Variation of g

The acceleration due to gravity, g, is often quoted as a fixed value of 9.81 m s⁻² at Earth's surface, but this is an idealisation. In reality, g varies depending on where you are — both in terms of your position on the globe and your distance from Earth's centre. Understanding these variations is essential for satellite engineering, geodesy, and even precision timekeeping, since pendulum clocks tick at different rates at different altitudes and latitudes. Three main factors cause g to deviate from its standard value. First, altitude: as you move away from Earth's centre, the gravitational pull weakens following an inverse-square law. Second, depth: descending below the surface means some of Earth's mass is now 'above' you, reducing the effective gravitational pull. Third, latitude: Earth is not a perfect sphere — it bulges at the equator due to rotation — so points on the equator are further from the centre, and the centrifugal effect of rotation also reduces the apparent value of g. These effects are small but measurable and technologically significant. GPS satellites must account for how g changes with altitude to maintain accurate timing. Geologists use sensitive gravimeters to detect slight local variations in g caused by underground ore deposits or oil reservoirs, turning gravity into a prospecting tool.

Key takeaways
  • Variation with Altitude
  • Variation with Depth
  • Variation with Latitude
  • The Shell Theorem — Key to Depth Calculations

Variation with Altitude

Above Earth's surface, Newton's law of gravitation gives g(h) = GM/(R+h)², where R is Earth's radius (~6.371 × 10⁶ m) and h is the height above the surface. As h increases, (R+h) increases, so g decreases. For small heights relative to R, we can approximate: g(h) ≈ g₀(1 − 2h/R). At the top of Mount Everest (h ≈ 8,848 m), g is approximately 9.77 m s⁻², noticeably less than at sea level. At the altitude of the International Space Station (h ≈ 400 km), g ≈ 8.69 m s⁻² — astronauts are not truly 'weightless'; they are in continuous free fall.

Variation with Depth

Below Earth's surface, assuming Earth has uniform density ρ, only the mass within radius r contributes to the gravitational force (by the shell theorem, the outer shell exerts no net force). The mass enclosed is M(r) = (4/3)πr³ρ. Substituting into g = GM(r)/r² gives g(d) = g₀(1 − d/R), where d is the depth below the surface. This means g decreases linearly with depth, reaching zero at Earth's centre. In reality Earth's density increases toward the core, making the actual profile more complex, but the linear model is a good first approximation for shallow depths.

Variation with Latitude

Earth is an oblate spheroid — its equatorial radius (6,378 km) is about 21 km greater than its polar radius (6,357 km). At the equator, you are further from Earth's centre, so gravitational attraction is weaker. Additionally, Earth's rotation creates a centrifugal pseudo-force directed away from the rotation axis. The component of this pseudo-force opposing gravity is greatest at the equator and zero at the poles. Combining both effects, g at the equator is approximately 9.780 m s⁻² while at the poles it is approximately 9.832 m s⁻² — a difference of about 0.5%. The latitude formula is: g(φ) ≈ 9.780(1 + 0.005302 sin²φ − 0.000006 sin²2φ) m s⁻², where φ is the geographic latitude.

The Shell Theorem — Key to Depth Calculations

Isaac Newton proved the shell theorem: a uniform spherical shell of mass exerts no gravitational force on any object inside it. This is the cornerstone of the depth calculation. Imagine descending a mine shaft to depth d. You can think of Earth as two parts: a solid sphere of radius (R−d) below you, and a shell of thickness d above you. By the shell theorem, the shell contributes nothing. Only the inner sphere pulls you, and since its mass is proportional to r³, while the gravitational formula divides by r², the net effect is that g falls linearly with depth.

The Key Equation

g(h)=GM(R+h)2≈g0(1−2hR)g(h) = \frac{GM}{(R+h)^2} \approx g_0\left(1 - \frac{2h}{R}\right)
  • g(h) — Gravitational acceleration at altitude h (m s⁻²): The free-fall acceleration experienced at height h above Earth's surface.
  • G — Universal gravitational constant (N m² kg⁻²): G = 6.674 × 10⁻¹¹ N m² kg⁻², a fundamental constant of nature.
  • M — Mass of the Earth (kg): M ≈ 5.972 × 10²⁴ kg, the total mass of Earth.
  • R — Radius of the Earth (m): R ≈ 6.371 × 10⁶ m, the mean radius of Earth.
  • h — Altitude above surface (m): The height above Earth's surface at which g is being calculated.
  • g_0 — Standard gravitational acceleration (m s⁻²): g₀ = 9.81 m s⁻², the accepted standard value at Earth's surface.

Worked Example

Problem. A communications satellite orbits at an altitude of 200 km above Earth's surface. Calculate the gravitational acceleration at this altitude. Use G = 6.674 × 10⁻¹¹ N m² kg⁻², M = 5.972 × 10²⁴ kg, and R = 6.371 × 10⁶ m.

g(200 km)≈9.23 m s−2g(200\,\text{km}) \approx 9.23\,\text{m s}^{-2}

Answer. The gravitational acceleration at 200 km altitude is approximately 9.23 m s⁻².

Figure
Diagram
Variation of g with Altitude and DepthEarthR = 6371 kmRh (altitude)g decreasesas 1/(R+h)²dg decreaseslinearly with dd (depth)Approximate valuesSurface (sea level):9.81 m s⁻²Everest top (8.8 km):9.77 m s⁻²ISS orbit (400 km):8.69 m s⁻²Equator (latitude 0°):9.78 m s⁻²Poles (latitude 90°):9.83 m s⁻²Equatorial bulge
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton formulated the Universal Law of Gravitation in his 1687 masterpiece Principia Mathematica, showing that the gravitational force between two masses falls off as the inverse square of the distance between them. This directly explains why g decreases with altitude. He also proved the Shell Theorem — that a uniform spherical shell exerts no net gravitational force on a body inside it — which is the mathematical foundation for calculating how g varies with depth inside Earth. Newton reportedly delayed publishing his gravitational theory for nearly 20 years partly because he did not have an accurate enough value for Earth's radius to verify his predictions — he needed the Shell Theorem to hold rigorously before he felt confident, and he had to invent calculus to prove it.

  • Newton's Law of Gravitation — The inverse-square law of gravitation is the direct source of the altitude formula for g; understanding F = GMm/r² is prerequisite to deriving g(h).
  • Gravitational Field Strength — g is defined as the gravitational field strength — force per unit mass — so understanding the field concept is essential before studying how g varies.
  • Orbital Motion and Satellites — Satellites orbit because of the value of g at their altitude; the variation of g with height directly determines orbital speeds and periods.
  • Gravitational Potential Energy — Gravitational potential energy depends on g, so its variation with altitude and depth links directly to how g changes in these regions.