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An elementary problem asks,

A book is at rest on a table top. In what direction is the net force of air pressure on the book?

Is this a meaningful question, and if so, what's the answer?

If we imagine that the book and table are completely smooth, there's no air between the book and table. Then the air pressure from above the book creates a large net downward force of hundreds of pounds. (This will be countered by an upward force from the table.)

On the other hand, if we imagine that due to the rough nature of the book and table, there is some air between the book and the table at most places, maybe there is enough air underneath the book to provide a net pressure force that is upwards. The scale height of the atmosphere is about 10^4m, so a 1cm book needs to have less than 1 part in 10^6 in contact with the table to have net upward force from air pressure.

How realistic are these approaches? Do we need a molecular view of the book, table, and air to understand the situation? For a typical, everyday book and table, is it meaningful to ask what direction the net force of air pressure pushes?

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  • $\begingroup$ It is meaningful; And the direction is(no surprise) downward! $\endgroup$
    – Ali
    Jul 17, 2013 at 8:54
  • $\begingroup$ The point(I think) is that we usually start with a book in the air, and then we put it down on the table(so there is more than often some air trapped under the book, with somewhat higher pressure to cancel the pressure from above). $ \\ \\ $However, if we start with the book on the table, and then we add the atmosphere; then we will have an obvious downward force from the air! $\endgroup$
    – Ali
    Jul 17, 2013 at 8:59
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    $\begingroup$ This is why a 1kg of lead is heavier than 1kg of feathers :) $\endgroup$
    – Bernhard
    Jul 17, 2013 at 9:31
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    $\begingroup$ @Ali The direction of this force is upward because there's always air between the book & the table. $\endgroup$
    – user5402
    Jul 17, 2013 at 9:44
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    $\begingroup$ If there was no air between the book and the table, the book would stick to the table, just like a vacuum sticker. $\endgroup$
    – udiboy1209
    Jul 17, 2013 at 10:38

3 Answers 3

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How realistic are these approaches? Do we need a molecular view of the book, table, and air to understand the situation?

It is realistic. There are no need to consider the molecular nature of air.

See http://www.nanovea.com/Application%20Notes/paperroughness.pdf. According to their data, the typical length in paper's surface variation is in order of $10^{-5}$ meters, while air's mean free path is $6.8\times 10^{-8}$ m (at room temperature, ambient pressure.) That means the room below the book cannot be thin enough to prevent significant amount of air sitting there (Babou actually reasoned his/her answer from wrong assumptions)

Considering the deformation of 'peaks' of surface of cover of a (heavy) book, it is likely that the book is in contact with the table more than 1 part in $10^6$. Therefore I'd guess that the net force is downward.

However, our static approach cannot be used if someone is picking up the book. S/he may experience the 'suction cup' effect (described in Babou's answer) when the book s/he lifted feels like "glued" to the table at one instant. When the book is lifted, air trapped below the book experiences a rapid, adiabatic expansion. Viscosity prevents surrounding air from entering the expanding room (below the book's surface) so rapidly. Hence the pressure below the book drops, and pressure from air above the book wins.

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  • $\begingroup$ Interesting, thanks for the data. I'm consistently surprised at how things I thought I understood long ago can still twist me around until somebody helps show me how to think about it. $\endgroup$
    – Mark Eichenlaub
    Jul 17, 2013 at 19:49
  • $\begingroup$ You cannot have it both way. Either the force exists, or it does not. If it exists, as you first state, regarding paper contact, then you have to overcome it before you move anything. It is true that a flexible object may slowly deform to let air and pressure in, but that is another issue, not incompatible with what I said. I was simply not specific about the the amount, and I did choose to neglect buoyancy. But your dynamic justification of suction effet is irrelevant. I would not be sure about other minute effects,though you may well be right. Still, very small scale often offers surprises. $\endgroup$
    – babou
    Jul 17, 2013 at 22:12
  • $\begingroup$ Thinking further about your analysis of paper surface, I was wondering about measuring the forces. I do not know how to do it, but there is one experiment that goes your way. It does happen that light paper may stay like glued to the underside of a book. This is probably a consequence of what you describe. The paper acts as a very light tabletop that is lifted with the book. $\endgroup$
    – babou
    Jul 17, 2013 at 22:42
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We will assume that the book and the air layer between book and table are thin enough, possibly no air in places, and the book dense enough (very important), so that atmospheric pressure may be considered constant, i.e., so that its variation can be neglected.

In a nutshell, the answer is that there is a downward force from atmospheric pressure. It can be null. The reason is that the pressure on the sides balances. The pressure downward applies to all of the top of the book. The pressure upward may apply to only part of the bottom surface of the book. This can be seen with a book that has a rubber cover and is placed on a glasstop.

Essentially the you have a suction cup effect that may make it very difficult to lift the book, much more than its weight would warrant. It can actually be measured by a dynamometre. So the net effect of atmospheric pressure is downward.

Actually it can be upward if you use atmospheric pressure to "glue" your book to the underside of a glass tabletop (better use one with a rubber cover).

It is the same phenomenon that happens when you need extreme force to pull the plug from a full bathtub. And it is also what has drowned a few people in swimming pools. They could not detach themselves from an open water exit at the bottom.

Note that you can also have trapped bubbles of compressed air that have the opposite effect.

I had made a complete model of the system (as you started discussing details), but I realized that was not what you asked.

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Is the book and table one object or two?

Either way, the large downward force on the top surface is matched by an equal force on the bottom surface.

If it weren't, the object(s) would experience acceleration.

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  • $\begingroup$ the large downward force on the top surface is matched by an equal force on the bottom surface. What about the buoyancy? $\endgroup$
    – user5402
    Jul 17, 2013 at 12:45
  • $\begingroup$ @metacompactness: touche'. I ignored the weight of the objects, and the weight of the displaced air. $\endgroup$
    – Mike Dunlavey
    Jul 17, 2013 at 12:56
  • $\begingroup$ You cannot neglect the weight of the displaced air if the question is "in what direction is the net force of air pressure on the book?" $\endgroup$
    – Sawi
    Jul 17, 2013 at 13:22

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