Why do astronauts float weightless in the International Space Station?

By NASA/Crew of STS-132 - http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/hires/s132e012208.jpg (http://spaceflight.nasa.gov/gallery/images/shuttle/sts-132/html/s132e012208.html), Public Domain, Link

We have seen on TV that astronauts float weightless in the International Space Station (ISS), and during spacewalks. In fact, the ISS serves as a microgravity research laboratory in which crew members conduct experiments in biology, physics and other fields. Microgravity is more or less a synonym of weightlessness and zero-g (zero gravitational field strength).

Nevertheless, the ISS maintains an orbit with an altitude h of between 330 and 435 km so, according to Newton's law of universal gravitation, the gravitational field strength g in the ISS is:

where G is Newton's constant, M mass of Earth and R is the radius of Earth.

Taking into account that the gravitational field strength at the surface of the Earth is g=9.8 m/s^2, we conclude that things and astronauts inside the ISS weigh only a 10% less than they do on Earth! The weight is almost the same! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Why do not the sizes of Venus and Mars as viewed from Earth change during the course of the year?

Venus, by NASA or Ricardo Nunes - http://www.astrosurf.com/nunes/explor/explor_m10.htm, Public Domain, https://commons.wikimedia.org/w/index.php?curid=338424

Just before his death, in 1543, Nicolaus Copernicus published in his book On the Revolutions of the Celestial Spheres a Heliocentric model of the universe, that is, a model of the universe that placed the Sun rather than the Earth at the center of the universe.  This is considered a major event in the history of science, triggering the Copernican Revolution and making an important contribution to the Scientific Revolution.

By Original image by Niko Lang SVG version by User:Booyabazooka - Own work, CC BY-SA 2.5, https://commons.wikimedia.org/w/index.php?curid=5502912

According to Copernicus' model, since the Earth circulates the Sun in an orbit outside that of Venus and inside that of Mars, the apparent size of both Venus and Mars should change appreciably during the course of the year. This is because when the Earth is around the same side of the sun as one of those planets it is relatively close to it, whereas when it is on the opposite side of the sun to one of them it is relatively distant from it. When the matter is considered quantitatively, as it can be within Copernicus's own version of his theory, the effect is a sizeable one, with a predicted change in apparent diameter by a factor of about eight in the case of Mars and about six in the case of Venus.

On the other hand, according to the Ptolemaic system (the Geocentric model) Venus and Mars should not change appreciably during the course of the year because its epicyclical motion implies only a small change in distance from the Earth.

However, when the planets are observed carefully with the naked eye, no change in size can be detected for Venus, and Mars changes in size by no more than a factor of two. This gives us strong evidence for the Geocentric model and refutes the Heliocentric model! How is this possible?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

Does the expansion of space apply in the solar system?

The prevailing cosmological model for the universe accounts for the fact that the universe expanded from a very high density and high temperature state, and that nowadays the expansion is even accelerating. This is an expansion of space, that is, the increase of the distance between two distant parts of the universe with time. It is an intrinsic expansion whereby the scale of space itself changes. This is different from other examples of expansions and explosions in that, as far as observations can ascertain, it is a property of the entirety of the universe rather than a phenomenon that can be contained and observed from the outside.

By NASA/WMAP Science Team - Original version: NASA; modified by Ryan Kaldari, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11885244

Since it is an intrinsic expansion, it is natural to think that the planets in our solar system are expanding with time, as universe is. Moreover, our measurement devices should be expanding too. But, taking into account that a measurement is the assignment of a number to a characteristic of an object by comparing with other objects, why were we able to measure the expansion of the universe if our devices are expanding too?

Does the expansion of space apply to the objects inside our solar system?

Please, explain your reasoning. You can post your attempted answers in the comment box below. Please, do not use Facebook or Twitter to give your answers.

The Moon is getting further away from Earth. Where does this extra energy come from?

All bounded orbits where the gravity of a central body dominates are elliptical in nature. In the case of the Moon orbiting the Earth, the eccentricity of the ellipse is so small (0.055) that it is almost a circle:

Therefore, the gravitational force F_g that the Earth exerts on the Moon is perpendicular to Moon's velocity v, so it is a centripetal force F_c, making the trajectory of the Moon bend:
$$ F_{g}=F_{c} \\
\frac{GMm}{r^2}=\frac{mv^2}{r} $$ where G is Newton's constant, M is Earth's mass, m is Moon's mass and r is the radius of the orbit.

This implies that the kinetic energy of the Moon is
$$
K=\frac{1}{2}mv^2=\frac{GMm}{2r}
$$ which is smaller than the absolute value of the potential energy
$$
U=-\frac{GMm}{r}
$$ So the mechanical energy of the Moon is
$$
E=-\frac{GMm}{2r}
$$

We know that at the time of its formation, the Moon sat much closer to the Earth, a mere 22,500 km away, compared with the 402,336 km between the Earth and the Moon today. So the Moon is getting further away from Earth, now at the rate of 3.78 cm per year. Nevertheless, according to the last equation, a larger r means that the Moon has more energy every year. Is its energy non conserved? Who is giving energy to the Moon?