In AP Physics C: Mechanics, a gravitational field is a model assigning a vector g to every point in space, equal to the gravitational force a test mass would feel there divided by that mass (g = F_g/m, units N/kg). Near a sphere of mass M, its magnitude is g = GM/r².
A gravitational field is how physicists answer the awkward question of how the Earth pulls on you without touching you. Instead of saying "mass A reaches across empty space to yank on mass B," the field model says mass A fills the space around it with a vector field g. Drop any test mass m into that space and the force on it is simply F = mg. The field strength is force per unit mass, g = F_g/m, with units of N/kg (which are the same as m/s², and that's not a coincidence).
For a point mass or a uniform sphere of mass M, the field at distance r from the center has magnitude g = GM/r² and points toward the mass. That r² in the denominator is the famous inverse-square law. The clean part is that g describes the source mass alone. It doesn't care whether the object sitting in the field is a feather or a freight train; the field is the same, only the force F = mg changes. That separation of "what creates gravity" from "what feels gravity" is the entire point of the field model, and it's the same logic you'll reuse for electric fields in Physics C: E&M.
Gravitational field lives in Topic 2.6 (Gravitational Force) in Unit 2 of AP Physics C: Mechanics. It's the bridge between the everyday g = 9.8 m/s² you've used since Unit 1 and Newton's universal law of gravitation. Once you see that 9.8 m/s² is just GM/r² evaluated at Earth's surface, a lot of the course snaps into focus. Weight is mg because g is the field. Orbits work because the field supplies centripetal force. Gravitational potential energy in Unit 3 is what you get by integrating the field-driven force over distance. The exam also loves pushing past the simple formula, asking for the field inside a planet (where only the enclosed mass counts, thanks to the shell theorem) or for planets with non-uniform density where you have to integrate ρ(r) to find the enclosed mass first.
Keep studying AP® Physics C: Mechanics Unit 2
Newton's shell theorem (Unit 2)
The shell theorem is what makes g = GM/r² legal for planets, not just point masses. Outside a spherical shell, the field looks like all the mass is at the center; inside the shell, the field is zero. That second part is the trick behind "field inside a planet" problems, where only the mass enclosed below your radius contributes.
Weight and apparent weight (Unit 2)
True weight is just the field doing its job, W = mg. Apparent weight is what a scale reads, and it differs from mg whenever you accelerate (elevators, orbiting astronauts). The field hasn't changed in those situations; your frame of reference has.
Equivalence principle and inertial mass (Unit 2)
The m in F = ma (inertial mass) and the m in F = mg (gravitational mass) are experimentally identical. That's why every object free-falls with the same acceleration regardless of its mass, and why g in N/kg equals g in m/s². The field strength and the free-fall acceleration are the same number wearing different units.
Gravitational potential energy and orbits (Unit 3)
Integrate the gravitational force from the field over distance and you get U = -GMm/r, the energy function behind escape speed and orbital energy. Field gives you the force picture; potential energy gives you the energy picture of the exact same physics.
Expect field questions in both MCQ and FRQ form, and expect them to go beyond plugging into g = GM/r². Classic moves the exam asks for include finding g at altitude h above a planet's surface (one practice problem hands you g = 9.5 N/kg at height h and asks you to work backward for h), deciding when two spheres can be treated as point masses (answer: their centers must be separated by at least R₁ + R₂ so neither is inside the other), and computing the field inside a planet with non-uniform density like ρ(r) = kr, which forces you to integrate dm = ρ(4πr²)dr to get the enclosed mass before applying g = GM_enc/r². Calculus-based setup is the differentiator here. You should also be ready to explain why gravitational and inertial mass being equal means all objects fall at the same rate. No released FRQ has leaned on the phrase "gravitational field" by itself, but field reasoning is the backbone of any orbit, weight, or shell-theorem FRQ.
The field g is a property of the source mass and the location in space; the force F = mg is what happens when you actually put an object there. The field at a point near Earth is 9.8 N/kg whether or not anything is sitting there. The force depends on what you place in the field, so a 2 kg object feels 19.6 N while a 10 kg object feels 98 N in the exact same field. If a question asks for the field, your answer should not contain the test object's mass.
The gravitational field is force per unit test mass, g = F_g/m, measured in N/kg, which is dimensionally identical to m/s².
Outside a uniform sphere of mass M, the field magnitude is g = GM/r², measured from the center of the sphere, and it points toward the mass.
Inside a planet, only the mass enclosed below your radius contributes to the field; the spherical shell of mass above you contributes zero (Newton's shell theorem).
For non-uniform density problems, find the enclosed mass by integrating dm = ρ(r) · 4πr² dr before applying g = GM_enc/r².
The surface value 9.8 m/s² is not a fundamental constant; it's just GM/R² evaluated at Earth's surface, so g drops as you climb to altitude.
Because inertial mass equals gravitational mass, the field strength in N/kg equals the free-fall acceleration in m/s², which is why all objects fall at the same rate.
It's a vector field created by mass, defined as the gravitational force per unit test mass at each point in space, g = F_g/m. For a sphere or point mass M, its magnitude is g = GM/r², where r is measured from the center.
Numerically yes, conceptually they're two readings of the same quantity. The field strength in N/kg equals the free-fall acceleration in m/s² because gravitational mass and inertial mass are equal (the equivalence principle). An object in free fall accelerates at exactly the local field value.
The field depends only on the source mass and your position (g = GM/r²), while the force also depends on the object placed in the field (F = mg). The field at a location exists whether or not anything is there to feel it.
Only at the exact center. At radius r inside a uniform planet, the shell of mass above you contributes nothing, but the sphere of mass below you still pulls inward, so g = GM_enc/r². For a uniform-density planet this makes g grow linearly from zero at the center to its maximum at the surface.
For the hardest ones, yes. If a planet's density varies with radius, like ρ(r) = kr, you have to integrate dm = ρ(r) · 4πr² dr to find the enclosed mass before computing the field. Uniform-sphere and altitude problems just need g = GM/r².
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