04 / CHAPTER 09
Newtonian gravity
Use inverse-square interaction and distinguish weight from mass.
ANCHOR EQUATION
Newtonian gravity
In symbols: F = GMm/r²
Gravity is an inverse-square interaction between two masses.
Gravity falls as inverse square
Newtonian gravity between point masses weakens because the same influence spreads over spherical area 4πr². The force is attractive and proportional to both masses.
Combine field and mass
The gravitational field of mass M at distance r has magnitude g(r) = GM/r². A test mass m experiences F = mg(r) = GMm/r². Here G = 6.674×10⁻¹¹ N·m²/kg², M and m are masses, and r is center-to-center separation.
At Earth’s surface, using GM ≈ 3.986×10¹⁴ m³/s² and r = 6.37×10⁶ m gives g = GM/r² ≈ 9.82 m/s². At twice Earth’s radius, g becomes one quarter, 2.46 m/s².
Weight is local; mass is not
The approximation treats Earth as spherical and the object as small compared with r. Close to an irregular body, altitude, latitude, and local mass distribution alter g.
Work it: Calculate the gravitational force between 1000 kg and 2000 kg separated by 10 m. Compare it with the weight of the smaller mass on Earth.