Oscillation Relation Cross Scans

Phase State Three-Row Scan

Quarter-cycle rows make the sign pattern visible without hidden trigonometry. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Phase zero is the right turning state

Phase 0 gives position 4 m, velocity 0 metres per second, and acceleration -4 metres per second squared.

ϕ=0⇒x=4 m, v=0 m/s, a=−4 m/s2\phi=0\Rightarrow x=4\ \mathrm{m},\ v=0\ \mathrm{m/s},\ a=-4\ \mathrm{m/s}^{2}
Phase state scanPosition, velocity, and acceleration are helper-checked.A=4 momega=1 1/sphase=0x=4 mv=0 m/sa=-4 m/s^2

One quarter cycle trades displacement for velocity

Phase 1/4 gives position 0 m, velocity -4 metres per second, and acceleration 0 metres per second squared.

ϕ=14⇒x=0 m, v=−4 m/s, a=0 m/s2\phi={1\over 4}\Rightarrow x=0\ \mathrm{m},\ v=-4\ \mathrm{m/s},\ a=0\ \mathrm{m/s}^{2}
Phase state scanPosition, velocity, and acceleration are helper-checked.A=4 momega=1 1/sphase=1/4x=0 mv=-4 m/sa=0 m/s^2

Half a cycle flips the endpoint signs

Phase 1/2 gives position -4 m, velocity 0 metres per second, and acceleration 4 metres per second squared.

ϕ=12⇒x=−4 m, v=0 m/s, a=4 m/s2\phi={1\over 2}\Rightarrow x=-4\ \mathrm{m},\ v=0\ \mathrm{m/s},\ a=4\ \mathrm{m/s}^{2}
Phase state scanPosition, velocity, and acceleration are helper-checked.A=4 momega=1 1/sphase=1/2x=-4 mv=0 m/sa=4 m/s^2