3.7.2.2 - Gravitational Field Strength

3.7.2.2 - Gravitational Field Strength

Gravitational field strength measures force per kilogram. Connect its vector definition to the inverse-square field outside a spherical planet, and decide when a uniform-field approximation is accurate enough.

Radial fields and the uniform-field approximation

A gravitational field line points along the force on a small test mass, not necessarily along the test mass’s velocity. Around an isolated spherically symmetric planet, the field is radial: inward towards the centre at every exterior point. Line separation increases outward, representing a weaker field.

[DIAGRAM: asset_name: 7.2.2 - Gravitational Field Strength - Diagram 1; asset_slug: 7.2.2 - Gravitational Field Strength - Diagram 1; recommended_method: retained_png; caption: Local uniform approximation and global radial field. Radial arrow lengths show direction only.; description: A small near-surface region can be treated as uniform; globally the field is radial. The radial arrows show direction only and are not drawn to a vector-magnitude scale.]
Local uniform approximation and global radial field. Radial arrow lengths show direction only.

A uniform field has the same magnitude and direction throughout the region: its lines are parallel and equally spaced. Earth's field can be treated as uniform only over a region whose height and horizontal extent are small compared with Earth's radius. A small height change makes little difference to field strength; a small horizontal extent makes little difference to direction. Merely being at the same altitude does not give the same field direction on opposite sides of Earth.

For example, 100 m100\,\mathrm m is only 1.6×10−51.6\times10^{-5} of Earth's radius of about 6.4×106 m6.4\times10^6\,\mathrm m. The inverse-square model predicts a fall in field strength of about 0.0031%0.0031\% over that height. This is negligible for many school calculations, although a sufficiently precise measurement can detect a change. “Uniform” is an approximation matched to the required accuracy.

Force per unit mass and free-fall acceleration

Gravitational field strength at a point is the gravitational force per unit mass on a small test mass placed there.

Definition of gravitational field strength

g=Fgm,Fg=mg in magnitude\mathbf g=\frac{\mathbf F_g}{m},\qquad F_g=mg\text{ in magnitude}

Its SI unit is N kg−1\mathrm{N\,kg^{-1}}. The test mass is small enough for its presence not to appreciably change the source arrangement. The field is a property of the sources and position, not of the chosen test mass.

At one point, a 2.0 kg2.0\,\mathrm{kg} mass experiencing 18 N18\,\mathrm N and a 5.0 kg5.0\,\mathrm{kg} mass experiencing 45 N45\,\mathrm N both measure g=9.0 N kg−1g=9.0\,\mathrm{N\,kg^{-1}}. More mass means more force in the same field, not a stronger field.

If gravity is the only force, Newton’s second law gives a=Fg/m=g\mathbf a=\mathbf F_g/m=\mathbf g. Consequently N kg−1=m s−2\mathrm{N\,kg^{-1}}=\mathrm{m\,s^{-2}}, and bodies released together have the same free-fall acceleration when air resistance is negligible. A resting supported object still has weight: its total acceleration is zero because its support force balances gravity.

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