eli5: What is an inverse square force and how is gravity an inverse square force?

In: Physics

An inverse square force just means that the strength of the force is proportional to the inverse of the square of the distance. What that means is that G is proportional to 1 divided by the distance squared(the distance multiplied by itself).

So, if you are twice as far away as something, you feel 1/4 of the force due to gravity. If you are 3 times as far away, you feel 1/9 the force of gravity.

So, a practical consequence of this is when rockets get enough velocity, they can get far enough away that the gravity due to earth is negligible.

An inverse square force is a force which varies in inverse proportion to the square of the distance between the object exerting the force and the object experiencing it. For example, if you get twice as far from earth (2X the distance), you will feel once fourth (1/2^2) of the gravitational pull. If you get 4x as far, you will feel 1/16 (1/4^2) the force. 16 times as far, and it will be 1/256 (1/16^2) as strong.

An inverse square is basically anything that decreases proportionally to an increase squared, or increases proportionally to a decrease squared. In math terms y=1/x^2 . Gravity is an inverse square force because the force of gravity decreases relative to an increase in distance squared. For example, if you built a tower on the surface of the earth with a height equal to the earths radius, the top would be 2x the distance and therefore have 1/4 the gravity.

The inverse square means that the force is proportional to 1/D^2 where D is the distance from the origin.

Let’s say you’re some distance away from a planet, and you get some amount of force acting on you.

If you get twice as far away, the force is a *quarter* as strong.

If you get three times as far away, the force is a *ninth* as strong.

If you get four times as far away, the force is a *sixteenth* as strong.

Now, why does it do this?

I used to read late into the night with just a lamp to read by – and my hands would get cold. This was in the bad old days of incandescent lightbulbs, which gave off a buttload of heat – so I’d warm up my hands by putting them close to the bulb.

Of course, the closer my hand got, the hotter it would get – within a few inches it would be painfully hot, but a foot away it would only be slightly warm, and from two feet I couldn’t feel it at all.

But here’s the thing: the closer my hand got, the darker the room would get, as well. Which makes sense – it was casting a bigger shadow… *because it was blocking a bigger share of the light.*

Oh my god.

The falloff in heat – the inverse square law – isn’t about *physics*, it’s about *geometry*.

If I didn’t mind burning my hand, I could have wrapped it right round the bulb, and covered the whole thing, using up absolutely all of the light.

The further away my hand got, the less area I could cover. At a foot away, I could only cover as much as a handprint on a basketball. At two feet away, I could only cover as much as a handprint on a gym ball.

The force doesn’t fall off with distance – the area it’s spread over just gets bigger.

The surface area of a sphere increases according to the *square* of the diameter.

The amount of that sphere covered by a given object, your share of the total force – is thus proportional to *one over the square* of the diameter.

An inverse square force is a force whose strength decreases with the square of its distance.

Say you are touching a magnet and you can feel it’s force. Now, if you move 2 metres away from the magnet, you will feel that it’s force is 4 times less (as 2² = 4).

Gravity is an inverse square force as it follows the same rule.

Why does Gravity follow this rule? No one knows for sure.