Physics I: Mechanics › Oscillations & gravitation › g at a planet's surface

g = GM/R²Say it: “g at a planet's surface equals G times the planet's mass over its radius squared”

g at a planet's surface

Why every planet has its own gravity — and why your mass never appears in it.

Notation on this page: g is the surface gravitational acceleration (m/s²), M the planet's mass, R its radius. Little m (your mass) cancels out.

Before this lesson: Newton's gravitation

Where it comes from

Stand on a planet of mass M and radius R. Newton's gravitation says the planet pulls you with F = GMm/R² — and Newton's second law says that same force accelerates you at a = F/m. Divide:

g = F/m = (GMm/R²)/m = GM/R²your mass m cancels — a feather and a hammer fall together
Before reading on: Planet X has twice Earth's mass but the same radius. Planet Y has Earth's mass but twice the radius. Which has the stronger surface gravity — and by what factors?

X: g ∝ M, so 2× the mass → 2g. Y: g ∝ 1/R², so 2× the radius → g/4. Radius beats mass: spreading the same mass over a bigger ball weakens gravity fourfold while doubling the mass only doubles it.

Derivation

One division turns Newton's universal law into every planet's personal gravity. The density form at the end is the power tool: it shows g cares about density times radius.

F
=
GMm/R²
Step 1 — Newton at the surface. Centre-to-centre distance at the surface is the radius R (shell theorem).
g
=
F/m = GM/R²
Step 2 — divide by your mass. a = F/m, and m cancels. Surface gravity depends only on the planet.
M
=
(4/3)πR³ρ
Step 3 — mass from density. A uniform sphere's mass is volume × density ρ.
g
=
(4/3)πGρR
Step 4 — the density form. R³/R² = R: g grows with density times radius. Same density, bigger ball → bigger g. ∎

Why 9.8 and not 9.82? GM/R² for Earth gives 9.82 m/s². The measured 9.80 is a touch less: Earth's spin flings you outward slightly and the equator bulges farther from the centre. 9.8 is the practical value.

How to use it

The procedure, every time:

  1. Look up M and R. Planet data tables give both — keep them in kg and metres.
  2. Compute GM/R². Watch the powers of ten: G is 10−11, M is ~1024, R² is ~1013.
  3. Weight = mg. The g you just found times any mass gives its weight there.
  4. Altitude: g(h) = GM/(R+h)² — add h to R before squaring.
  5. Ratios beat arithmetic. gX/gEarth = (MX/MEarth) × (REarth/RX)² — G cancels entirely.

The ratio shortcut

gX/gE = (MX/ME) · (RE/RX)²compare any planet to Earth with no G and no powers of tenSay it: “the g ratio equals the mass ratio times the radius ratio, squared and flipped”
Common mistake: using the planet's diameter for R. Squaring a doubled radius quarters g — the diameter slip makes every answer 4× too small. Radius, not diameter.

Worked examples

Four problems, easiest first. In each one, read every step — the why of each move is the lesson.

Example 1 — Earth: M = 5.972×1024 kg, R = 6.371×106 m

  1. Substitute. g = 6.674×10−11 × 5.972×1024 / (6.371×106)².
  2. Evaluate. = 3.986×1014 / 4.059×1013 ≈ 9.82 m/s².
  3. Sanity check. Matches the familiar 9.8 ✓ — the formula reproduces the gravity you live in.
Common mistake: getting 9.82×1012 or similar — a power-of-ten slip in R² (writing (6.371×106)² = 40.59×106 instead of 1012). Square the mantissa and double the exponent.
Your turn — the Moon: M = 7.348×1022 kg, R = 1.737×106 m. g?

Answer: ≈ 1.63 m/s². g = 6.674×10−11 × 7.348×1022/(1.737×106)² = 4.904×1012/3.017×1012 ≈ 1.63 m/s² — about one-sixth of Earth's, as the Apollo footage shows.

Example 2 — Mars: M = 6.417×1023 kg, R = 3.390×106 m

  1. Substitute. g = 6.674×10−11 × 6.417×1023 / (3.390×106)².
  2. Evaluate. = 4.283×1013 / 1.149×1013 ≈ 3.73 m/s².
  3. Ratio check. 3.73/9.82 ≈ 0.38 — Mars gravity is 38% of Earth's. A 70 kg astronaut weighs 70 × 3.73 ≈ 261 N there (vs 686 N here) ✓
Common mistake: “Mars is half Earth's size, so half the gravity.” Size alone says nothing — g needs both M and R. Mars has 11% of Earth's mass but 53% of its radius: 0.11/(0.53)² ≈ 0.38.
Your turn — Venus: M = 4.867×1024 kg, R = 6.052×106 m. g?

