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The Law of Universal Gravitation is a fundamental principle that explains how objects attract each other with a force proportional to their masses and inversely proportional to the square of the distance between their centers. This law, formulated by Sir Isaac Newton, is expressed as:

• Formula: \( F = G \frac{m_1 m_2}{r^2} \)
• \( F \) is the gravitational force between two objects.
• \( m_1 \) and \( m_2 \) are the masses of the objects.
• \( r \) is the distance between the centers of the two masses.
• \( G \) is the gravitational constant.

This principle is crucial to understanding gravitational acceleration on Earth. Gravitational acceleration at a point is given by the formula \( g = \frac{G M}{r^2} \), where \( M \) is the Earth's mass, and \( r \) is the distance from the Earth's center. Thus, the further you move away from the Earth’s surface, the smaller the value of \( g \) becomes.
This relationship helps in calculating how gravity affects objects differently depending on their distance from the Earth.

The Gravitational Constant, denoted by \( G \), is a fundamental constant used in the calculation of gravitational forces between two masses. It is the key to quantifying the strength of the gravitational force and appears in Newton's equation for the Law of Universal Gravitation:

• G = 6.674 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2

This constant is essential when calculating how different masses interact due to gravity. In the context of gravitational acceleration \( g \), \( G \) helps us understand how the Earth’s mass exerts a force on objects, pulling them towards its center. Since \( G \) is a constant, it helps ensure that the universal gravitational equation remains consistent and applicable across various situations, whether calculating gravity on the Earth’s surface or high above it.
In our problem, \( G \) is used alongside the Earth’s mass \( M \) and the changing distance \( r \) to find how acceleration due to gravity changes with distance from the Earth's surface.

The radius of the Earth is a crucial measurement in calculations related to gravitational acceleration. In the context of the given exercise, the radius of the Earth \( R \) helps determine how gravity weakens with height.

• The average radius is approximately \( 6,371 \text{ km} \).
• It represents the distance from the Earth's center to its surface.
• Influences the calculation for \( g = \frac{G M}{r^2} \) at different distances.

When solving the problem to find the height at which gravity becomes \( 0.980 \text{ m/s}^2 \), the Earth’s radius becomes a baseline reference for the distance \( r \) used in the gravitational acceleration formula.
By understanding that \( g \) decreases with increased distance \( r \), one can deduce the height above the surface where gravity is reduced to a specified value. Solving for \( r \) involves equating known gravitational conditions at the Earth's surface to those at the desired height, illustrating the direct link between gravitational acceleration and the Earth’s radius.