In this relatively old age of scientific enlightenment, astronomical data such as planetary masses, sizes, planet-sun distances are just a book or a click away from the inquiring minds. But many a times that we are seem content at knowing those numbers and never bothered at how they were obtained, when in fact any direct means of such massive task of planetary data measurements are self-evidently impossible. In Newton’s time, it demands genius and massive sense of curiosity, but today it just require a little push of our  little minds and inner thirst for knowledge.

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The mass of the Earth taken into consideration, Newton’s Universal Law of Gravitation provides the tool. And the use of it enabled Newton himself to arrive at a not-so-complex formula for the determination of the mass of the Earth. Here is how he did it.

Any object close to the Earth’s surface, its weight is just equal to the force of gravity that the earth exerts on it. That is,

GmM/r2 = mg

Where G is the universal gravitation constant, m is for the mass of the any object, M and r are for the mass and radius of the Earth respectively, and g the acceleration due to the action of the Earth’s gravity.  Performing the necessary algebra, that is by dividing both sides of the equation by m (which will cancel out), then cross multiply and divide by the universal constant G yields a very elegant formula for the mass of the Earth, given by

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M = gr2/G

It was first derived by Sir Isaac Newton but failed to obtain the numerical result for he lacked the value of G, a physical constant that he introduced himself. The Earth’s radius r was not a problem for it was been already known centuries before Newton. Eratosthenes of Alexandria, Egypt, did it. In addition, the acceleration due to gravity g posed no problem  either; it was most likely obtained by Galileo, who lived a lifetime ahead of Newton.

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In a long interval of time, it took 100 years later before the numerical value for G was determined for the first time. Henry Cavendish, a compatriot of Newton obtained it through his delicate and high-precision torsion balance experiment (shown above). Cavendish in his painstaking and momentous experimental effort found G to be numerically equal to 6.67X10-11 N. m2/kg2. Substitution of this value together with the known values of g = 9.8 m/s2 and the radius of the Earth r = 6,378,000 meters to

M = gr2/G

     gives,                                      

M = 5.97×1024 kg                                                     

It is logical; it’s a pretty big number: 597 followed by 22 zeroes. On ground of this feat, Henry Cavendish is now known as the first person to weigh the Earth. But it’s worth-taking note that this extraordinary scientific leap is one of the early triumphs of Newton’s Law of Gravitation. Newton provided the entire recipe; Cavendish did the cooking.

In the Earth-bound setting, weight is the quantitative measure of the gravitational force the Earth exerts on any object; thus to talk about the weight of the Earth is scientifically unsound. And what was obtained by Cavendish is the mass of the Earth, the bulk of matter contained in, and an unchanging property. So, “how to weigh the Earth” may likewise sound awkward; “how to mass the Earth” would be proper. The title simply bears the influence of the pervading culture of meaning mass to weight in the same unsoundly manner to talk about Henry Cavendish as the first person to “weigh” the Earth (found in books) rather than the first  to “mass” the Earth.