Does gravity vary across the surface of the Earth? (Intermediate)

I have a debate with my friend regarding variations of gravity on Earth. We read in an article that gravity on the poles of the Earth and the equator are not the same, so I continued with this line of thinking to speculate that gravity also varies depending on the altitude of the place you measure it; for example, the gravity in the Dead Sea (which is the lowest place on Earth) will be stronger than at the top of Mount Everest (the highest place on Earth).

My friend does not agree with me. He claims that due to the movement of Earth, on its own axis and orbiting around the sun, every place on Earth, that is to say any point within the atmosphere, has the same gravity. I do not understand how the motion of a planet has anything to do with gravity within the planet, but I'm not an expert, so I hope you can help us resolve the dispute.

You are right - gravity does change across the surface of the Earth and throughout its atmosphere, due to several effects.

First, there is the variation of gravity with latitude that you alluded to: you weigh about 0.5% more at the poles than on the equator. There are two effects that contribute to this, and they are discussed in more detail in a previous question. (It should be noted, however, that only one of these effects is due to an actual difference in the gravitational force between the equator and poles - the other effect is due to the fact that the Earth is spinning, which affects the weight you would see when you stepped on a scale but does not actually represent a change in the value of the gravitational force.)

Second, gravity does indeed change with altitude. The gravitational force above the Earth's surface is proportional to 1/R2, where R is your distance from the center of the Earth. The radius of the Earth at the equator is 6,378 kilometers, so let's say you were on a mountain at the equator that was 5 kilometers high (around 16,400 feet). You would then be 6,383 kilometers from the Earth's center, and the gravitational force would have decreased by a factor of (6,378 / 6,383)2 = 0.9984. So the difference is less than 0.2%.

Finally, there are very small differences (on the order of 0.01% or less) in gravity due to differences in the local geology. For example, changes in the density of rock underneath you or the presence of mountains nearby can have a slight effect on the gravitational force.