Does a particular object have the same weight on every part of Earth or does it vary?
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There is an entire field of Geophysics called gravimetry dedicated to measuring the magnitude of the gravitational field.
First, we should distinguish between weight (a force) and gravity (an acceleration). Gravity is the acceleration that Earth gives to objects near its surface due to the gravitational force.
The acceleration of an object near the surface of Earth is due to the combined effects of gravity and the centrifugal acceleration from the rotation of Earth. The resulting acceleration is weakest at the equator and maximum at the poles, with a difference in magnitude of about 0.5%, because the outward centrifugal force produced by rotation is larger at the equator than at the poles.
The standard gravity, $ɡ_n$ or $ɡ_0$, (the expected or mid-range value of the gravitational acceleration of an object near the surface of Earth) is 9.80665 $m/s^2$.
Gravimetry measures and studies gravitational anomalies: local variations in topography (e.g., mountains) and geology (rock density) that cause changes in the gravitational field. These gravitational anomalies are measured with instruments called gravimeters. Rocks with lower density, such as sedimentary rocks, result in negative anomalies.
Another effect affecting gravity is altitude, as an increase in altitude will result in a change in gravity of about 0.03% per kilometer. The maximum difference in gravity is about 2 Gal (0.02 $m/s^2$) from sea level to the top of Mount Everest.
The gravity field of Earth has been measured recently with a variety of satellites (GRACE, CHAMP, GOCE). The resulting data can be used to generate global geoids such as EGM96 (geoid calculator). Gravitational anomalies can be extracted by calculating differences with respect to the geoid. The range of variations is of order ±300 $mGal$ (±0.003 $m/s^2$).
Other factors have a small effect on gravity, such as the gravitational force from the Moon and the Sun that causes the tides. The fluctuations associated with these effects are of order 0.2 $mGal$ (2 $µm/s^2$).
The weight that an object with a constant mass experiences on different parts of the Earth's surface will be slightly different for different locations, largely dependent on the height above sea level and the latitude.
This is due to differencies in the gravitational strength on the Earth's surface at different locations.
The accelation due to gravity, $g$, on Earth varies by $0.7\%$ from $9.7639$ $m/s^2$. In Kuala Lumpa (near the equator) the gravitational accelation is $9.766$ $m/s^2$, whereas in Helsinki (nearer the North Pole) it is $9.825$ $m/s^2$.
The Earth is not a perfect sphere, there is a bulge around the equator. Because of this, the surface of the Earth, at the equitorial regions, is farther from the centre of gravity than the surface of the Earth at the poles. This reduces the effect of gravity at the equator.
To add to the previous answer the gravity will vary locally due to the density of the rocks (or other materials) in the area. Gravity meters are used in mineral exploration to find dense rock types associated with certain ore bodies.
Just for completeness changes in the groundwater saturation at a location can have a small effect on the gravity. Filling the pores in unsaturated sediments with water due to increased recharge from precipitation, irrigation, etc. increases the bulk density and thus the gravity.