Question

) 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.

Answer 1

Answer:

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 = $a_{c}$  

We replace

         G m M / r² = m  $a_{c}$  

         G =  $a_{c}$   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]