Acceleration due to Gravity

Section 5.3 gave the gravitational force between two point masses as $F=\dfrac{GmM}{r^2}$. Since the Earth is, to a very good approximation, a uniform sphere, its entire mass M can (by the shell theorem of section 5.3) be treated as though concentrated at its centre when computing the force on an external point object of mass m at distance r from that centre: $F=\dfrac{GMm}{r^2}$.

If no other force acts on the object, Newton's second law $F=ma$ lets this force be converted directly into an acceleration: $a=\dfrac{F}{m}=\dfrac{GM}{r^2}$. This acceleration -- towards the centre of the Earth -- is called the acceleration due to gravity, denoted g. Crucially, the object's OWN mass m cancelled out of the expression entirely: $g=\dfrac{GM}{r^2}$ depends only on the Earth's mass M and the distance r, never on the mass of the falling object itself. This is exactly the property Galileo discovered experimentally: bodies of different mass, dropped from the same height, fall with identical acceleration.

When the object is close to the Earth's surface, $r\approx R$ (the Earth's radius), giving the familiar surface value $g_{surface}=\dfrac{GM}{R^2}\approx9.8$ m/s^2 -- treated as effectively constant for objects near the surface, since the small variation in their distance from the Earth's centre over ordinary heights (a building, a hill) is negligible compared to R itself. …