According to Newton's gravitational law, the magnitude of the gravitational force is:
[tex]F=-G\frac{m_1m_2}{r^2}[/tex]
In this case, we have: [tex]m_1'=2m_1,m_2'=2m_2, r'=4r[/tex]
Therefore, replacing this values and solving for F' in function of F:
[tex]F'=-G\frac{m_1'm_2'}{r'^2}\\F'=-G\frac{2m_1(2m_2)}{(4r)^2}\\F'=-G\frac{4m_1m_2}{16r^2}\\F'=\frac{1}{4}(-G\frac{m_1m_2}{r^2})\\F'=\frac{1}{4}F[/tex]
[tex]\frac{1}{4}[/tex] F
Newton's law of gravitation states that the force, F, that exists between the two objects of masses m₁ and m₂ is directly proportional to the product of the masses of the objects and inversely proportional to the square of the distance, r, between the bodies. i.e
F ∝ (m₁m₂ / r²)
F = Gm₁m₂/r² ------------------(i)
Where;
G = proportionality constant called the gravitational constant.
Now, when the masses are changed to 2m₁ and 2m₂, and the distance is changed to 4r, then according to equation (i) the new force (say F₂) becomes;
F₂ = G (2m₁)(2m₂) / (4r)²
F₂ = 4Gm₁m₂ / 16r²
F₂ = Gm₁m₂ / 4r²
F₂ = [tex]\frac{1}{4}[/tex] Gm₁m₂/r² -------------------------(ii)
Comparing equations (i) and (ii), equation (ii) can be re-written as;
F₂ = [tex]\frac{1}{4}[/tex] F
Therefore, the magnitude of the new gravitational force is a quarter of the old gravitational force, F. i.e [tex]\frac{1}{4}[/tex] F
