Change of g Equation: How to Calculate It

This calculator works out the change in gravitational acceleration (Δg) between two points, usually one at a reference location and one at a different altitude or latitude. You enter the starting gravity, the ending gravity, or the heights if you prefer, and it returns the change in m/s² and as a percentage. The change of g equation is simply Δg = g₂ − g�, where g� is the initial gravitational acceleration and g₂ is the final one.

Formula

The change in gravitational acceleration is the difference between two values of g:

Δg = g₂ − g�

where:

• Δg = change in gravitational acceleration (m/s²)
• g� = initial gravitational acceleration (m/s²)
• g₂ = final gravitational acceleration (m/s²)

If you want the percent change, use:

Percent change = (Δg / g�) × 100%

When you don’t have measured g values, you can compute g at a height h above sea level using the approximation:

g(h) ≈ g₀ × (R / (R + h))²

where g₀ is the standard gravity at sea level (9.80665 m/s²) and R is Earth’s mean radius (6,371 km). This formula is accurate for small heights compared to R.

Worked Example

Using the calculator’s default values:

• g� = 9.81 m/s²
• g₂ = 9.80 m/s²

Step 1: Subtract g₂ − g�:

Δg = 9.80 − 9.81 = −0.01 m/s²

Step 2: Compute percent change:

Percent = (−0.01 / 9.81) × 100 = −0.1019%

The calculator rounds to 4 decimal places for Δg, giving −0.0100 m/s², and 2 decimals for percent, giving −0.10%. A negative Δg means gravity decreased.

When to Use the Change of g Equation

You use this equation whenever you need to quantify how much gravity differs between two locations or two altitudes. Common situations include:

• Adjusting a scale or sensor that was calibrated at sea level and is now used at a higher elevation.
• Comparing gravity readings from a gravimeter taken at different times or places.
• Estimating the effect of altitude on a pendulum clock or a mass measurement.
• Teaching or lab exercises that measure g with a pendulum or free-fall apparatus.

The equation itself is trivial, but the difficulty lies in obtaining accurate g� and g₂. You can measure them directly with a gravimeter, or you can calculate them using the altitude formula above.

How to Choose the Right Values

To get a meaningful result, you need to be consistent about what g� and g₂ represent. Ask yourself:

• Are both values measured at the same latitude? If not, the difference includes latitude effects.
• Are both values at the same elevation? If not, the difference is mostly due to altitude.
• Are you using local gravity or standard gravity? Standard gravity (9.80665 m/s²) is a defined constant, not a measurement.

For altitude changes, use the formula g(h) = g₀ × (R/(R+h))². For example, at the top of Mount Everest (8,848 m), g ≈ 9.80665 × (6371/(6371+8.848))² ≈ 9.779 m/s². The change from sea level is about −0.027 m/s², or −0.28%.

Common Mistakes and Misconceptions

A frequent error is mixing up the sign. If you move to a higher altitude, gravity decreases, so Δg is negative. If you move downward (e.g., into a mine), gravity can increase slightly, so Δg is positive—though the relationship is not linear deep inside the Earth.

Another mistake is using standard gravity instead of local gravity. The standard value 9.80665 m/s² is an average; actual gravity at sea level varies from about 9.78 at the equator to 9.83 at the poles. If you use the wrong baseline, your change will be off.

Also, don’t confuse the change in g with the change in weight. Weight = mass × g, so a 1% change in g means a 1% change in weight, but mass stays the same.

Limits of the Simple Equation

The simple Δg = g₂ − g� works for any two points, but it doesn’t tell you why the change happened. To interpret it, you need a model. The altitude formula assumes a spherical, uniform Earth and ignores:

• Earth’s rotation (centrifugal effect)
• Non-spherical shape (equatorial bulge)
• Local density variations (mountains, ore deposits)
• Height above ground vs. altitude above sea level

For most everyday purposes—like adjusting a scale or understanding altitude effects—the simple formula is accurate enough. For geophysical surveys, you need the full free-air and Bouguer corrections.

Frequently Asked Questions

Why does gravity decrease with altitude?

Gravity follows an inverse-square law: the force between two masses weakens with the square of the distance between their centers. As you move farther from Earth’s center, the distance increases, so g decreases. The formula g(h) = g₀ × (R/(R+h))² captures that effect.

What is the difference between Δg and g-force?

Δg is a change in gravitational acceleration, measured in m/s². “g-force� is a dimensionless ratio of an acceleration to standard gravity (9.80665 m/s²). For example, a pilot experiencing 3 g has an acceleration of 3 × 9.80665 = 29.42 m/s², but that’s not a change in g itself.

Can I use this calculator to find gravity on another planet?

Yes, if you enter the surface gravity of that planet as g� and the gravity at some altitude as g₂ (or compute both using the planet’s radius and mass). The change equation works universally, but the altitude formula must use that planet’s radius.

How much does gravity change per meter of altitude?

Near Earth’s surface, gravity decreases by roughly 3.1 × 10�� m/s² per meter of altitude. So climbing 1,000 m reduces g by about 0.0031 m/s², or 0.031%. This is a useful rule of thumb for quick estimates.

Why does my scale give different readings at high altitude?

A spring scale measures force (weight), which is mass × g. If g is lower at altitude, the force is lower, so the scale reads a smaller mass even though the actual mass hasn’t changed. A balance scale that compares masses directly is not affected.

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in g1 | Initial g | m/s² | 9.81 | 0
in g2 | Final g | m/s² | 9.80 | 0
out dg | Change in g | m/s² | g2 - g1 | 4
out pct | Percent change | % | (g2 - g1) / g1 * 100 | 2