physics, rotational motion, gravitation, question is in description

Question

By how much does the gravitational force between two objects decrease when the distance between their centers is doubled? Tripled? Increased tenfold?

Answer 1

In general, when comparing distances where r₂ can be expressed as some multiple of r₁:

r₂ = n*r₁,

Then since F = Gm₁m₂ / r²,

F₁ / F₂ = Gm₁m₂ / r₁² / Gm₁m₂ / r₂²

F₁ / F₂ = r₂² / r₁² = n²r₁² / r₁² = n²

So for doubling, n=2, the force decreases by a factor 2² = 4

For Tripled, n=2, the force decreases by a factor of 3² = 9

n=10, 100