Relation Cross-Scans

Circular V-Square over Radius Cross-Scan

A circular orbit closes only when speed-square over radius equals the field. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Candidate acceleration 1

Radius 4 and speed-square 4 give centripetal acceleration 1.

v2r=44=1,a=1\frac{v^{2}}{r}=\frac{4}{4}=1,\quad a=1
Circular row 1The circular check cites the field and radius source.speedSquared=4 m^2/s^2centripetalAcceleration=1 m/s^2acceptedBit=1 bit

Candidate acceleration 4

Radius 2 and speed-square 8 give centripetal acceleration 4.

v2r=82=4,a=1\frac{v^{2}}{r}=\frac{8}{2}=4,\quad a=1
Circular row 2The circular check cites the field and radius source.speedSquared=8 m^2/s^2centripetalAcceleration=4 m/s^2acceptedBit=1 bit

Candidate acceleration 2

Radius 2 and speed-square 4 give centripetal acceleration 2.

v2r=42=2,a=0\frac{v^{2}}{r}=\frac{4}{2}=2,\quad a=0
Circular row 3The circular check cites the field and radius source.speedSquared=4 m^2/s^2centripetalAcceleration=2 m/s^2acceptedBit=0 bit

Circular closure requires v-square over radius to equal the field

The first two rows close exactly. The third row uses the same radius as row two but halves the speed-square, so its acceleration is too small.

rv⋅vv⋅v/rga441112844124240\begin{array}{c|c|c|c|c}r&v\cdot v&v\cdot v/r&g&a\\4&4&1&1&1\\2&8&4&4&1\\2&4&2&4&0\\\end{array}
Circular closure cross-scanThe displayed row is the rejected candidate.speedSquared=4 m^2/s^2centripetalAcceleration=2 m/s^2acceptedBit=0 bit