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?

• "Newton's Laws are assumed to work because you know apriori the mass of Earth" - why do you say that? I don't think that's the case. – user2357112 supports Monica Apr 29 '15 at 21:51
• – David Hammen Apr 29 '15 at 22:43

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.

• Cavendish did not determine the gravitational constant G. The intent of his experiment was to measure the mass of the Earth, as indicated by the title of his 1798 paper was "Experiments to determine the Density of the Earth." Newton's law of gravitation wouldn't be expressed in its modern form, $F=G\frac{m_1 m_2}{r^2}$, until 75 years later. – David Hammen Jul 9 '17 at 1:12
• @DavidHammen : Thank you for pointing this out. I edited the answer. An additional link I included notes that in Cavendish's experiment, "virtually all calculations are done via ratios so that proportionality constants are, in general, dropped out..." He avoided G. – Ernie Jul 9 '17 at 23:18
• @Eddie -- Cavendish didn't avoid using $G$. How could he? The concept of a universal gravitational constant didn't even exist in Cavendish's time. – David Hammen Jul 9 '17 at 23:32
• @DavidHammen : Yes, he didn't avoid it. My choice of words was faulty. – Ernie Jul 11 '17 at 18:57

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.

• But the gravitation force was deduced based on the fact that the mass of the Earth was known. But it wasn't. So how can you be sure that Newton's gravitation law is correct if you don't have the masses apriori so that you can first test that law ? – Abc2000ro Apr 29 '15 at 21:19
• "But the gravitation force was deduced based on the fact that the mass of the Earth was known." No, not really. The gravitational force was measured with a scale. The reason that didn't immediately give the value of $M_{earth}$ is that when you start $G$ is also unknown. – dmckee --- ex-moderator kitten Apr 29 '15 at 21:55
• I think Newton estimated the mass of the Earth by taking the average density of rocks and multiplying by the volume of the Earth which was known since the time of Eratosthenes. – user78939 Apr 29 '15 at 21:59
• Newton didn't deduce that g=9.8, was Gallileo. Mesuring the time that takes an object to fall, Gallileo estimmate that the freefall aceleration is 9.8 for all object (doesn't matter the mass of the object). The gravitation law was not based in the fact that the mass of the earth. I don't know exactly how Newton deduced it, but the law tell us that the gravitional force between 2 object is proportional to the mass of the onjects and the invert of square of the distance. – José Jorge Medina Serna Apr 30 '15 at 1:01

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.

• But how did Newton deduced his law then if at his time the mass of the Earth was not known ? – Abc2000ro Apr 29 '15 at 21:39
• I don't know the specifics of how Newton worked out his law (it involved the observed movements of planets), but his law is certainly not specific to the Earth's mass - it's true for, and therefore the general form could be derived from, the behavior of any two masses. – Brionius Apr 29 '15 at 22:00

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.