How do scientist calculate the weight of the earth?

First of all let us make a correction. It is not weight of the earth it is only MASS of the earth. By gravitational field concept we arrive at one expression for acceleration due to gravity as g = G M/ R2

G - universal gravitational constant. R-radius of the earth. g-of course measured by the simple pendulum experiment. Knowing g, G, R, we can compute M ie mass of the earth as M = gR2/G

Cavendish was the first scientist who found the mass of the earth. But we say so colloquially that he was the first man who weighed the earth.