I'm guessing I'm being sold a bridge with his numbers, but can't tell for sure.
You're being sold a bridge with his numbers. If he had done the math correctly he would have obtained a deflection of 33.58 cm at 45° geocentric latitude rather than 17.245 cm. The youtube comments show a number of errors. Two key errors are that the commenter assumes that
- Local vertical points directly toward the center of the Earth.
- Gravitational acceleration due to the Earth's mass points in the same direction.
What those youtube comments do show is that the Earth is not quite a sphere. A closer approximation is an oblate spheroid. That said, his comments fall into the wronger than wrong category. Quoting Isaac Asimov,
When people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.
Note that I qualified latitude in the above as geocentric. Geodesists have multiple concepts of latitude. Geocentric latitude is the angle between a line to the center of the Earth and the Earth's equatorial plane. Geodetic latitude is the angle between the normal to the reference ellipsoid and the Earth's equatorial plane. Yet another concept is astronomical latitude, which is the angle between true vertical and the Earth's equatorial plane.
The difference between geocentric and geodetic latitude is at its greatest at 45°. A geocentric latitude of 45° is equivalent to a geodetic latitude of 45° 11' 32.7". Converting that 11' 32.7" difference to radians and multiplying by 100 meters results in 33.58 cm. The difference between geodetic and astronomical latitude is called the deflection of the vertical (or vertical deflection). The deflection of the vertical is rather small, less than 2' (two minutes of arc) even in extremely mountainous terrain. Typical values are a few tens of arc seconds.
What this means is that an oblate spheroid is a better model of the Earth's shape than a sphere. That does not mean a flat Earth model is correct. The Earth's flattening (how squished the Earth is at the poles versus the at the equator) is rather small. The poles are 22 km closer to the center of the Earth than is the equator. Percentage-wise, that's a 0.345% difference.
The reason an oblate spheroid is a better model of the Earth's shape than a sphere is because the Earth is spinning. That spin does two things: It results in a centrifugal acceleration from the perspective of an observer fixed with respect to the rotating Earth, and it results in an equatorial bulge. If the Earth was a uniform density liquid, the spin would make the Earth naturally relax into an oblate spheroidal shape. "Straight down", the combined effects of gravitational and centrifugal acceleration (geodesists call this "gravity" to distinguish it from gravitation) would be exactly normal to the surface.