Gravitational Field & Force Lab

H2 Physics · Gravitation — g = F/m, F = Gm₁m₂/r², and the inverse square you can watch

g against r · gₛ = 9.82 N kg⁻¹ at r = R

Drag anywhere on the graph — or drag the pink mass in the scene.

Zoom: the first 100 km above the surface

ⓘ more

Same vertical scale as the graph above, so nothing is exaggerated. Over the whole 100 km g changes by -3.07 % — that is why g ≈ constant near the surface, and why the same number doubles as the acceleration of free fall. Zoom far enough out and the same curve is the 1/r² fall.

Try first: press r ×2 — watch g fall to a quarter.

Mode

Arrow length ∝ g at that point. Thickness is cosmetic.

Preset

Controls

Planet mass M5.97×10²⁴ kg

Planet radius R6.371 Mm

Same M, bigger R → weaker surface field.

Test mass m1 kg

Changes F. Never changes g.

Distance r2.00 R = 12.74 Mm

At the test mass

r: 2.00 R = 12.74 Mm
F = GMm/r² (N): 2.45
F / m (N kg⁻¹): 2.45
g = GM/r² (N kg⁻¹): 2.45
g / gₛ: 0.2500
(R/r)²: 0.2500

Rows 2 and 3 never disagree — that equality is the definition of field strength.

At the surface

gₛ = GM/R² (N kg⁻¹): 9.82
free-fall accel. (m s⁻²): 9.82
weight of m (N): 9.82
Δg over first 100 km: -3.07 %

Constants used

G (N m² kg⁻²): 6.674×10⁻¹¹
M (kg): 5.972×10²⁴
R (m): 6.371×10⁶

What this confirms

›LO (a) — field strength is the force per unit mass at a point: the F/m row and the g row stay identical however you drag the test mass slider. m is only in the force, never in the field.
›LO (c) — Newton’s law F = Gm₁m₂/r²: the F readout is that product, and two-mass mode gives the real Earth–Moon pull of about 1.98×10²⁰ N straight from the data-booklet constants.
›LO (d) & (e) — g = GM/r² is not a separate law: divide F = GMm/r² by m and m cancels. The whole outer curve is that one equation plotted.
›Inverse square — press r ×2 and the printed ratio is 4.00 every time, at any M, any R, any preset. Same reason the field arrows shrink to a quarter one shell out.
›LO (f) — g is constant near the surface and equals the acceleration of free fall: the zoomed inset drops only 3.07 % over 100 km on the current body, on the same vertical scale as the full graph.
›LO (b) — the electric-field analogy — E = F/Q mirrors g = F/m and both fall as 1/r², but every arrow here points inward: gravity is only ever attractive, so there is no repulsive case to draw.
›Equipotentials — the shells are drawn at equal steps of φ, radii recovered by inverting φ = −GM/r, so their spacing is 1/g: 1.154R to 1.364R at the bottom of the ladder, but 3R to 5R at the top, for the same 8.34×10⁶ J kg⁻¹ each time. Field lines cut them at 90°, and moving along one does no work because Δφ = 0 — hold r, drag the mass round, and the Δφ readout stays at exactly zero while θ sweeps. In two-mass mode the contours warp towards each other and pinch at the neutral point, but never cross: one point can only have one potential.
›Notes: the g–r graph — inside a uniform body only the enclosed mass pulls, so g = GMr/R³ rises linearly; outside it falls as 1/r². The maximum sits exactly at the surface, which is the shape examiners ask you to sketch.
›Tutorial: the neutral point — fields are vectors and add. Bisection on GM₁/x² − GM₂/(d−x)² lands on the same answer as x = d/(1+√(M₂/M₁)); the two contributions there cancel to the last digit double precision can hold, which the residual readout shows live.