Question

What happens to the gravitational force between two objects when the distance between them is:
(i) doubled?
(ii) halved?

Answer 1

Every particle in the cosmos attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres, according to Newton's law of universal gravitation. The theory's publication was dubbed the "first great unification" since it brought together previously reported gravity events on Earth with known celestial tendencies.

Complete step by step answer:
Every particle in the cosmos attracts every other particle in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres, according to Newton's law of universal gravitation. The theory's publication was dubbed the "first great unification" since it brought together previously reported gravity events on Earth with known celestial tendencies.
The law says that every point mass attracts every other point mass by a force acting down the line crossing the two points in today's terminology. The force is proportional to the product of the two masses, and it is inversely proportional to the square of their distance.
$F = G\dfrac{{{m_1}{m_2}}}{{{r^2}}}$
As a result, the universal gravity equation has the form. where F is the gravitational force between two objects, ${m_1}$ and ${m_2}$ are their masses, r is the distance between their mass centres, and G is the gravitational constant.
Since F and r are inversely proportional
i) If r is doubled
$r’ = 2r$
$F’ = G\dfrac{{{m_1}{m_2}}}{{r{`^2}}}$
$F’ = G\dfrac{{{m_1}{m_2}}}{{4{r^2}}}$
$F’ = \dfrac{1}{4}F$
The gravitational force is reduced by one-fourth when the distance between two bodies is doubled.

ii) If r is halved
$r’ = \dfrac{1}{2}r$
$F’ = G\dfrac{{{m_1}{m_2}}}{{r{`^2}}}$
$F’ = G\dfrac{{{m_1}{m_2}}}{{{{(\dfrac{1}{2}r)}^2}}}$
$F’ = 4G\dfrac{{{m_1}{m_2}}}{{r{`^2}}}$
$F’ = 4F$
When the distance between two bodies is cut in half, the gravitational force is multiplied by four.

Note:
Coulomb's law of electrical forces, which is used to determine the amount of the electrical force originating between two charged substances, is similar to Newton's law of gravitation. Both are inverse-square laws, meaning that force is proportional to the square of the distance between the bodies. The product of two charges replaces the product of masses in Coulomb's equation, and the Coulomb constant replaces the gravitational constant.