Deeper Oscillation Budget Scans

Omega-Square Acceleration Three-Row Scan

At a fixed endpoint displacement, omega changes acceleration through omega squared. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

One-per-second oscillator gives three acceleration units

At the right endpoint, displacement is 3 m. Omega 1 per second has square 1, so acceleration is -3 metres per second squared.

a=(−ω2)x=−1×3=−3a=(-\omega^{2})x=-1\times3=-3
Omega-squared acceleration rowAmplitude, omega, position, and acceleration are checked.A=3 momega=1 1/sphase=0x=3 mv=0 m/sa=-3 m/s^2

Doubling omega squares the acceleration factor

At the right endpoint, displacement is 3 m. Omega 2 per second has square 4, so acceleration is -12 metres per second squared.

a=(−ω2)x=−4×3=−12a=(-\omega^{2})x=-4\times3=-12
Omega-squared acceleration rowAmplitude, omega, position, and acceleration are checked.A=3 momega=2 1/sphase=0x=3 mv=0 m/sa=-12 m/s^2

Tripling omega makes nine times the factor

At the right endpoint, displacement is 3 m. Omega 3 per second has square 9, so acceleration is -27 metres per second squared.

a=(−ω2)x=−9×3=−27a=(-\omega^{2})x=-9\times3=-27
Omega-squared acceleration rowAmplitude, omega, position, and acceleration are checked.A=3 momega=3 1/sphase=0x=3 mv=0 m/sa=-27 m/s^2