Weight¶
The weight of an object is the total gravitational force excreted on the object by all the other objects in the universe.
If an object is at the surface of the earth we can neglect all the other forces.
If the earth would be a symmetric sphere with radius \(R_E\) and mass \(m_E\) the weight of an small object with mass \(m\) at the surface of the earth would be.
\[
w = F_g = \frac{G m m_E}{R^2_E}
\]
From this we can derive the gravitational constant:
\[\begin{split}
w = mg \\
mg = \frac{Gmm_e}{R^2_E} \\
g = \frac{Gm_E}{R^2_E}
\end{split}\]
Here we can see that the gravitational constant is independent of the objects weight and depends only on the mass of Earth (or other celestial object) and its Radius.
The weight of an object decreases with the square of its distance from earth center.