Weightlessness in Satellites The following is written in my textbook as the reason for weightlessness felt in satellites:

The gravitational pull is counterbalanced by the centripetal force.

This introduces two problems:


*

*For satellites orbiting around Earth, gravity is, in fact, the centripetal force.

*If the gravitational pull and centripetal force on a satellite are both in the same direction, then how do they cancel each other out?
 A: I think the statement in your book is false. The reason why one feels weightlessness in a satellite orbiting the earth can be understood in two ways:


*

*In an inertial frame attached with some distant stars.

*In the non-inertial frame of the satellite.
(Strictly speaking, all the explanation I am giving is pre-relativistic.)


Looking from the frame of distant stars, when one is in a satellite, the gravitational force on the person is exactly equal to the acceleration times mass of the person. So the normal reaction force from outside or from one part of the body to the other part of the body is zero. That is why you should feel weightless as analyzed from a distant frame.
In your satellite frame, you feel a pseudo force as it is a non-inertial frame. This pseudo force is equal to your acceleration in an inertial frame multiplied by your mass - which will be exactly equal to the force of gravitation on you as stated before. In your frame, you are in equilibrium and the two forces - pseudo and gravitation balances each other and any normal reaction forces from outside or from one part of your body to the other part of your body don't come into the picture. This is why you feel weightless. 
Edit
Do have a look at the comment by John Dvorak. The statement in the textbook makes sense if we read "centrifugal" instead of "centripetal". It then simply becomes the explanation of the weightlessness in the non-inertial frame of the satellite. 
A: Let's assume for now that the item is in a circular orbit (just to keep things simple)

The Gravitational Pull is counterbalanced by the Centripetal Force

This sounds to me like a case of the anti-centrifugal brigade "correcting" a statement by replacing the word "centrifugal" with the word "centripetal" and, in doing so, turning the statement into nonsense.
We can look at the orbit problem from two reference frames.
First let's look at things from a rotating reference frame that rotates with the satellite's orbit. Since this is a rotating reference frame we have a centrifugal force and we can make the statement,


*

*The Gravitational Pull is counterbalanced by the Centrifugal Force.


Now let's look at things from a non-rotating frame.


*

*To keep something moving in a circle, a centripetal force is needed. In the case of an orbit that, centripetal force is provided by gravity.


It's important to realise that these are just two ways of describing the same situation from different reference frames.
However this has little to do with "weightlessness". The key to understanding weightlessness is that we don't feel gravity or acceleration (and centrifugal force is just another way of looking at acceleration) directly. When we talk about feeling our weight what we are really feeling is not gravity but the contact forces, tensions and compressions on/in our body that are acting against gravity. When we talk about experiencing "G-forces" what we are really feeling is not the acceleration but the contact forces, tensions and compressions on/in our body that are causing the acceleration.
If those contact forces (and therefore the resulting tensions/compressions) are negligible* we don't feel gravity. This applies equally to being in orbit, the NASA Vomit Comet or the early stages of skydiving (before your body has accelerated enough for air resistance to be non-negligible).
Orbit is interesting mostly because it can be sustained for much longer periods of time then the other weightlessness scenarios.
*They aren't actually zero in a satellite because the gravitational field is not quite uniform across the satellite and because satellites in low orbit do experience some air resistance.
A: The statement:
The Gravitational Pull is counterbalanced by the Centripetal Force
is rubbish.
The satellite undergoes a centripetal acceleration because it is acted on by the gravitation force.  Some people call the force which causes a centripetal acceleration the centripetal force.  So in such a case the gravitational force and the centripetal force are one and the same thing.  This is your statement 1.
The reason for thinking that you are weightless when in an orbiting satellite is that the satellite and your good self would be accelerating towards the Earth at exactly the same rate.  So there is no normal reaction between you and the satellite.
You must have a force acting on you because otherwise you would not accelerate and hence not be in orbit.
A: It's a mistake to think of a body in orbit having its forces 'balanced' in some way - the forces are not balanced, because the body is accelerating towards the centre of the orbit!
See my BBC article here, which explains this and weightlessness in general terms.  I hope it helps.  http://news.bbc.co.uk/1/hi/magazine/4625150.stm
A: Consider the ISS orbiting at an altitude of around 400 km every 90 minutes. In the frame of the satellite:
(Radius of Earth $R = 64.10^6 \text{m}$)
Gravitational acceleration $ = g \left(\dfrac R{R+h}\right)^2 ≈ 8.66 \text{m/s²}$
Centrifugal (not centripetal) acceleration $= ω^2 (R+h) 
= \left(\dfrac{2π}{(90)(60)}\right)^2(R+h) ≈ 8.66 \text{m/s²}$
Thus the gravitational force is counterbalanced by the centrifugal force and the astronauts feel weightless.
Note that this is not the case when the ISS is using thrusters. Also, calling this as microgravity is a misnomer as the astronauts still experience around 0.9g.
Calculation
