers) recorded on a standard Wood-Anderson torsion seismograph at 100 km from the epicenter, corrected for distance. Each unit increase in magnitude corresponds to a 10-fold increase in recorded amplitude and approximately a 31.6-fold increase in energy released.
Modern seismology uses the moment magnitude Mᵣ, which is based on the seismic moment M₀ = μAD, where μ is the rock’s shear modulus, A is the fault rupture area, and D is the average slip. This scale does not saturate for large earthquakes as the Richter scale does. The 2011 Tōhoku earthquake was Mᵣ 9.1; the 1960 Chile earthquake, the largest ever recorded, was Mᵣ 9.5.
The seismograph you see here is a simplified mechanical model, but the physics is identical to the instruments that first revealed the internal structure of the Earth — by tracking how seismic waves refract through layers of different density and rigidity, seismologists mapped the crust, mantle, outer core, and inner core without ever digging deeper than a few kilometers.