Would an astronaut in this spacecraft feel weightless? 
A spacecraft is placed in orbit around Saturn so that it is
Saturn-stationary (the Saturn equivalent of geostationary – the
spacecraft is always over the same point on Saturn’s surface on the
equator).
Information the question provided:
mass of saturn  = $5.68\times 10^{26} kg$
mass of spacecraft = $2.0 \times 10^{3}kg$
period of rotation of Saturn = $10$ hours $15$ minutes

Information I calculated:
radius of orbit = $1.1 × 10^8m$
Now part d)
Would an astronaut in this spacecraft feel weightless? Explain your answer.
I am unsure how to answer this question. I guess I first need to define "weightlessness"? From what I know the sensation of weightlessness is the absence of normal force? Or its the sensation that you feel that you weight less than your normal weight? I also calculated $g = 3.13m/s^{2}$ if thats any useful?
 A: Weightlessness is when your proper acceleration is zero.
The proper acceleration is an important concept in general relativity because it is a scalar invariant but, despite the fearsome reputation for complexity that general relativity enjoys, the proper acceleration has a simple physical interpretation.
To determine your proper acceleration simply drop an object and measure the acceleration of the object relative to you. Then your proper acceleration is the negative of the acceleration you've just measured. For example if I drop a pen it accelerates away from me at $a = -9.81 \mathrm{m/s}^2$ (the minus sign means the acceleration is downwards) so my proper acceleration is $a = +9.81 \mathrm{m/s}^2$.
Suppose now I leap off a cliff and I drop a pen. The pen and I fall at the same rate (ignoring air resistance) so the pen remains stationary alongside me. In this case my proper acceleration is zero, and therefore I would be weightless.
A less fatal example would be an astronaut aboard the International Space Station. The ISS and all its contents orbit the Earth with the same period. If an astronaut aboard the ISS drops a pen it remains stationary beside them because the astronaut and pen are both orbiting the Earth in the same orbit. Hence the astronauts aboard the ISS are weightless (as countless YouTube videos testify).
So in your case you need only consider whether the spaceship, the astronaut aboard it, and any items the astronaut drops are following the same orbit.
A: We all know a  fundamental principle from physics which says, " freely falling body are weightless"
That is, their Apparent weight is zero. So what is a freely falling body.....?
Any object on or around a planet (like earth or Saturn), whose acceleration is equal to acceleration due to gravity at that point (in magnitude as well as direction BOTH), is a freely falling body.
So in the given example, first you have to calculate the acceleration due to gravity at that point where the body is located. Then you have to find the centripetal acceleration of its circular motion. The direction of both of these acceleration is a towards the centre of the planet. If their magnitudes are equal, then that spacecraft is a freely falling body and therefore it will be in a state of weightlessness.
A: Feeling weightless is like saying that you feel not a normal force on yourself preventing free-falling.
I think that this is the starting point for formulating Einstein Weak Equivalence Principle.
