The gravitational force of earth acts in the space also till some certain distance from its surface. Then why does an astronaut in an international space station feel weightlessness and floats inside the station ?

Explanation

For an astronaut of mass m in an international space station, orbiting the earth with orbital speed `v = sqrt((GM)/(R+h))` where R is the radius of earth and h is the height of international space station above th earth.s surface, the gravitational force of attraction due to earth is `(GMm)/((R+h)^(2))`. Let N be the normal reaction of the surface inside international space station, radially outwards. Then, net radial force on the astronaut will provide the necessary centripetal force.
`F - N = (mv^(2))/(R+h)`
`rArr (GMm)/((R+h)^(2)) - N = (m(GM)/(R+h))/(R+h)`
`rArr N = 0`
Thus, there is no normal reaction acting on the astronaut and hence, she feels weightlessness ad floats inside the station.