When the distance between two bodies is doubled and the mass of each body is also doubled. Then which of the following will be the correct statement about the gravitational force?

We know that according to the law of gravitation, the gravitational force between any two bodies that have some mass is given as:

${\text{F}}_{ }=\text{G}.\frac{{\text{m}}_{\text{1}}{\text{m}}_{\text{2}}}{{\text{r}}_{ }^2}$
Where m₁ and m₂ are the masses of the two bodies, r is the distance between them and G is the universal gravitation constant.

Let's now consider F₁ as the initial force and F₂ as the final force.

${\text{F}}_{\text{1}}=\text{G}.\frac{{\text{m}}_{\text{1}}{\text{m}}_{\text{2}}}{{\text{r}}_1^2}$ 

Since, both the masses and the distance are doubled:

${\text{F}}_{\text{2}}=\text{G}.\frac{{\text{2m}}_{\text{1}}{\text{2m}}_{\text{2}}}{{\left(\text{2r}\right)}_1^2}$

${\text{F}}_{\text{2}}=\text{G}.\frac{{\text{4m}}_{\text{1}}{\text{m}}_{\text{2}}}{{\text{4r}}_1^2}$

${\text{F}}_{\text{2}}=\text{G}.\frac{{\text{m}}_{\text{1}}{\text{m}}_{\text{2}}}{{\text{r}}_1^2}$

${\text{F}}_{\text{2}}={\text{F}}_{\text{1}}$

Therefore, the gravitational force will remain unchanged.

Hence the correct option is D.