Tides and distributed gravity

Tides are differential gravity

A tide is not caused by gravity being ‘strong’ alone; it is caused by gravity varying across an extended body. The near side is pulled slightly more than the center, and the far side slightly less.

Expand the inverse-square field

For g(r) = GM/r², a small separation Δr changes the field by Δg ≈ |dg/dr|Δr = 2GMΔr/r³. The factor r⁻³ is steeper than the ordinary field’s r⁻² because a tide compares neighboring values.

For Earth–Moon distance r = 3.84×10⁸ m and Earth radius Δr = 6.37×10⁶ m, the lunar differential acceleration scale is about 2GM_mΔr/r³ ≈ 1.1×10⁻⁶ m/s², small but persistent and amplified by ocean geometry.

The simple gradient is not a tide-height prediction

Ocean basins, rotation, continents, and the Sun modify actual water levels. The gradient estimate predicts the forcing scale, not the final shoreline amplitude.

Work it: Compare the tidal gradient from the Moon and Sun using their masses and distances. Explain why the closer Moon can dominate despite the Sun’s larger mass.