13. Gravitational fields

A gravitational field is a region in which a mass experiences a force due to gravity; field strength g is force per unit mass.

Use the field as a property of the source distribution, then calculate the force on a test mass with F=mg.

Near Earth, g≈9.8 N kg⁻¹ means a 2 kg mass experiences about 19.6 N downward.

The field exists even when no test mass is present, and g is not universally constant everywhere.

A field line points in the direction of force on a small positive test mass; closer lines represent a stronger field qualitatively.

Lines are a visual model, not physical tracks. Around an isolated mass they point radially inward and do not cross.

Parallel equally spaced lines model an approximately uniform field near Earth’s surface; radial lines model a point mass.

Field-line density is not an exact numerical scale, and crossing lines would imply two field directions at one point.

For points outside a spherically symmetric mass distribution, gravity acts as if the total mass were concentrated at the sphere’s centre.

Use centre-to-point distance r, not distance from the surface, when applying point-mass equations.

The external gravitational field of a spherical planet can be calculated with its total mass and the distance from its centre.

The point-mass equivalence does not apply inside a nonuniform shell or to an arbitrary irregular body.

Two point masses attract with F=Gm₁m₂/r² along the line joining their centres, where G is the universal gravitational constant.

Use centre-to-centre separation and treat the force as attractive. For several masses, add force vectors.

Doubling one mass doubles force; doubling separation reduces force to one quarter.

The inverse-square law is not inverse distance, and the force pair acts on both masses with equal magnitude in opposite directions.

For a circular orbit, gravitational attraction provides F=mv²/r, so the orbital speed satisfies GMm/r²=mv²/r.

Use the distance r from the attracting body’s centre and remember that gravity is inward while velocity is tangential.

A lower circular orbit has greater speed because v=√(GM/r) increases as r decreases.

Gravity is not absent in orbit; an orbiting object is continuously falling while its tangential motion carries it around.

A geostationary satellite stays above one longitude because it orbits eastward in Earth’s equatorial plane with the same angular speed and period as Earth’s rotation.

All three conditions matter: circular orbit, equatorial plane and synchronous period. Its apparent fixed position is relative to the rotating Earth.

Communications dishes can point at a geostationary satellite without tracking it across the sky.

Any satellite with a 24-hour period is not automatically geostationary if its orbit is tilted or elliptical.

Equating gravitational force GMm/r² with weight mg gives g=GM/r² for a point mass or spherical source outside its surface.

The test mass cancels, so g depends on source mass M and centre distance r, not on the mass being tested.

Doubling distance from a planet’s centre quarters g; changing the test object’s mass does not change g.

g is a field strength, not a universal constant; G is constant while g varies with location.

For a spherical source outside its surface, g=GM/r² in N kg⁻¹ or m s⁻².

Use M for the source and r from its centre. The inverse-square relation is a scaling law, not a claim that g changes linearly with height.

At twice the centre distance, a satellite experiences one quarter of the original field strength.

Do not use altitude alone when the planet’s radius is significant; r is radius plus altitude.

Although g=GM/r² varies with distance from Earth’s centre, the change is negligible over small everyday height ranges, so g≈9.8 m s⁻² is a useful approximation.

Use the constant-g model for local motion and mg∆h calculations, but switch to inverse-square reasoning for satellites or large altitude changes.

A building’s roof and ground have almost the same g compared with a spacecraft thousands of kilometres away.

“Constant near Earth” is an approximation, not evidence that the gravitational field has no spatial dependence.

Gravitational potential at a point is the work done per unit mass by an external agent bringing a small mass from infinity to that point, with zero potential at infinity.

For an attractive field around a point mass, potential is negative because the field releases energy as the mass moves inward.

Moving a test mass farther from a planet raises its gravitational potential toward zero.

Potential is energy per unit mass, not force; equal potential does not imply equal field strength everywhere.

For a point mass, gravitational potential Φ is work done per unit mass from infinity and equals Φ=−GM/r.

The negative sign reflects attraction: moving a mass inward releases energy, while moving it outward requires work. Use centre distance r.

At twice the distance from a planet’s centre, potential is half as negative and closer to zero.

Potential is not the same as field strength g; Φ can be negative while g is a positive magnitude directed inward.

A potential difference ∆Φ is energy transferred per unit mass; a mass m changes gravitational potential energy by ∆E_P=m∆Φ.

Use the sign to determine whether the field or an external agent supplies energy, and choose a reference consistently.

Moving outward from a planet raises Φ toward zero, so an external agent does positive work against gravity.

Equal potential difference does not imply equal force over a path; field strength is the spatial rate of potential change.