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The force of gravitational attraction that one body exerts on another.
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The weight of a body is the total gravitational force exerted on the body by all other bodies in the universe. When the body is near the surface of the earth, all other gravitational forces can be neglected and only the weight can be considered as just the earth's gravitational attraction. At the surface of the moon a body's weight would be considered to be the gravitational attraction of the moon, and so on.
If the earth is assumed as a spherically symmetric body with radius RE and mass mE, the weight w of a small body of mass m at the earth's surface (distance RE from its center) is:
Which is the equation for the weight of a body of mass m at the earth's surface. The weight w of a body is the force that causes the acceleration g of free fall, so by Newton's second law of motion, w = mg. Equating Eq1 with the previous equation and dividing by m:
This is the equation for the acceleration due to gravity at the earth's surface. The acceleration due to gravity g is independent of the mass m of the body because m doesn't appear in the equation. It can be seen how it follows from the law of gravitation.
All quantities in Eq2 can be measured except for mE, so this relation allows the mass of earth to be computed. Solving Eq2 for mE and using R = 6380 km = 6.38 × 106 m and g = 9.80 m/s2:
This value is very close to the currently accepted value of 5.974 × 1024 kg.
At a point above the earth's surface a distance r from the center of the earth (a distance r − RE above the surface), the weight of a body is given by Eq1 with RE replaced by r:
The weight of a body decreases inversely with the square of its distance from the earth's center.
The apparent weight of a body on earth differs slightly from the earth's gravitational force because the earth rotates and is therefore not precisely an inertial frame of reference. This effect has been ignored in the above discussion and it has been assumed that the earth is an inertial system.