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

G = 6,786 10⁻¹¹ m³ / s² kg

Step-by-step 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]