Quote:weight over the center as we got closer. Pressure from above, to me, wouldn't add. I would think that pressure would increase to a point somewhere about half way down and then start to decrease as approaching the center.
F = ma
weight is the FORCE that an object exerts, it is equal to the mass of an object times the acceleration of that object due to the FORCE of gravity acting upon that object...
...so how do you know the force of gravity?
F = G * M1 * M2 / r^2
The force of gravitational attraction between two objects is a constant G times the mass of object1 and the mass of object2, divided by the radius between them squared.
why is the acceleration due to gravity always -9.8 m/s^2? this isn't a constant...but there is a force between every particle on earth and every other particle. you can use superposition to treat every single particle as 1 particle in order to calculate the force that any object will have towards the center, rather than computing the force of a gazillion particles and adding them all up...
if you combine the two equations, you can cancel out mass of the first object...and you know mass of the second object can be treated as a constant on Earth because the mass of the earth is constant, and the radius of the earth is constant, so you end up getting a constant acceleration of -9.8 for any object on the surface of the earth...note that this value is going to be different if you move farther or closer from the center of the Earth
you can see by the equation that as the radius (distance from the center of the Earth) decreases (Gets closer to the center) the gravity will increase.
note: we can only treat all the particles of earth as 1 giant particle if we are OUTSIDE of the radius of earth
when you are at the center of the Earth, the radius=0...you obviously can't divide by zero. but you dont need to, because you can realize that you have an equal number of particles all around pulling in all directions, and their force will cancel out...which is why the particle in the center doesn't move out from the center (well volcanos are a different story

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PRESSURE is just a measure of how much force is acting on an object, unlike gravity it does not cancel out if you have a force from two opposing directions, it adds. since we already established that particles CLOSE to the center have the strongest gravitational force towards the center, we know that the particle at the very center has all the strongest forces pushing against it from all sides.
so, pressure can only increase as you travel inside the earth towards the center. just like being under water, the more water you have pushing down on top of you, the more pressure you have...imagine a millions of tons of rock "falling" on you from all directions, yeah the pressure at the center is going to be high...
gravity is a different story...once we penetrate the surface of the Earth, you can no longer treat all the particles as a single particle, im not going to try to explain the math to you but if you just think about it...it's going to be decreasing as you get closer and closer to the center because you have rock at the surface trying to pull you back from the center. finally at the very center, it's pulling from all directions equally so you won't be moving...that means net force = 0