Gravity Variation: How Altitude, Depth and Spin Change g

The value g = 9.8 m/s² from school problems is only an average. The acceleration due to gravity changes with altitude, with depth, and with the Earth's spin. Gravity on top of Mount Everest is measurably weaker than at sea level, it weakens as you go down a deep mine, and it is slightly weaker at the equator than at the poles.

This matters for real measurements: surveyors, satellite engineers and oil prospectors all correct for local g. Your weight would genuinely differ by a few grams between the poles and the equator. These variations also prove the Earth is a spinning, slightly squashed ball rather than a perfect static sphere.

Key Concepts

• Surface g = GM/R² ≈ 9.8 m/s², with M = 5.97 × 10²⁴ kg (Earth's mass), G = 6.674 × 10⁻¹¹ N·m²/kg², R = 6.371 × 10⁶ m.
• Altitude: climbing weakens gravity by the inverse-square law: g(h) = GM/(R + h)².
• Depth: going down, rock above you pulls upward and cancels part of the pull below. For a uniform Earth, g(d) = g(1 − d/R).
• Rotation: Earth's spin flings you slightly outward. At latitude λ, g is reduced by ω²R cos²λ (ω = 7.292 × 10⁻⁵ rad/s) — maximum at the equator, zero at the poles.

Formulas

• Altitude h: g(h) = GM/(R + h)²; for h much smaller than R, g(h) ≈ g(1 − 2h/R).
• Depth d: g(d) = g(1 − d/R) (uniform-density model; g = 0 at the centre).
• Rotation at latitude λ: g′(λ) = g − ω²R cos²λ, with ω²R ≈ 0.034 m/s².
• Units: m/s²; h, d, R in metres.

Worked Example

How much weaker is gravity on the Mount Everest summit (8848 m) than at sea level?

1. Fractional change: 2h/R = 2 × 8848 / 6.371 × 10⁶ = 17696 / 6.371 × 10⁶ ≈ 0.002778.
2. g at summit = 9.8 × (1 − 0.002778) = 9.8 × 0.997222 ≈ 9.77 m/s².
3. Difference ≈ 0.027 m/s², about 0.28% weaker.

A 70 kg climber weighs 686 N at sea level but about 684 N on the summit — roughly 200 g lighter on a scale. Earth's rotation alone trims equatorial g by about 0.034 m/s², a similar-sized effect from a completely different cause.

Common Misconceptions

• "Gravity is 9.8 everywhere on Earth." It runs from about 9.78 m/s² at the equator to 9.83 m/s² at the poles — small but measurable.
• "Gravity keeps increasing as you dig deeper." In a uniform Earth it decreases linearly with depth, because the shell of rock above pulls back. It reaches zero at the centre.
• "Astronauts float because there is no gravity in space." At the ISS altitude g is still ~8.7 m/s². They float because they are in continuous free fall, not because gravity vanished.

Real-World Applications

• Geophysical surveys: tiny local g variations reveal oil, minerals and groundwater; GRACE satellites track water loss from space.
• Satellite orbits: launch planners use exact local g and Earth's bulge to hit precise orbits.
• Precision metrology: standards labs correct for latitude and altitude; pendulum clocks tick at different rates at different latitudes.

Gravity is not one fixed number: it fades with altitude as g(1 − 2h/R), falls with depth as g(1 − d/R), and is trimmed by Earth's spin by up to 0.034 m/s² at the equator. Together these small variations map the Earth's shape, rotation and hidden underground mass.