Wednesday, July 29, 2020

Gravity of a flat Earth

        First let me remind you, if I may, that the force of gravity is the attraction that exists between any two masses. We feel it as "weight", but it's expressed mathematically as the force of attraction between our body mass and the mass of the planet beneath us.

        If you placed two cricket balls on a flat surface, not touching, there would be an attractive force between them. To be sure, it would be negligible, but mathematically it would exist. We can calculate it, in fact.

Gravitational constant G = 6.673 x 10-11 N m3 kg-2
m, Mass of each ball 0.163 kg
d, Distance between their centers (say) 0.12 m

f = G mm/d2

f = 6.673 x 26.5 x 10-14/1.44 x 10-2
f = 12.28 x 10-11 newtons.

        Of course, the force of attraction between a cricket ball and planet Earth is not negligible at all. It is the ball's weight, or its mass multiplied by the acceleration due to gravity.

0.163 x 9.8 = 1.6 newtons

        We could calculate the atraction between a cricket ball here on Earth and the planet Mars. It would be even more ridiculously negligible but, again, mathematically it exists.

Expat rides a hobby-horse:
        If you're American, you've been taught that an object of mass 10lb also has a weight of 10lb by definition, and isn't that convenient? Yes, it's convenient but it's also confusing. It accounts for the misunderstanding of the difference between mass and weight in the average American mind.

          All right, enough of the Grade 6 physics. Let's think about a flat Earth. Such a concept is usually depicted roughly thus, an obtuse cone:

Consider a man standing in the center:

        That's him, standing proudly erect, with the force of gravity as a vector acting vertically. It's perfectly possible for him to feel the same gravitiational force as he does on a spherical Earth.

        But now, get him to walk half way to the edge of the flat Earth.

        If he tries to stand vertically, he's got a problem. Consider the planet beneath him as two pieces, separated by the vertical line. The right piece is way larger than the left piece. It is therefore also way more massive, and exerts much greater attractive force on our man than does the left piece.

        The gravitational vector is not, in this case, vertical, and our man is going to have to lean left to avoid falling down.

        I've attempted to draw the vector such that it bisects the flat planet.

        Now let the man walk all the way to the edge. The gravitational vector is now almost horizontal.

        This model may be flat, but to its inhabitants, it would seem like a rather steep-sided bowl.

       Of course, we don't need any more reasons to refute the claims of flat Earthers. We already know that the Earth is a solid sphere. But if you should happen to find yourself having a beer with some of those maniacs, try telling them about the gravity vector on their hypothetical planet.


David Evans said...

Your diagrams are not quite right. The gravitational force due to a non-spherical object does not in general point towards its centre of mass. The direction of gravity would tilt, but more gently than you show. The video at this link shows how it would be.

expat said...

Thanks David, that's a nice vid. But it simply says what I'm saying in slightly different words. "It would feel, to a runner heading for the edge, as if he was fighting to climb up a steeper ansd steeper hill."

That's it exactly. I wrote that the subjective effect would be of living in a bowl. Same thing.

Steve M said...

I'm afraid your argument is meaningless to the Flatearthers. They don't believe in gravity insisting that objects fall down because the earth is constantly moving upwards.

David Evans said...

Steve M That's really quite clever of them. But the Earth would have to be accelerating upwards at 1 g, which means after a few years it would be moving at close to the speed of light and the stars would look radically different. They deny this, of course, because they don't believe in any aspect of physics that isn't convenient to them.

Two Percent said...

David Evans said...

"But the Earth would have to be accelerating upwards at 1 g, which means after a few years it would be moving at close to the speed of light"

Is that really true? I understood that as an object approaches the speed of light, its mass increases. Would that not increase the gravitational field strength(s) of the flat earth (and its occupants)? Of course, time would also slow down, though the occupants would not notice that, but surely they would notice themselves getting "heavier". And presumably, their world would "shrink" to an area in the middle, outside which they could not climb (if they could even stand up). To compensate for this, the upward acceleration would need to decrease. (And maybe it has!)

Of course, flat earthers don't believe in relativity either.

But I have seen some arguments that are difficult to counter. There's a canal somewhere which is supposedly long enough that the curvature effects should be observed, but reportedly are not.

David Evans said...

Two Percent: that's ingenious, but it's not how special relativity works. The mass increases in the reference frame of those left behind (if any) but it doesn't increase in its own reference frame. That has to be true, if you think about it, because we are all moving at close to the speed of light in the reference frame of the most distant galaxies, and that fact doesn't affect us at all.

The canal experiment was done back in 1870. It's a complicated story but part of it is that the flat-Earther first didn't allow for refraction and in a later experiment simply refused to accept the observations. Observations close to water are always tricky because of mirage effects. I've seen yachts on a lake seeming to float completely out of water.

This is my favourite picture showing the curvature.

expat said...

Great picture. Brings to mind the time I inadvertently drove onto the bridge northbound. It must've been at least 20 minutes before I could recover.

Dee said...

Hi David, 2%,

The flat earth theory not only maintains upward acceleration of 1G but also rejects any General Theory of Relativity. Since these both are directly related it's no surprise but in that context the argument of star observations seems not that relevant. They do hold special relativity in esteem though, see:

But yeah the many issues around curvature are mostly circumvented and remote sensing data from space are flat out dismissed as being fraudulent. Forgive me the puns!