How to measure the mass of Earth? I was wondering how you can measure the mass of Earth. From what I find on the internet, people are using Newton's Laws. But how can you do that ? Newton's Laws are assumed to work because you know a priori the mass of Earth. But you don't know that! Isn't this a circular calculation ? So how can you really measure the mass of Earth?
 A: This is a description of the experiment Cavendish performed at the end of the 18th  Century to measure the density of the Earth:
Cavendish put two lead balls on either end of a long bar.  He hung the bar at its center from a long twisted wire with known torque.   Then, he placed two really massive objects at exactly identical fixed distances from the center of the torsion bar, in the plane of the torsion bar and at right angles to the bar at rest.  The balls were attracted and started the wire twisting, but their inertia caused them to overshoot the equilibrium position of the wire.  The bar wound up oscillating, and Cavendish measured the rate of oscillation to determine the torsion coefficient of the wire.
With this, he was able to determine the force attracting the balls to each other, which he used to set up a proportion to derive the density of the Earth.  Here is a description of the experiment: http://large.stanford.edu/courses/2007/ph210/chang1/, as well as a derivation of the gravitational constant, Big G, that you can perform: http://www.school-for-champions.com/science/gravitation_cavendish_experiment.htm#.VUFS80uiKlI.
One can use the density derived by Cavendish, and the diameter of the Earth (which has been known since Eratosthenes in ancient Greece) to compute the mass of the Earth.
To find the mass of the Earth using the modern form of Newton's Law of Gravitation, you may enploy Little g, the Earth's gravitational acceleration, which is determined by dropping an object, any object, and measuring its acceleration toward the Earth.  You do not have to know the mass of the Earth to measure an object's acceleration toward the Earth.  Then, you plug the acceleration (9.81 m/sec^2), and the mass of the dropped object into Newton's definition of Force (F=ma), to find the force (F) that the Earth exerts (gravitational acceleration) at the height from which you dropped the object.
Now you know everything in the equation F = g * (m1*m2)/r^2, except for m2, the mass of the Earth.  Solve for m2!
Although Newton did not know the magnitude of the gravitational constant (Big G), the form of his equation, which sets the force of gravity inversely proportional to the square of the distance between objects, was rapidly accepted by scientists because it agrees with the motions of the planets as measured by Keppler.
A: You dont know the mass of earth, but you know the force earth apply to you. This is F=mg, where g is 9.8 m/seg^2
Buut this is equal to the gravitation force, F=GmM/r^2. G is the gravitation constant, M the mass of earth, m your mass and r the radio of earth.
Cavendish was who messure the earth mass. In that time, Cavendish knew the radio of earth and G (he messured it with his balance).
Then yo equal the ecuation to mg=GmM/r^2 =>  g=GM/r^2  => M=(gr^2)/G.
A: This is explained in the Wikipedia article on the Cavendish Experiment.  Cavendish measured the gravitational attraction between two known masses using a torsion balance apparatus, which allowed him to calculate $G$ (to a surprising degree of accuracy given that it was done in the 1790s).  That measurement did not involve the mass of the Earth.
Once $G$ was known, the mass of the Earth could be calculated by measuring the force on a known mass from the Earth.
A: Brionius dealt with the value of G. Cavendish's experiments also confirmed the product of masses term.
The inverse-square portion of Newton's Theory of Universal Gravitation was immediately accepted, since it straightforwardly produces Keppler's Laws. Further thought shows that, in order to produce stable orbits, the exponent must be exactly 2 - no more, no less. It's integral calculus, and I'm not going to do it here.
Once you know G and the inverse square law, you can determine the mass of the earth by measuring g, as Jose stated. For very large mass ratios (like the earth/sun, for instance) you can take advantage of the fact that orbital period is essentially independent of the mass of the smaller, so measuring the orbital periods of the smaller planets allows determination of the sun's mass. Comparing the various orbital radii also confirms the inverse square part. 
Finally, looking at the sun's motion allows confirmation of the values by looking at the displacement of the sun by Jupiter. (The central axis of motion is actually slightly outside the surface of the sun - see sun/Jupiter barycenter).
While any of these findings might be coincidence with some other law, the fact that they are entirely consistent makes Newton's theory very well established.
