The Cosmologies of Galileo and Newton

Example 3: An apple of my eye…litte $g$ and big $G$

The Question:

Place your apple on the ground—notice that it’s distance from the center of the Earth is $R_E$. Let’s calculate the force on that little apple with mass $m$ due to the big Earth, with mass $M_E$. Here’s a picture:

The Answer:

Newton taught us that the force between them is from Newton’s Gravitational law

Now isolate the little m outside of the other terms:

and can you see that we’ve discovered an acceleration buried in the middle term by recognizing $F=ma$ in it:

Since this situation is an apple on the surface of the Earth, what we’ve really found is a derivation for Galileo’s $g$! So we can just identify:

All constants. Try it:

• $M_E=5.972 \times 10^{24}$ kg
• $R_E= 6378 \times 10^3$ m
• $G=6.67 \times 10^{-11}$ m$^3$kg$^{-1}$m$^{-2} *$

Bingo.