how can we calculate the pressure at the centre of the earth? And what is the pressure there?

Question

how can we calculate the pressure at the centre of the earth? What are the forces that acts on it? Is the centre of the earth is very hot because of the very high pressure

Answer 1

Pressure in a fluid (yes, I know the earth is mostly solid, but the equation works) is the weight of the fluid above a surface divided by the area of the surface. The surface can have any area and, through the magic of algebra, disappears from the equation so that we are left with the product of density (ρ), gravity (g), and height (h in swimming pools and blood vessels, r in astronomical situations like this). Now for the calculus. You can't assign a value for gravity in this situation. It varies from 9.8 m/s2 on the surface to zero at the center. We can reduce the amount of variation if we examine just a part of this total distance (Δr). We can reduce it even more if we examine an even smaller part. And we can reduce the variation to nothing if we examine an infinitesimal part (dr). Now the product of density (ρ), gravity (g), and height (dr) works again. All we have to do is add up the contributions to the pressure made by the infinite number of infinitesimal parts from the surface of the earth down to its center. The process of adding infinitesimals is called integration.
P = ∫ R ρg(r)dr = ∫ R 3m Gmr dr = 3Gm2 [ r2 ] R
r r r
4πR3 R3 4πR6 2

and here's our equation …
P = 3Gm2 ( R2 − r2 )

8πR6

which reduces to …
P0 = 3Gm2

8πR4

at the center where r = 0.
Let's do it.
For the earth …
P0 = 3Gm2 = 3(6.67 × 10−11 Nm2/kg2)(5.98 × 1024 kg)2

8πR4 8π(6.34 × 106 m)4

P0 = 1.7 × 1011 Pa = 170 GPa = 1.7 million atmospheres

The actual value is closer to 360 GPa or about twice the value calculated above, which is annoyingly big, but at least we got the right order of magnitude. To do this correctly, we'd have to account for variations in density with depth. The density of the earth starts at about 2300 kg/m3 at the crust, increases (nonuniformly) with depth in the mantle, jumps drastically at the outer core where it nearly doubles, and keeps increasing (nonuniformly) hitting a maximum of 12,580 kg/m3 at the center.