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In the solar system, things float around. We are so certain that the gravity of our sun exists. Still, why does that gravity not influence satellites and other objects there?
Our solar system is also orbiting the center of our galaxy, and the gravity of our black hole, but still many objects float around. I wonder why.
It is not true that "objects float around" in the solar system.
Perhaps you have seen video from the space station, and you can see things floating. This is not because there is no gravity, but because everything in the space station going at the same speed in the same direction. This makes it look as if things are floating. In fact the space station and everything in it, is moving at about 7km per second around the Earth.
The Earth and everything on it is is also moving at 30 km/s around the sun. The satellites around the Earth are also affected by the sun's gravity, but because they are affect the same as the Earth, they are also moving at about 30km/s around the sun. If two things are moving at the same speed and in the same direction, it will look as if they are floating. In fact they are "whizzing".
The Sun, Earth and satellites are also whizzing around in the gravity of the galaxy. The black hole is only a very small part of this; most of the gravity of the galaxy is in the dark matter that we can't see. But we don't feel this pull because the Sun, Earth, the satellites and you are all being pulled at the same time.
To help with James K's excellent answer, a visual representation might help. Let's look at a thought experiment - Newton's Cannonball.
Let's say you have a cannon, high enough that it's being held above Earth's atmosphere.
You fire it, and it falls to Earth a little ways away ("D" in the below diagram).
You fire another one with more power so it's moving quicker, so that it falls to Earth farther away. ("E" in the below diagram)
Eventually, you fire a cannonball with such extreme velocity, that it's "falling" around the Earth fast enough that it never reaches the ground.
This is orbit. Orbit isn't necessarily being really high up and moving slowly. More often than not, orbit is going sideways fast enough that you're falling without losing height.
So, why on places like the ISS, does it look like things are just floating around? Let's quickly go back to the cannonball.
Imagine the cannon fires two cannonballs at the same time, both going fast enough to be in orbit. These cannonballs are going blisteringly fast... but they were fired at the same time, going the same speed, so they stay together. If you can imagine being one of the cannonballs, the other cannonball would look like it's just floating next to you as you fly around the Earth. This is because, relative to each other, the cannonballs have next to no relative velocity.
The ISS, similarly, is traveling about 7.66 km/s, or around 27,600 km/h (about 17,150 miles per hour for those using imperial measurements). It's going fast. But when you're on there, everything is going the same speed, because you're all on the ISS together.
So if you let go of a pen on the ISS, it's still traveling the same speed as you - around 27,600 km/h. But because it's going the same speed as you, relative to you it just looks like it's floating.
Earth isn't just floating in orbit around the Sun, it's in orbit at (on average) 107,000 km/h (or 67,000 miles per hour).
Our Solar System isn't just floating around the center of our galaxy, it's in orbit at around 828,000 km/h (or around 514,500 miles per hour).
These are all hard to comprehend speeds - we can all agree though, that this isn't just floating around. Things are moving fast.
Things can appear to be just "floating" because their relative velocity to the observer is small. But I hope this all gives an explanation of how just because something appears to be slowly floating around, that doesn't mean it's not still moving fast from a different point of view.
Ok, gotta quote XKCD on this.
This is not how space works:
This is:
Gravity in low Earth orbit is almost as strong as gravity on the surface. The Space Station hasn't escaped Earth's gravity at all; it's experiencing about 90% the pull that we feel on the surface.
To avoid falling back into the atmosphere, you have to go sideways really, really fast.
The reason things in space tend to stay in space, or why things "float" in the international space station is because they're going so fast gravity doesn't do anything more than keep those objects moving around the planet: the parabolic trajectory that all falling objects have is so wide that it misses the horizon and comes back around again. Aka, orbit. As the space station and all of its contents are moving at the same speed, and are in constant in free fall, things appear to float: there's no object at rest for things to fall towards; everything is falling and everything is falling at the same speed, all the time.
If the ISS had an arm long enough1 and someone put something perfectly still in the center of the space at the far end, that object wouldn't stay there. It would be ever so slightly off the orbital path that the station would either catch up to it, or fall away from it. But it would be very, very slow. And it would happen because the ISS as a whole and the "floating object" would be in slightly different, intersecting, orbits.
Still, why does that gravity not influence satellites and other objects there.
It does. It just isn't very strong. Gravity falls off at an inverse-square relationship with distance, just like light does. Except that gravity is way way weaker of a force than electromagnetism is ("no its not, I can feel the Earth right now!" Yes, and the Earth is a few trillion-trillion times heavier than you are, how about a table? Can you feel the table's gravity?). The gravitational acceleration towards the sun is on the order of about 0.005m/s2 (and that's still 50 million times greater than the gravitational pull from all the spiders everywhere).
The real answer is that, locally (strictly speaking, in an infinitesimal region), gravity is not a thing. Whatever your movement is relatively to earth or the sun (even without going "fast to the side" as previously stated), you won't feel any gravity in a local experiment. Even when forces enter into account (like air friction or the reaction of the ground), you can't say without looking at the window if you're act on by gravity or inertia coming from, say, thrusters from your rocket.
This is the equivalence principle, the one responsible for the miracle that astronauts in the ISS and the ISS fall at the same rate, leaving astronauts weightless, in Newton's theory, through the dubious force = mass x acceleration and gravitational force = mass x gravity field, hence m a = m g, so a = g for any mass.
Newton himself knew that the strict equality between inertial and gravitational masses for any body, whatever its mass, composition or whatever, could not come from something random. This laid the groung for general relativity.
