Gravity predictions

Here is the process I use to predict the downward acceleration for my $10,000 gravity challenge. The specific values for the inputs are listed at the end of the page. The process and the values are the same as specified in the official challenge. This data is also available in a spreadsheet.

The measurements, locations, and elevations are part of this specific set. A large collection of IGSN71 measurements can be found here.

Here is the two step process to predict the total downward acceleration on any location on the surface of the earth.

Start with the Law of Gravitational Attraction:

F = G * M1 * M2 / r2

  • F = Force due to gravity
  • G = Gravitational constant = 6.674215 × 10-11 m3 / kg s2
  • M1 = Mass of the earth = 5.972245 x 1024 kg
  • M2 = Mass of an object on the earth
  • r = distance between centers of mass

We are interested in acceleration because it applies to any mass equally. We can use the Second Law of Motion:

F = m * a

  • F = Force
  • m = mass of an object being accelerated, in out case, an object on the surface of the earth, the same as M2 in the first equation.
  • a = acceleration

The variables M₂ and m are the mass of an object on the surface of the earth, they cancel giving a final equation:

m * a = G * M1 * M2 / r2

a = G * M1 / r2

The other value we need is the acceleration away from the earth due to centrifugal acceleration. This is the equation:

a = v2 / r

  • a = centrifugal acceleration
  • v = tangential velocity
  • r = distance to the center of rotation

Location 1: Kauai, Hawaii, USA

  • Coordinates: 22.12611111, -159.6682222
  • Elevation: 1130 meters
  • WGS84 distance from center of earth: 6376255.1 meters
  • Tangential speed: 1552.071 km/h
  • Gravitational field formula: a = GM/r^2
    • Predicted acceleration applying formula: 9.80407084m/s^2
  • Centrifugal acceleration formula: a = v^2/r
    • Predicted centrifugal force applying formula: 0.02915093461 m/s^2
  • Total predicted downward acceleration: 9.77491990 m/s^2
  • Measured acceleration: 9.787882079 m/s^2
  • Difference between predicted and measured: 0.01296217823 m/s^2

Location 2: International Falls, MN, USA

  • Coordinates: 48.58472222, -93.16222222
  • Elevation: 339.8 meters
  • WGS84 distance from center of earth: 6366494.218 meters
  • Tangential speed: 1109.762 km/h
  • Gravitational field formula: a = GM/r^2
    • Predicted acceleration applying formula: 9.84315223m/s^2
  • Centrifugal acceleration formula: a = v^2/r
    • Predicted centrifugal force applying formula: 0.01492637503 m/s^2
  • Total predicted downward acceleration: 9.81923001 m/s^2
  • Measured acceleration: 9.808252073 m/s^2
  • Difference between predicted and measured: 0.01097793685 m/s^2

Location 3: Fortaleza, Brazil

  • Coordinates: -3.876944444, -38.42583333
  • Elevation: 23.5 meters
  • WGS84 distance from center of earth: 6378063.551 meters
  • Tangential speed: 1670.564 km/h
  • Gravitational field formula: a = GM/r^2
    • Predicted acceleration applying formula: 9.79851189m/s^2
  • Centrifugal acceleration formula: a = v^2/r
    • Predicted centrifugal force applying formula: 0.03376232863 m/s^2
  • Total predicted downward acceleration: 9.76474956 m/s^2
  • Measured acceleration: 9.780744882 m/s^2
  • Difference between predicted and measured: 0.01599532522 m/s^2

The value for the gravitational constant and the mass of the earth are from this measurement:

https://doi.org/10.48550/arXiv.gr-qc/0006043

This is the value of G to be used: 6.674215 × 10^-11 m^3 * kg^−1 * s^−2
The value of the earth’s mass to be used: 5.972245 * 10^24 kg

I use WGS 84 for the distance from sea level and the given elevation from the IGSN71 measurement to obtain the distance to the center of the earth and the tangential velocity. This web page does both, be certain to select “Don’t Adjust” for the Geoid height.

https://www.summitpost.org/distance-to-the-center-of-the-earth/849764