That’s about the thickness of aluminum foil, or the diameter of a human hair. If the ball were out of round by this much, of course you couldn’t tell. But if there was a small irregularity of this height (e.g. a scratch or small bump), I think you’d be able to tell pretty easily.
Those appear to both be about double the thickness. A hair is about a tenth of a millimeter and aluminum foil about 2 tenths. So it probably would be detectable as a scratch or whatever.
Since the Earth is an oblate spheroid, perhaps a better comparison might be with a bowls ball? (NB: not bowling ball.)
Regardless, I recall reading somewhere (offline - and Google isn’t helping) that the human sense of touch is sensitive to within several microns.
I said differently? Sure, 5 mile mountain, 4000 mile radius planet, insignificant bump - I’ve no disagreement there.
Otherwise I see we’ve had some interesting points of view put forward as to whether you measure bumpiness by height above the local mean or by distance from the centre of the spheroid, in which case Kilimanjaro’s not necessarily a bad choice and Everest is not necessarily in pole position on any criterion. 
And of course we’ve been to the top of either Everest or Kilimanjaro - or to the Moon - many times more often than we’ve been to the - Ow! Stop that!
You’re a very naughty boy; I think we’ll start with the comfy chair. 
This is closer to my mental model. I don’t think the Earth is smoother than a new billiard ball, then. This would seem to suggest that it is not.
The equatorial bulge represents a difference of only .003 percent from the polar circumference, so while technically oblate, it is still within the tolerances of a billiard ball.
ETA: Never mind, math is not my strong point. .3 percent is more like it, so it is outside the tolerances of the billiard ball.
But not by much; after calculating the billiard ball tolerances as percent of the circumference, it turns out it is .2 percent, compared to .3 percent for the equatorial bulge of the Earth. Damn close.
Aluminum foil is not, in general two tenths of a millimeter. Aluminum foil, in general, has a thickness less than two tenths of a millimeter. Household aluminum foil is in the range of .02mm, a full order of magnitude less.
As to whether you could feel features of this size on a pool ball, I think it would depend more on the steepness of the feature than the absolute height. Something like the Grand Canyon, with its steep walls, you could probably feel. Mount Everest, I don’t know.
Due respect to the Bad Astronomer, but he’s wrong on this one. “Smoothness” is a measurement of surface finish, not a measurement of sphericity or deviation from a standard diameter. The specification quoted (diametral tolerance of +/- 0.005 inches) covers only the deviation from standard size. It does not, necessarily, cover how spherical the ball may be, and it certainly says nothing whatsoever about surface roughness.
So the Earth isn’t more smooth than a billiard ball or a reasonably-sized ball-bearing. Whether it’s close is a matter of opinion, but I’d say not really.
The total volume of water on earth is 1 360 000 000 km³ [1] which weighs 1.3610^21kg [2]. The mass of the earth is 5.9710^24 kg[3] so water makes up 0.23% [5].
A billiard ball weighs 150 grams [6] so 0.23% of that is 0.034 grams[7] which is roughly equivilant to:
~ 0.4 x typical large sand grain mass (~ 9x10^-5 kg )
~ 10 x mass of a typical snowflake (~ 3x10^-6 kg )
~ 20 x mass of a typical mosquito (~ 1x10^-6 kg )
Radius r of a drop of water from m=rho 4pir^3/3:
| 2 mm (millimeters)
| (assuming water density ~~ 1000 kg/m^3)
So no, I’m betting it wouldn’t leave a wet trail. There’s slightly less than a single drop of water on your earth billiard ball.
That sure isn’t much water to get spread over three quarters of the billiard ball’s surface.
I would think that even the grand canyon would be to narrow to notice.
Would it be possible, considering the highly accurate topological data of the surface of our planet we now have, thanks to satellite measurements, to take a metal sphere, machined and highly polished to the scale/dimensions of our planet (say about 2 or 3 feet in diameter?), and laser-etch that topology into the sphere’s surface.
It sounds like something that would need incredibly sensitive and precise equipment and technology. I would think we could do something like that. But I wonder what materials and technologies we could use to get thoe most accurate result. I’m thinking a super hard/brittle metal like tungsten carbide or something, but I don’t know. Maybe aluminum would be more than adequate.
I’d also ignore the oceans. I think it’d be far more interesting to see/feel the very sight edge of the continental shelfs, and all the other interesting things that consists of 70% of the surface.
I guess it is the nerd in me that I find this thread utterly fascinating.
One drop of water on a scale model of earth on a billiard ball? geez, ~70% of the earth surface is water. Considering the ocean is deep (I am not going there) that is one thin film of water on the billiard ball.
The Grand Canyon ranges from 4-18 miles across. At pool ball scale, that’s about 0.001" to 0.005" – in the range of the thickness of a piece of paper, and definitely something you could catch the edge of your fingernail on.
I think you’re a factor of ten out there. I make it that water is 0.023% of the total mass of the Earth.
It is startling, how thin the layer of water on the Earth’s surface is. For something that affects the climate and the biology so much to be little more than a film of water on the surface.
It always seems fortuitous to me that there is just the right amount of water on the Earth. Not enough to make the planet a Waterworld, but enough so that the place is not an arid desert punctuated by the occasional lake. Maybe it’s an example of the anthropic principle. I doubt that desert planets have much going for them, but could civilisation have arisen on a planet entirely covered by water?
(Kevin Costner notwithstanding.)
Though I suspect that the bigger issue would be whether the Earth, shrunk down to the size of a billiard ball and subjected to the gravitational pull of a full sized Earth, could survive without falling apart and leaking molten goo all over the place.
We hashed all this out in a thread I killed about six weeks ago.
Depends on which pancake we’re talking about, doesn’t it? The pancake in the linked article doesn’t look like a standard ‘pour the batter in the pan’-type pancake; it looks more folded than layered (metamorphic rather than sedimantary).
:: ears perk up with interest ::
So are we talking ‘rotten kiwifruit left on the kitchen counter for three days’ or ‘raw egg’ or ‘cream-filled chocolate egg’ or what?