Weight at latitude calculator
A 70 kg mass reads 686.4655 N at the poles and a little less toward the equator, where Earth's spin takes about 0.034 m/s² off gravity. Enter mass and latitude to get the scale reading in newtons. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Mass
- 70
- Latitude
- 45
- G effective
- 9.7897111225 N
- Weight
- 685.2797785742
- Centrifugal accel
- 0.0169388775
- G equator
- 9.772772245
A bathroom scale does not report your mass. It reports the contact force that balances your apparent weight, and that force changes with geographic latitude because Earth is spinning. This weight at latitude calculator takes a mass in kilograms and a latitude in degrees and returns the measured weight in newtons after subtracting the local-vertical component of centrifugal acceleration from standard gravity. The model is the first-order rotating-sphere correction used in introductory geophysics and mechanics: effective g equals g-zero minus omega-squared times R times cosine-squared of latitude. Poles keep the full reference g; the equator is lighter by about 0.034 metres per second squared. Latitude outside minus ninety to ninety is rejected as invalid input.
What a scale measures when Earth is spinning
A spring scale or load cell reports the contact force that keeps a mass from accelerating through the floor, not the intrinsic mass of the object. On a rotating planet that contact force is smaller than the true gravitational attraction because part of the attraction is spent supplying the centripetal acceleration of co-rotation with Earth. In the first-order spherical model the centrifugal acceleration has magnitude omega squared times the perpendicular distance to the spin axis, which is the mean radius R times the cosine of geographic latitude.
Only the component along the local vertical, omega squared R cosine-squared of latitude, subtracts from g. At the poles the axis distance is zero, so a scale reads mass times the reference g-zero. At the equator the full omega-squared R, about 0.0339 metres per second squared with the IERS conventional sidereal spin and a mean radius of 6371000 metres, is subtracted from standard gravity 9.80665 metres per second squared.
Mid-latitudes interpolate with cosine squared, so forty-five degrees is not a linear halfway point: cosine of forty-five squared is exactly one half, and that is the equator-to-pole split for this rotational term. Mass never changes when you move north or south. What changes is measured weight, the product of mass and effective g.
This calculator isolates that rotational correction and does not add Earth's flattening, local geology, height above the geoid, or buoyancy. Those effects matter for a gravimeter survey; they are out of scope for a rotation-only weight-at-latitude tool. The response therefore answers one question: given this mass and this latitude, what would a scale read if the only latitude dependence were Earth's spin.
Input contract, formula, and the numbers you get back
Send two required finite numbers: mass in kilograms, which must be non-negative, and latitude in degrees, which must lie between minus ninety and ninety inclusive. A latitude of ninety-one, minus one hundred, or any other value outside the closed interval is rejected with a message that latitude must be between minus ninety and ninety degrees. Negative mass is rejected because measured weight has no meaning for negative mass in this calculator.
Zero mass is valid and returns zero weight at every latitude. The engine evaluates W equals m times (g-zero minus omega-squared R cosine-squared of phi) with omega 7.292115 times ten to the minus five radians per second, R 6371000 metres, and g-zero 9.80665 metres per second squared. At the exact poles the cosine is forced to zero so floating-point cosine of ninety degrees cannot leak a residual.
Where a latitude-dependent weight belongs in study and software
Students meet this effect when a textbook asks why you weigh slightly less at the equator than at the poles even if mass is unchanged, or when a lab compares a spring scale to an equal-arm balance after a field trip. Teachers can generate answer keys for a list of cities without trusting opaque apps that silently mix flattening into a problem that only asked for rotation.
Survey and geophysics courses use the same first-order term as the starting point before they introduce the Somigliana formula and the free-air gradient; this page stays on the rotation term so the lesson does not smuggle in an ellipsoid. Metrology notes can show that a one-part-in-three-hundred shift from equator to pole is small next to a cheap scale's resolution and large next to a laboratory balance. On this page the work runs entirely in your browser.
Pair it with the elementary weight-force helper when you already know local g, and with planetary-weight tools when the attracting body is not Earth. Keep requests small, pin the rounded newtons in tests, and you get a reproducible rotation-only weight without standing up a gravity library of your own.
What you can do with it
Homework and lecture checks
Compute measured newtons from mass and latitude so students can see the equator-to-pole split without mixing in flattening.
Travel and city comparisons
Compare the scale reading for the same mass at Quito, Cairo, and Helsinki using only the rotational cosine-squared term.
FAQ
How does Earth rotation change measured weight with latitude?
A scale reads W = m · (g0 − ω² R cos²φ). The subtracted term is the local-vertical centrifugal acceleration. It is largest at the equator and zero at the poles.
What latitude range is accepted?
Latitude must be a finite number from −90 to 90 degrees inclusive.
Is mass the same as the weight this calculator returns?
No. Mass is an invariant input in kilograms. The returned weight is the apparent force in newtons after the rotational correction at the given latitude.
Does the model include Earth's flattening?
No. It isolates the first-order centrifugal term on a sphere. Flattening, geology, and height above the geoid are out of scope.
What does the weight at latitude calculator cost?
It is free in your browser on this page.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/mech/weight-at-latitude \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"mass":70,"latitude":45}'const res = await fetch("https://api.kit.forhosting.com/mech/weight-at-latitude", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"mass": 70,
"latitude": 45
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/weight-at-latitude",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"mass": 70,
"latitude": 45
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/weight-at-latitude", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"mass":70,"latitude":45}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"mass":70,"latitude":45}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/weight-at-latitude", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"mass": 70,
"latitude": 45
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.weight_at_latitude",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Limits
max_abs_mass | 1000000000000000 |
min_latitude | -90 |
max_latitude | 90 |
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Input contract, formula, and the numbers you get back
Aliases m and M are accepted for mass; lat, lat_deg, and phi are accepted for latitude, so spreadsheets and query strings can reuse the same contract. Missing fields, empty strings, non-numeric tokens, infinities, and NaN fail with invalid_input and name the field. The JSON response echoes mass and latitude, then reports g_effective, the measured weight in newtons, the centrifugal acceleration that was subtracted, the equatorial and polar reference g and weights for the same mass, the signed delta from the polar weight, the constants omega, earth_radius, and g0, plus a formula string, a method tag, and a unit hint N. Failed validation is not billed; a successful API call costs $0.002.
Where a latitude-dependent weight belongs in study and software
Through the API each successful lookup costs $0.002, invalid input is not charged, and nothing is stored after the response returns.
Integration use cases
Deterministic test fixtures
Generate reproducible apparent-weight values for CI, tutoring software, and game-physics demos that must not invent g(φ) by hand.