✔ 最佳答案
1) The g.p.e. U is derived by integrating the gravitational force F = -GMm/r2 from r = ∞ to r = R which initiates from work done = force x displacement.
The negative sign has a meaning of "the gravitational pull between the planet and the object is attractive in nature"
And this formula, F = -GMm/r, takes the planet's centre (or infinity where g.p.e. = 0) as reference point while the formula U = mgh takes the planet's surface (where g.p.e. = 0) as reference point. Also when the height is comparable to that of the planet's radius, the formula U = mgh is NOT applicable.
2) Indeed, when the astronaut + spacecraft is orbiting around the earth, a centripetal force is needed to keep them running on the orbit. Thus when the gravitational force is exactly equal to the centripetal force required, according to the formula:
Weight = Gravitational force - centripetal force
The astronaut can feel "weightlessness" since weight = 0 at that time (But should note that the astronaut + spacecraft is still accelerating towards the earth centre)
3) At the pole, the object is NOT performing any circular motion about the earth's centre and hence NO centripetal force is needed to keep. Therefore at this state:
Weight = Gravitational force.
Hence:
Weight mg = GMm/r2
g = GM/r2
= M/(4πr3/3) x 4πrG/3
= 4πrGM/(3ρ) where ρ = Density
So from the formula, we can deduce that when r increases, g also increases since other parameters remain unchanged.
2011-04-29 16:22:20 補充：
Yes, R is counted from the centre of the earth.
