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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:

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

  2. If the gravitational pull and centripetal force on a satellite are both in the same direction, then how do they cancel each other out?

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    $\begingroup$ Maybe the fact that satellites are in the state of free fall might clear your doubt? $\endgroup$
    – N.S.JOHN
    Jan 24, 2016 at 11:16
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    $\begingroup$ Which textbook? Which page? $\endgroup$
    – Qmechanic
    Jan 24, 2016 at 11:36
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    $\begingroup$ +1: IMHO the statement from your book is simply wrong, and your analysis is right. $\endgroup$ Jan 24, 2016 at 11:58
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    $\begingroup$ I really wish books would stop using the words "centripetal force" and only use the words "centripetal acceleration"! The use of the first causes so much unnecessary confusion by giving the wrong impression that "centripetal" is the source of the acceleration rather than it being the consequence of some other unbalanced forces... $\endgroup$ Jan 24, 2016 at 14:52
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    $\begingroup$ @honeste_vivere although I respect your opinion, I don't think it is the reason which creates unnecessary confusion. But I can think of a book which says that Gravitational Force or other attractive forces are the source of Centripetal Force which tend to hold the body along circular path. For me, this sentence is an example of something which can create some unnecessary confusion. In particular, the writer should not have used the word Source. $\endgroup$
    – user104909
    Jan 24, 2016 at 17:36

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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.

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    $\begingroup$ I have been reading books by books for hours but it (weightlessness problem) was never so clear as it is now with just three lines. +1! $\endgroup$
    – user104909
    Jan 24, 2016 at 17:48
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    $\begingroup$ The key is we don't actually feel gravity directly because it affects all of the matter in our body equally (or at least close enough to equal that the differences are too small to perceive). What we feel is forces fighting against gravity. In free fall (and orbit is pretty much free fall) there is nothing fighting against gravity and therefore we don't feel it. $\endgroup$ Jan 24, 2016 at 23:39
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    $\begingroup$ Worse, If the satellite itself has a non-zero mass, it has some gravity of its own and a passenger would be attracted to it if not at the centre-of-mass, ie. would have some weight:) Anyway, the text book is indeed sucky and should be burned, (or possibly accelerated to escape velocity). $\endgroup$ Jan 25, 2016 at 8:09
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    $\begingroup$ @MartinJames: If the passenger is inside the satellite, the gravitational pulls from different parts of it will tend to cancel out even if he is not at its exact center. (Famously, they cancel out completely everywhere inside a uniform spherical shell). And even if they don't cancel out exactly due to irregularities of shape or density, the sum will not tend to pull him towards the center-of-mass. $\endgroup$ Jan 25, 2016 at 11:42
  • $\begingroup$ @HenningMakholm: Yes. But in mitigation, H.G. Wells made the same mistake in The First Men in the Moon. $\endgroup$
    – TonyK
    Jan 25, 2016 at 15:59
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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:

  1. In an inertial frame attached with some distant stars.
  2. 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.

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    $\begingroup$ "This pseudo force" - would be a centrifugal force, not a centripetal force. Perhaps the former was actually meant by the book? $\endgroup$ Jan 25, 2016 at 13:59
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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.

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    $\begingroup$ Actually, we do sense acceleration and not just the contact forces that are accelerating us. We do that by sensing the relative motion of the fluid in our inner ears. $\endgroup$ Jan 25, 2016 at 1:35
  • $\begingroup$ Well, you seem to be arguing that we don't "feel acceleration" but, rather, we "feel the effect of acceleration on our bodies". I would say that's a distinction without a difference, like saying that I don't "see that table over there"; rather, I "see the light rays that it reflects into my eyes." The reason you don't feel gravitational acceleration in a satellite is that your ear fluid is equilibrated with your body and both are being accelerated in the same way by gravity. $\endgroup$ Jan 25, 2016 at 2:37
  • $\begingroup$ Conversely, if you're accelerated by being shoved by one of your fellow astronauts, you do feel that acceleration, and not just because of the force being applied to your body. You also feel it because your body is being accelerated by the physical contact but the fluid in your ears is being accelerated more slowly by friction against the walls of the tubes it sits in. $\endgroup$ Jan 25, 2016 at 2:38
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    $\begingroup$ Right, what you feel is the internal stresses resulting from the external force. Whether that force is fighting gravity or causing acceleration makes no difference. No external force (and negligable non-uniformities in the gravitational field), nothing to feel. $\endgroup$ Jan 25, 2016 at 2:42
  • $\begingroup$ @DavidRicherby "Relative motion" is the key part. When you're being accelerated more or less equally over the body, there is no longer any relative motion (relative acceleration). We can only feel a change in acceleration, not acceleration itself. Standing on Earth, there is no net acceleration - that's why you don't fall through the ground. Gravity in Earth's orbit acts on every part of your body almost exactly, so you don't feel its acceleration, even though it's still pretty close to 1 g. And this isn't about humans either, it's a fundamental part of how the relativistic universe works. $\endgroup$
    – Luaan
    Jan 25, 2016 at 10:44
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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

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  • $\begingroup$ @NickAllen that's absolutely great! $\endgroup$
    – user104909
    Feb 9, 2016 at 13:21
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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

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