Welcome! The pendulum experiment is refuted by Rowbowtham in "Earth Not a Globe" as seen here: http://www.sacred-texts.com/earth/za/za39.htm His refutation basically boils down to claiming it's all due to weather. He also discusses how the experiment itself is NOT a 'direct proof' of a globular Earth here: http://www.sacred-texts.com/earth/za/za52.htm Basically saying that (at minimum at the time) the experiment wasn't conclusive enough because of too many variable outcomes. Thus, no round Earth.
The first article talks about the variable speed of a pendulum depending on latitude. He's talking about a "Focault pendulum" but this is actually about pendulums in general.
In RET, gravity varies from pole to equator, in part because the earth isn't a perfect sphere - so the poles are closer to the center than the equator - and in part because the earth is spinning, resulting in centrifugal force (technically, a pseudo-force) that opposes gravity to some small degree. Since the period of a pendulum varies with the force of gravity, this explains the small changes in the period of a pendulum.
In FET, Rowbotham claims that the higher temperatures at the lower latitudes is the explanation - perhaps he imagines the air resistance would be a little different - or that expansion and contraction of the pendulum arm itself would make the difference.
Well - back when Rowbotham was thinking about this - that MIGHT have been a valid consideration. However, these days there are Focault pendulums in dozens, perhaps hundreds of museums - many of which are heated and airconditioned. These pendulums are also checked for their length and they are EXCEEDINGLY heavy - I've seen them with 2' diameter brass weights on the ends. With temperature control and careful checking of the length of the pendulum, there can be no doubt that the effect of subtle changes in gravity on such devices by FAR outweighs any other effects.
Furthermore - we have confirmation from simply weighing things in different places that gravity varies from place to place. We don't even need a pendulum to demonstrate that.
In fact, since we know things weigh different amounts in different places - then if temperature variations mattered then we'd have to explain why the pendulum's swing doesn't change even more than it does.
So Rowbotham is for 100% sure incorrect about this. Gravity DOES vary from place to place - and FET has a very tough time explaining that adequately.
OK - so what about the second paper?
This *IS* talking specifically about a Focault Pendulum. This is a kind of pendulum that is free to swing in any direction.
If you visit any one of the museums and other locations listed here:
https://en.wikipedia.org/wiki/List_of_Foucault_pendulums - you'll be able to see a demonstration of such things.
Typically, they are very long and have very heavy 'bobs' at the end, typically made from brass or some other non-magnetic material. They are swung from the tops of towers or inside tall stairwells. The one in the London Science Museum is 22 meters long and takes a leisurely 9 seconds to swing from one end to the other.
With the length and weight of these things - they swing all day long without anyone having to pull them back to provide more momentum. Air resistance is negligible - and with a brass bob - the Earth's magnetic field is irrelevent.
What's interesting about these pendulums is that the axis of their swing gradually changes throughout the day - you can actually tell time by them - and most places that have them put a convenient scale on the floor so you can do exactly that.
What you find is that through the day, the direction of the pendulums swing moves by 15 degrees per hour. Therefore, it's swing direction moves through a complete circle over the course of a day. They do this completely reliably - but the effect is only strongly obvious for places far from the equator. Focault pendulums don't work on the equator.
The RET reason for this is that the earth is rotating - but the pendulum is continuing to swing as close to the direction it was set off in as it can. At the North or South pole, this would be extremely obvious - if you pulled the bob in the direction where the sun is and then released it - the pendulum would continue to swing towards the sun throughout the day. That also works when you're further from the pole - and the visible result is very compelling.
According to Rowbotham - the effect is "magnetic". Well - that would work if any significant fraction of the mass of the thing were made of a magnetic material like iron. But most that I've seen are brass - so that doesn't work. Not only that - but the position of the magnetic pole doesn't change through the day - so if these devices worked as he claims, they'd always swing North/South...and they don't.
Now - he points to a couple of examples where this effect didn't happen...but we don't know whether the experimenter screwed up in some way. It's very important (for example) that the top of the structure is fixed VERY firmly. In the example he discusses, it was fixed to the frame of a skylight. So it's quite possible that some sympathetic vibration was set up in that one very specific experiment that screwed up the results.
HOWEVER:
https://en.wikipedia.org/wiki/List_of_Foucault_pendulums lists several HUNDREDS of these things displayed in museums around the world that change their swing axis by 15 degrees per hour. The fact that he found ONE that didn't - hardly disproves the fact that hundreds of carefully constructed pendulums DO NOT swing towards the poles and DO track the rotation of the Earth.
So picking one dusty old experiment and comparing it to literally hundreds of modern experiments is hardly the kind of compelling evidence we need here.
Honestly - if I was an FE'er - I'd be trying to claim that the Focault pendulum swing axis was driven by some bizarro property of the Sun itself. That's a better fit for FET than the ramblings of Rowbotham.
So, no. Rowbothams ramblings about pendulums are TRIVIALLY disproven.
You'll find Rowbowtham's work spoken of in high regard around here, so gaining a familiarity with it isn't a bad idea. Though be aware he is one of few voices speaking out, and his experiments must be taken at his word since his work was all done in the 1800's, and I've found little corroborating evidence to the soundness of his experiments so far.
I've read his stuff - it's nonsense.
I don't doubt he was an honest man - but his tendency to pick one single data point (a badly set up focault pendulum) and to use that to make such insanely large sweeping claims...that's not how science works.
So by all means treat him as a religious figure - but please don't claim he has a scientific explanation for the world.
Go look at a REAL Focault pendulum and come to your own conclusions...or get a pair of binoculars and look at the mountains on the moon. Either way.