 Quote by ihaveabutt
If a spaceship is in a fixed orbit around the earth, wouldn't it be bound to earth's axis by its gravitational pull?
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Yep.
 Quote by ihaveabutt
And if so, since the earth rotates at 1,600 km/h, wouldn't there be a point where the angular momentum would be enough to mimmick the effects of gravity onboard?
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The rotation of the earth has only a small effect on the perceived gravity at the Earth's surface. In other words, the Earth is just spinning too slowly to have much effect (except maybe on the tides). So almost all of the perceived gravity at the Earth's surface is due to actual gravity caused by the mass of the Earth.
 Quote by ihaveabutt
And if that's the case than why do astronauts always appear weightless?
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They have nothing else to compare their motion to. Here on earth, the surface of the earth is stationary, while we are getting pulled down by gravity. But astronauts get pulled down by gravity at the same rate as their spaceship, so they do not experience gravity on the floor of their spaceship.
Also, the example of the spinning tube, any person inside will naturally be falling away from the axis of rotation, while the tube stays rigid, so they do experience a 'perceived' gravity on the outer end of the tube.