Gravitational Field Concept

Newton's law of universal gravitation states that a mass M exerts an attractive force on a mass m given by F = GMm/r², where r is the separation and G is the universal gravitational constant (6.674 × 10⁻¹¹ N m² kg⁻²). The field approach asks: what property of the space near M causes this force? We define the field so that F = mg, separating the 'source' contribution (the field g) from the 'test mass' contribution (m). This lets us characterise the region around M without needing a second mass to be present.

The gravitational field strength g at a distance r from a point mass M is defined as g = GM/r². It is a vector directed toward M. Notice that m cancels when we write g = F/m = GMm/(r²·m) = GM/r². This cancellation is profound: it means the field depends only on the source mass M and the position in space, not on the test mass. Every object at the same location in the field experiences the same field vector, regardless of its own mass.

Gravitational field lines (also called lines of force) are drawn pointing in the direction of g at each point—that is, toward the centre of the source mass. For a spherically symmetric body like Earth, the field lines are radial and point inward. Near Earth's surface, over small regions, the field lines are approximately parallel and equally spaced, representing a uniform field of magnitude g ≈ 9.81 N kg⁻¹. The density of field lines indicates field strength: closer spacing means stronger field.

Using Newton's second law, the acceleration of any object in field g is a = F/m = mg/m = g. The mass m cancels completely. This mathematical fact explains Galileo's famous observation that all objects—heavy or light—fall with the same gravitational acceleration in the absence of air resistance. The gravitational field strength g at Earth's surface (≈ 9.81 N kg⁻¹ = 9.81 m s⁻²) is therefore simultaneously the field strength and the free-fall acceleration.

For a uniform spherical body of mass M and radius R: outside the sphere (r ≥ R), the field is g = GM/r², identical to that of a point mass M at the centre. At the surface, g_surface = GM/R². Inside the sphere (r < R), only the mass enclosed within radius r contributes (shell theorem), so g = GMr/R³, which decreases linearly to zero at the centre. This distinction is important when studying variation of g with depth and altitude.

• \vec{g} — Gravitational field strength (N kg⁻¹ (= m s⁻²)): The gravitational force per unit mass at a given point in space; a vector pointing toward the source mass.
• \vec{F} — Gravitational force on test mass (N): The gravitational force experienced by the test mass m placed at distance r from mass M.
• m — Test mass (kg): A small mass placed at the point of interest; small enough not to disturb the field of M.
• G — Universal gravitational constant (N m² kg⁻²): A fundamental constant equal to 6.674 × 10⁻¹¹ N m² kg⁻².
• M — Source mass (kg): The mass creating the gravitational field (e.g., Earth, Moon, or Sun).
• r — Distance from source mass centre (m): The radial distance from the centre of the source mass M to the point where g is evaluated.
• \hat{r} — Unit radial vector (dimensionless): A unit vector pointing radially outward from M; the negative sign ensures g points toward M.

Problem. The Moon has mass M_Moon = 7.35 × 10²² kg and radius R_Moon = 1.74 × 10⁶ m. (a) Calculate the gravitational field strength at the Moon's surface. (b) A 70 kg astronaut stands on the Moon's surface. What gravitational force does the Moon exert on the astronaut?

Answer. g_Moon ≈ 1.62 N kg⁻¹; Force on astronaut ≈ 113 N

• Inverse-Square Law — The gravitational field strength g = GM/r² follows directly from the inverse-square law; understanding how g varies with distance requires this relationship.
• Gravitational Constant (G) — G appears in the definition of the gravitational field g = GM/r²; knowing its value and units is essential for numerical calculations of field strength.
• Acceleration Due to Gravity (g) — The gravitational field strength and the acceleration due to gravity are numerically equal (both ≈ 9.81 near Earth's surface) and share units N kg⁻¹ = m s⁻²; this concept explains why.
• Variation of g — Since g = GM/r², the field strength changes with altitude and depth; variation of g is a direct application of the gravitational field concept.
• Free-Fall as 1D Motion — Free fall occurs when gravity is the only force acting; the gravitational field g provides the acceleration in the kinematic equations of free fall.
• Upward & Downward Motion Under Gravity — Both upward and downward projectile motion use g as the constant acceleration, which is defined by the gravitational field at Earth's surface.
• Distinction Between Mass & Weight — Weight is defined as W = mg, where g is the gravitational field strength; this concept clarifies why mass is intrinsic but weight depends on the local field.
• Weight on Earth, Moon & Space — Different celestial bodies have different gravitational field strengths; calculating weight using W = mg on the Moon or other planets requires the field concept.
• Apparent Weight & Free-Fall — Apparent weight depends on the normal force, not the gravitational field alone; understanding the field is necessary to distinguish true weight (mg) from apparent weight.
• Gravitational Potential Energy Near Earth — Gravitational potential energy ΔE = mgh arises from work done against the gravitational field; the field concept underpins the energy stored in raised objects.