Answer: ≈ 8.87 m/s². g = 6.674×10−11 × 4.867×1024/(6.052×106)² = 3.248×1014/3.663×1013 ≈ 8.87 m/s² — 90% of Earth's. Venus is Earth's near-twin in gravity.

Example 3 — altitude: g at the ISS, h = 400 km

  1. Add h to R first. r = 6.371×106 + 0.400×106 = 6.771×106 m.
  2. Scale by ratio. g(h)/g = (R/r)² = (6.371/6.771)² ≈ 0.885.
  3. Evaluate. g ≈ 0.885 × 9.82 ≈ 8.69 m/s² — 89% of surface gravity!
  4. The lesson. Astronauts float from freefall, not from zero gravity. The ISS and its crew fall around Earth together ✓
Common mistake: g(400 km) ≈ 0 (“space has no gravity”). Gravity at the ISS is nearly 90% of Earth's — “zero-g” is zero weight, not zero g.
Your turn — g at h = R (one Earth radius up)?

Answer: ≈ 2.46 m/s². r = 2R → g = GM/(2R)² = gsurface/4 ≈ 9.82/4 ≈ 2.46 m/s². Double the centre distance, quarter the gravity.

Before reading on: squeeze Earth to half its radius, keeping its mass. Does g double (half the radius) or change by a different factor? Decide before computing.

Example 4 — judgment call: Earth squeezed to half its radius

  1. Scale by ratio. g ∝ 1/R²: R′ = R/2 → g′ = g/(1/2)² = 4g.
  2. Evaluate. g′ ≈ 4 × 9.82 ≈ 39.3 m/s².
  3. The lesson. Radius is squared — it dominates. This is why white dwarfs (Earth-mass, Earth-sized... no: Sun-mass squeezed planet-sized) have crushing surface gravity: g ∝ M/R² punishes small R brutally.
Common mistake: “half the radius, double the gravity.” The square strikes again: half the radius is four times the gravity. Whenever R changes, square the factor.
Your turn — same mass, double the radius. g?

Answer: g/4 ≈ 2.46 m/s². g′ = g/2² = 9.82/4 ≈ 2.46 m/s². (Equivalently via density: same density, double R → g = (4/3)πGρ(2R) doubles — wait, that contradicts! No: doubling R at fixed mass means density drops 8×, so (4/3)πG(ρ/8)(2R) = g/4 ✓. Keep track of what's held fixed.)

Memorization tips

  • Chant it: “g equals G M over R squared.” Big G is universal; little g is local.
  • Your mass cancels. g has no little m — the most-tested fact. If a problem gives your mass, it's for the weight (mg), not for g.
  • Radius is squared, mass isn't. R changes hit four times harder than equal M changes. “Square the radius factor” is the mantra.
  • The ratio shortcut: gX/gE = (MX/ME)(RE/RX)² — no G, no powers of ten, exam-speed.
  • Density form: g = (4/3)πGρR — same density, bigger ball, bigger g. Rebuilds the formula from scratch.
  • Earth anchor: GM/R² = 9.82 ≈ 9.8. Every other planet is a ratio away from this one number.

Final challenge

Five mixed questions — computations, ratios, and the traps, all in one. Score 5/5 and surface gravity is yours.

← Back to the Physics I formula sheet

How to learn a formula here

  1. Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
  2. Work the examples with the answers covered, then uncover one step at a time and compare.
  3. Finish with the final challenge — five mixed questions including the classic traps.
  4. Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.

Frequently asked questions

Why is g = 9.8 m/s² and not 9.82?

9.82 m/s² is the pure GM/R² value for a spherical Earth. The measured 9.80 is slightly less because Earth's spin flings you outward a touch (centrifugal effect) and the planet bulges at the equator. 9.8 is the practical value.

Does g change with altitude?

Yes: g(h) = GM/(R+h)². At the ISS's 400 km altitude it's about 8.7 m/s² — 89% of surface gravity. Astronauts float because they're in freefall, not because gravity is gone.

Why is the Moon's gravity so weak?

g = GM/R²: the Moon has only 1.2% of Earth's mass, and its smaller radius only partly compensates (dividing by a smaller R² helps). Net: 1.63 m/s², about one-sixth of Earth's.

What's the difference between g and G?

G = 6.674×10−11 is the universal gravitational constant — same everywhere, in every formula. Little g = GM/R² is the resulting surface acceleration for one specific body — different on every planet.

Does a heavier person feel a larger g?

No. g = GM/R² has no little-m in it: a heavier person feels a larger force (mg) but accelerates identically. Galileo's leaning-tower result, built into the formula.

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