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Gravitational force is a fundamental concept in mechanics, representing the attractive force exerted by the Earth on objects in its vicinity. This force is mathematically expressed by Newton's law of universal gravitation. It states that any two masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers:

• \( F = \frac{G \, m_1 \, m_2}{r^2} \)

Here,

• \( G \) is the gravitational constant \( (6.674 \times 10^{-11} \, \text{N} \, (\text{m/kg})^2) \),
• \( m_1 \) and \( m_2 \) are the masses of the objects, and
• \( r \) is the distance between the centers of the two masses.

On Earth, this force manifests as the weight of an object that pulls it toward the center of the planet. Understanding gravitational force is critical for analyzing how objects behave under the influence of gravity.

The acceleration due to gravity, denoted by \( g \), is the rate at which objects accelerate when falling freely under the influence of Earth's gravity. On Earth's surface, \( g \) has an approximate value of \( 9.8 \, \text{m/s}^2 \). However, \( g \) is not a constant throughout space. Instead, it diminishes with altitude above the Earth's surface. This is because as you move away from the Earth's center, the effective radius \( r \) in the equation:

• \( g' = g \left( \frac{R}{R + h} \right)^2 \)

increases, as \( R \) is Earth's radius and \( h \) is the height above Earth's surface. As a result, \( g' \), the gravitational acceleration at height \( h \), becomes smaller. This decrease affects how forces like weight change, influencing phenomena such as spring extension at higher altitudes.

Spring extension refers to the elongation of a spring when a force is applied. According to Hooke's Law, the force exerted by a spring is directly proportional to its extension:

• \( F = kx \)

where \( k \) is the spring constant, and \( x \) is the extension. In the context of gravity, if a mass is suspended from the spring, the extension occurs due to the gravitational force:

• \( x = \frac{F}{k} \)

When the gravitational force decreases, as it does with altitude, the extension \( x \) reduces as well. The exercise illustrates this by showing a reduced spring extension at a height of 2624 km above the Earth's surface, compared to the extension at Earth's surface. This reduced spring extension signifies less gravitational pull at that height.

Earth's radius is a fundamental measurement that plays a significant role in calculations involving gravity. It is approximately \( 6400 \, \text{km} \). The radius is crucial for determining how gravity changes with altitude, as seen in the gravitational acceleration formula:

• \( g' = g \left( \frac{R}{R + h} \right)^2 \),

where \( R \) is Earth's radius and \( h \) is the height above Earth's surface. Earth's radius also factors into many scientific and engineering calculations when considering the Earth's gravitational field. For example, as an object moves further from the Earth's surface, the force of gravity acting on it decreases. This affects both the object's weight and the extension of springs subjected to gravitational forces, as evident in the original exercise scenario.