Variation of g With Latitude: Formula and Explanation

The standard approximate formula for gravitational acceleration at latitude φ is g(φ) = 9.780327[1 + 0.0053024 sin²φ − 0.0000058 sin²(2φ)] m/s². Enter latitude in degrees to estimate local standard gravity; the result reflects Earth’s rotation and flattened shape, but not local elevation or geological variations.

Formula

The calculator uses the International Gravity Formula 1980, also called the GRS 80 normal-gravity formula:

g(φ) = 9.780327[1 + 0.0053024 sin²φ − 0.0000058 sin²(2φ)] m/s²

• g(φ) is the calculated standard gravitational acceleration, in metres per second squared (m/s²).
• φ (phi) is geographic latitude, in degrees. The calculator’s degree-based sine function is used.
• 9.780327 m/s² is the formula’s equatorial reference value.

Latitude ranges from −90° at the South Pole through 0° at the equator to +90° at the North Pole. The squared sine terms make the result the same at matching north and south latitudes. The formula estimates normal gravity on the reference ellipsoid, rather than the gravity at every actual point on Earth.

Worked Example

Using the default latitude of 45°:

1. sin(45°)² = 0.5.
2. sin(2 × 45°)² = sin(90°)² = 1.
3. Substitute: g = 9.780327 × [1 + 0.0053024 × 0.5 − 0.0000058 × 1].
4. The bracket is 1.0026454, so g = 9.80619920 m/s².

Rounded to the calculator’s six decimal places, the answer is 9.806199 m/s². This is about 980.620 cm/s², or 32.172 ft/s². The customary-unit figure is a conversion of the metric result, not a separate measurement.

Why Gravity Changes With Latitude

Gravity is slightly weaker near the equator and stronger near the poles. Two main Earth-wide effects explain this pattern.

• Earth’s rotation: In a rotating frame, the outward centrifugal effect reduces effective weight. It is greatest at the equator and falls to zero at the poles.
• Earth’s shape: Earth is an oblate spheroid, wider around the equator than from pole to pole. The equatorial surface is farther from Earth’s centre, which also contributes to lower gravity there.

The latitude formula represents the combined pattern through measured coefficients. It is not simply a calculation of centrifugal acceleration subtracted from one universal gravity value: Earth’s changing distance from its centre and the distribution of its mass matter as well.

How to Use and Read the Result

Enter a geographic latitude between −90° and +90°. Use degrees, not radians; for example, enter 45 for 45°, not 0.785. North and south latitudes of equal magnitude give the same result in this model. The output is acceleration, not force or the mass of an object.

To estimate an object’s weight force near the surface, multiply its mass in kilograms by g in m/s²; the resulting force is in newtons. For instance, a 1 kg mass at the worked-example latitude has a weight force of about 9.806 N under the model. An object’s mass does not change with latitude, although its weight force does.

For a quick comparison, the formula gives about 9.780327 m/s² at the equator (0°) and about 9.832186 m/s² at either pole (90°). The difference is roughly 0.052 m/s², or about 0.53% of the equatorial value. These figures are normal-gravity estimates, not promises that a gravimeter at any particular location will read exactly that value.

Limits and Common Mistakes

Latitude alone does not determine measured local gravity to high precision. Elevation, nearby mountains and rock density, underground mass distribution, tides, and variations in Earth’s shape can affect observations. The formula describes normal gravity on a reference ellipsoid; it does not apply an altitude correction or account for local geology.

• Mixing up degrees and radians: This calculator expects degrees. A radian value entered as though it were degrees produces the wrong latitude.
• Confusing latitude and longitude: Longitude does not go into this formula. Latitude measures position north or south of the equator.
• Using g as a force: The output is acceleration. Weight requires multiplying by mass.
• Overinterpreting decimal places: Six digits are retained for a consistent calculation, but local gravity can differ from the normal model. Extra displayed precision does not remove those physical differences.

For ordinary school calculations, the approximation g ≈ 9.81 m/s² is often sufficient. Use the latitude formula when comparing locations or when the latitude-dependent difference matters. For precise surveying or geophysics, use an appropriate local gravity model or measured value and account for height and local conditions.

Frequently Asked Questions

What is the formula for g at latitude?
A common standard formula is g = 9.780327[1 + 0.0053024 sin²φ − 0.0000058 sin²(2φ)] m/s², where φ is latitude in degrees.

Where is g greatest, at the equator or the poles?
Normal gravity is greatest at the poles and smallest at the equator. Earth’s rotation and equatorial bulge both contribute to the difference.

Does the formula use latitude in degrees or radians?
The formula is commonly stated with latitude in degrees when using degree-based sine values. This calculator takes degrees directly.

Does altitude change g too?
Yes. Gravity generally decreases with height above Earth’s surface, but this latitude-only formula does not include an altitude correction.

Why does the same latitude give the same answer north and south?
Latitude appears in squared sine terms, so changing its sign does not change the result. Local geography can still make actual measured gravity differ between two locations.

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in lat | Latitude | deg | 45 | -90
out g | Gravity acceleration | m/s^2 | 9.780327 * (1 + 0.0053024 * sind(lat)^2 - 0.0000058 * sind(2 * lat)^2) | 6