If F₁ is the force exerted by the earth on the moon and F₂ is the force exerted by the moon on earth. Which force is greater according to newton’s law of gravitation?

CONCEPT:

• Newton's law of gravitation states that everybody in this universe attracts every other body with a force, which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
• The direction of the force is along the line joining the particles.
• The magnitude of the gravitational force F is

\(F = G\frac{{{M_1}{M_2}}}{{{R^2}}}\)

Where G = universal gravitational constant, M₁ = mass of 1^st body, M₂ = mass of 2^nd body, and R = distance between the two bodies.

EXPLANATION:

Given - F₁ = Force exerted by the earth on the moon and F₂ = Force exerted by the moon on earth.

The magnitude of the gravitational force acting on the moon by the earth is F₁

\({{\rm{F}}_1} = {\rm{G}} \times {\rm{\;}}\frac{{{{\rm{M}}_{{\rm{moon}}}}{\rm{\;}}{{\rm{M}}_{{\rm{earth}}}}}}{{{{\rm{r}}^2}}}\)      .. (1)

The magnitude of the gravitational force acting on earth by the moon is F₂ is

\({{\rm{F}}_2} = {\rm{G}} \times {\rm{\;}}\frac{{{{\rm{M}}_{{\rm{earth}}}}{\rm{\;}}{{\rm{M}}_{{\rm{moon}}}}}}{{{{\rm{r}}^2}}}\)      .. (2)

Divide equation 1 and 2, we get

\(\Rightarrow \frac{{{{\rm{F}}_1}}}{{{{\rm{F}}_2}}} = \frac{{\frac{{{\rm{G}} \times {{\rm{M}}_{{\rm{moon}}}} \times {{\rm{M}}_{{\rm{earth}}}}}}{{{{\rm{r}}^2}}}}}{{\frac{{{\rm{G}} \times {{\rm{M}}_{{\rm{earth}}}}{\rm{\;}}{{\rm{M}}_{{\rm{moon}}}}}}{{{{\rm{r}}^2}}}}} = 1\)

∴ F₁ = F₂