14.6:
Acceleration due to Gravity on Earth
An object is considered to be free-falling when the only force acting on it is the Earth's gravitational force. The acceleration of such a free-falling object is called acceleration due to gravity, denoted by g.
According to Newton's second law of motion, the magnitude of the force acting on the object is equal to its mass times its acceleration. It is also known as the object's weight.
Equating mg with the gravitational force equation, the acceleration due to gravity of the object can be expressed as the product of the gravitational constant and the Earth's mass, divided by the square of its distance from the Earth's center.
The average measured value of g close to the Earth’s surface is 9.8 m/s2. Since g is independent of the object's mass, all masses near the Earth's surface free-fall with the same acceleration.
Therefore, substituting the values of acceleration due to gravity, the gravitational constant, and the distance as the Earth's radius, the Earth’s mass is estimated to be 5.97 × 1024 kilograms.
14.6:
Acceleration due to Gravity on Earth
According to Newton's law of gravitation, the gravitational force on a body is proportional to its mass. According to Newton's second law of motion, the acceleration produced by an external force is inversely proportional to the force. Hence, the acceleration of an object under an external force of gravitation is independent of its mass.
The acceleration of an object close to the Earth, because of the Earth's gravitational pull, is called the acceleration due to gravity. It is always directed towards the Earth's center.
Of course, any object would apply an equal and opposite pull on the Earth. However, since the Earth's mass is much larger than ordinary objects close to it, its acceleration is negligible. Hence, the Earth serves as an inertial frame of reference, appropriate for studying the dynamics of objects placed on it.
The acceleration due to gravity felt by any object close to the Earth's surface is measured to be about 9.80 m/s². As long as an object is relatively near the Earth's surface, its value can be assumed to be constant. For example, the acceleration due to gravity on Mount Everest, the tallest mountain on the Earth's surface, is not significantly different for all practical purposes. However, at distances hundreds of kilometers above the Earth's surface, its value is considerably different. Moreover, the acceleration due to gravity varies from the equator to the poles because of the Earth's rotation about its axis.
Since the mass and average radius of the Earth are related to the gravitational constant and the acceleration due to gravity on Earth, if one is known, the other can be estimated. For example, if we know the Earth's radius through geometrical measurements, we can measure its mass.
This text is adapted from Openstax, University Physics Volume 1, Section 13.2: Gravitation Near Earth's Surface.