If both masses in a gravitational force equation are doubled and distance between them is also doubled, what happens to the force?

The gravitational force between two masses is described by Newton's law of universal gravitation, which states that the force ( F ) is directly proportional to the product of the two masses ( m_1 ) and ( m_2 ), and inversely proportional to the square of the distance ( r ) between them. The formula can be expressed as:

[ F = G \frac{m_1 m_2}{r^2} ]

where ( G ) is the gravitational constant.

When both masses ( m_1 ) and ( m_2 ) are doubled, the new masses are ( 2m_1 ) and ( 2m_2 ), leading to an increase in the product of the masses:

[ \text{New mass product} = 2m_1 \times 2m_2 = 4m_1m_2 ]

At the same time, if the distance ( r ) between them is also doubled, the new distance becomes ( 2r ). The gravitational force equation now looks like this:

[ F' = G \frac{4m_1 m_2}{(2r)^2} ]

[ F'