Gravitational Force Near Earth

Main Idea

The gravitational force, as it acts near Earth's surface, is a simplified version of Newton's Law of Universal Gravitation. As a reminder, Newton's Law of Universal Gravitation states that any two bodies attract each other with a force that is dependent on the product of the two bodies' masses and is inversely dependent on the square of the distance between the two. Near Earth's surface, however, this can be simplified. The Earth is one of the bodies that we are interested in for determining this force, and because we know the mass of the Earth, the radius of the Earth, and the gravitational constant, we can simply this force to an easier form. This is only applicable at distances near the Earth's surface, as we are using the radius of the Earth as the distance between the two objects. Generally this is acceptable because the distance between the object of interest and Earth's surface is [math]\displaystyle{ \lt \lt }[/math] the distance from the center of the Earth to Earth's surface. Thus this extra distance wouldn't affect the unsimplified calculation.

A Mathematical Model

The gravitational force in general is equal to:

[math]\displaystyle{ {F}_{grav}=G \frac{m_1 m_2}{r^2}\ }[/math]

where,

• F is the gravitational force between the masses;
• G is the gravitational constant, [math]\displaystyle{ 6.674×10^{−11} \frac{N m^2}{kg^2}\ }[/math];
• m₁ is the mass of the first object;
• m₂ is the mass of the second object;
• r is the distance between the centers of both masses.

Near Earth's surface, however, [math]\displaystyle{ \frac{GM_{Earth}}{R_{Earth}^2} }[/math] is equal to the gravitational constant near Earth's surface, g, which is equal to [math]\displaystyle{ 9.8 \frac{m}{s^2} }[/math]. This calculation is shown below:

[math]\displaystyle{ g = \frac{GM_{Earth}}{R_{Earth}^2}= \frac{(6.67*10^{-11} \frac{m^3}{kgs})(5.972*10^{24} kg)}{(6.378*10^{6} m)^2} = 9.8 \frac{m}{s^2} }[/math]

This then puts the force near Earth due to gravity into a much simpler form:

[math]\displaystyle{ |\vec{\mathbf{F}}_{grav}|= mg }[/math]

where,

• g is the near-Earth gravitational constant, as defined above to be [math]\displaystyle{ 9.8 \frac{m}{s^2} }[/math]
• m is the mass of the object whose behavior we are interested in