Infer a field cancellation from signed source contributions

Distinguish field and force

A point source of mass M produces a gravitational field of magnitude GM/r² directed toward itself. A passive test mass m experiences force m times that field. The test mass changes force magnitude but cancels from acceleration in this ideal gravitational model. Weight is a gravitational interaction, whereas an apparent weight reading from a support is a separate contact force that can differ during acceleration.

Compare gravitational mass, inertial mass and a support reading

Gravitational mass sets attraction strength, while inertial mass sets the response to net force. Their experimentally established equivalence makes gravitational free-fall acceleration independent of the test object’s mass in this model. For a locally uniform field, a small enclosed laboratory cannot distinguish the support effects of uniform acceleration from those of gravity by a purely local mechanical test; spatial tidal differences limit that equivalence statement.

With upward positive, a supported 2 kg object in g=10 obeys N−20=2a. At a=+3 its apparent weight is 26 N; at a=−3 it is 14 N. In free fall a=−10, N=0 even though gravity remains 20 N. Predict the reading for a=0 while moving upward at constant speed: it is 20 N, because acceleration rather than velocity determines the change. These are paper comparisons, not instructions for an elevator or falling apparatus.

Add signed contributions

Place a left source at x=0 and right source at x=D. Between them, the left field is negative and the right field positive: gnet=−GML/x²+GMR/(D−x)². Adding positive magnitudes would describe neither the net field nor its zero. Outside both sources, both fields point toward the source pair, so they cannot cancel on that outer line for positive source masses.

Derive the interior cancellation

Set ML/x²=MR/(D−x)². Distances are positive inside the interval, so taking positive square roots gives √ML/x=√MR/(D−x). Solve x=D√ML/(√ML+√MR). This is a geometric consequence of the inverse-square law, not the center-of-mass formula. The two locations generally differ.

Check the original normalized packet

The model fixes normalized G=1, ML=4,D=4 and allows MR to vary. With MR=1, the null is 8/3 m from the left. At x=2 the left contribution is −1 and the right+.25, totaling−.75 in declared model field units. At the null each nonzero contribution has the same magnitude; zero net field does not imply no gravitational interactions.

Relate a continuous source carefully

A spherically symmetric source has an external field equivalent to a point mass at its center; that statement depends on symmetry. For a uniform solid sphere, only the enclosed mass contributes to the net interior field, yielding magnitude GM r/R³ inside radius R. It approaches zero at the center rather than following the exterior inverse-square expression to a singularity. Inside an ideal thin spherical shell the contributions cancel everywhere, not only at the center; outside, the shell behaves as a central point mass. For the uniform solid sphere at r=R/2, enclosed mass is M/8 and field magnitude is half its surface value. Predict the field at R/4: one-quarter the surface value, not sixteen times it. An arbitrary irregular body cannot be replaced by a point source at every nearby position merely because its center of mass is known.

Transfer and assumptions

Equal sources at 0 and 6 m cancel at 3 m. Double the test mass and both opposing force contributions double while the field null remains fixed. The workbench holds the sources fixed and omits their mutual motion; a moving-source orbit is a different problem. Measurements of one field coordinate do not uniquely identify an entire source distribution.

Public numerical transfer practice

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Two equal fixed point masses are at x0 and 6 m. Find the between-source zero-field coordinate.

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Original prose, models and cases, checked 30 October 2026 against the current official AP Physics C Mechanics CED and OpenStax University Physics. Values are supplied or generated, not measured apparatus records. No physical launch, suspension, moving machinery or practical safety validation is requested. Virtual captures and public answers do not certify independent mastery or laboratory participation.