Use center distance in gravity and orbital comparisons

Define the recipient and model

For an ideal spherical source, exterior gravitational force has magnitude GMm/r², where r is center-to-center separation. Altitude above the surface is h=r−R, not r. Substituting altitude into the denominator would predict a singular field at the surface even though the source-center distance remains finite.

Build the algebraic relationship

The synthetic source has GM=4×10^14 m³/s² and radius R=6×10^6 m. At center distance2R, gravity is one quarter of its surface value. Doubling the small test mass doubles gravitational force but leaves acceleration GM/r² unchanged. This comparison assumes the source is overwhelmingly more massive and its motion and other bodies can be neglected.

Check the changed state

For a circular orbit, gravity supplies mv²/r, so v=√(GM/r). Combining this with period T=2πr/v gives T²/r³=4π²/(GM). A larger circular orbit has smaller speed but longer period. An orbiting point is continuously falling under gravity; an absent support force does not imply that gravity has become negligible.

Interpret evidence and limits

General bound two-body orbits are ellipses with the source at one focus, with equal areas swept in equal times; the circular case is a special ellipse with constant speed. For common-source elliptical comparisons, the period relation uses semimajor axis, not arbitrary instantaneous radius. This workbench calculates circular states only. At r=R it shows an ideal mathematical limit, not a physically accessible orbit through atmosphere or surface terrain.

Worked original packet

At r=2R=1.2×10^7 m, g≈2.77778 m/s² and circular speed≈5773.50 m/s. A 2 kg test mass feels≈5.55556 N. Compared with r=R, speed is divided by√2 and period multiplied by 2√2.

Bounded investigation

Compare radii R,2R,3R at one test mass; retain field, force, speed and period ratios. Then change mass alone. Explain why equal periods need not imply equal instantaneous speeds at every point of a noncircular orbit.

Changed-case numerical practice

Try the changed case before feedback

Two ideal circular orbits around the same dominant source have radii r and 4r. What is the ratio of the larger orbit’s period to the smaller orbit’s period?

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and boundaries

Original prose, data and diagrams checked 30 October 2026 against OpenStax College Physics 2e. This is an algebra-based virtual evidence exercise, not a physical experiment, equipment certification, transport recommendation or independent mastery result. Save assumptions and a competing explanation with the course evidence; supervised practical requirements remain external.