)by observing that the centripetal acceleration of the moon around the earth is ac = 2.7 × 10-3 m/s2, what is the gravitatonal constant g, in cubic meters per kilogram per square second? assume the earth has a mass of me = 5.99 × 1024 kg, and the mean distance between the centers of the earth and moon is rm = 3.88 × 108 m.

G = 6,786 10⁻¹¹ m³ / s² kg

Explanation:

The law of universal gravitation is

         F = G m M/ r²

Where G is the gravitational constant, m and M are the masses of the bodies and r is the distance from their centers

Let's use Newton's second law

         F = m a

The acceleration is centripetal

          a =  

We replace

         G m M / r² = m    

         G =     r² / M

Let's replace and calculate

         G = 2.7 10⁻³ (3.88 10⁸)² / 5.99 10²⁴

         G = 6,786 10⁻¹¹ m³ / s² kg

Let's perform a dimensional analysis

[N m²/kg²] = [kg m/s²   m² / kg²] = [m³ / s² kg]

one "light year" is the distance that light travels, through vacuum, in one julian year.   that's 365.25 days, of 24 hours each.