Question

What happens to the gravitational force between two objects when the distance between them is
A. Doubled?
B. Halved?

Answer 1

The Newton’s law of gravitation states that the force acting between two objects is
Directly proportional to the mass of both the objects $F \propto {m_1}{m_2}$
Inversely proportional to the square of distance between the two bodies $F \propto \dfrac{1}{{{r^2}}}$.

Complete step by step solution:
We know that Newton’s law of gravitation states that the gravitational force acting between two bodies is inversely proportional to the square of the distance between the two bodies.
The force is given by $F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ .
When the distance between the bodies gets doubled, the new distance becomes $r' = 2r$
Plugging the values in the formula we get,
$F' = \dfrac{{G{m_1}{m_2}}}{{r{'^2}}}$
But we know that $F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ and $r' = 2r$
Substituting these values in the equation,
$F' = \dfrac{{G{m_1}{m_2}}}{{{{(2r)}^2}}}$
Further solving this equation,
$F' = \dfrac{{G{m_1}{m_2}}}{{4{r^2}}}$
$\Rightarrow F' = \dfrac{1}{4}F$
So, the gravitational force between two objects is reduced to one fourth of its values when the distance is doubled.
When the distance between the bodies gets halved, the new distance becomes $r'' = \dfrac{r}{2}$
Plugging the values in the formula we get,
$F'' = \dfrac{{G{m_1}{m_2}}}{{r'{'^2}}}$
But we know that $F = \dfrac{{G{m_1}{m_2}}}{{{r^2}}}$ and $r'' = \dfrac{r}{2}$
Substituting these values in the equation,
$F'' = \dfrac{{G{m_1}{m_2}}}{{{{(\dfrac{r}{2})}^2}}}$
Further solving this equation,
$F'' = \dfrac{{4G{m_1}{m_2}}}{{{r^2}}}$
$\Rightarrow F'' = 4F$
So, the gravitational force between two objects is increased to four times its values when the distance is halved.

Note:
The distance is taken from the centers of the two bodies along the line of action of force. When the radius of the bodies is appreciably large, we take into account their radii as well while calculating the force. Else we take point masses and neglect their radii. The value of G is universal and does not depend on any factors like temperature, pressure and hence it is also called the universal gravitational constant.