Circular V-Square over Radius Cross-Scan - Rotational Dynamics and Orbital Mechanics Step by Step

A circular v-square-over-radius cross-scan varies the radius across rows, checking that v^2 / r matches the field mu / r^2 at each row, the centripetal balance that closes a circular orbit. The arithmetic is exact only for the stated speeds, radii, gravitational parameter, and rows in the ideal circular two-body model. Real observations add noncircular orbits, perturbations, extended bodies, reference-frame choices, instrument calibration, finite precision, and measurement uncertainty.

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.

\frac{v^{2}}{r}=\frac{4}{4}=1,\quad a=1

Candidate acceleration 4

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

\frac{v^{2}}{r}=\frac{8}{2}=4,\quad a=1

Candidate acceleration 2

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

\frac{v^{2}}{r}=\frac{4}{2}=2,\quad a=0

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.

\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}