Recently, I was thinking about the acceleration due to gravity on Earth when I remembered that I was taught that you weigh less at the equator because a fictitious centrifugal force "throws" you off the surface of the Earth a little, thus making you lighter. After looking around a bit, I saw that many sources say that this is the main reason as to why you weigh less at the equator.
However, I thought about this for a while, and here is why I think this is wrong. The way your weight is measured is by the normal force you apply onto your scale. This means that if your scale was accelerating upward at about the same rate as you, your measured weight would be about the same as if you just sat still on the scale since you have zero acceleration relative to the scale. Now, think about what happens with our centrifugal force example. A force is applied on you, accelerating you upwards, but a force is also applied on the scale, accelerating it upwards at about the same rate. You and the scale are so close together that the "throwing" of a centrifugal force should be negligible.
Of course, the gravity at the equator is still weaker than at the poles because of Earth's bulge, but the fact is that the main reason that is usually taught seems off.
I was wondering if this thinking is correct, and if there is anything I made a mistake on or should know.
Feedback is greatly appreciated.