## How Big is the Earth?

 Last time we discussed Isaac Newton’s Law of Gravitation and how he used it to arrive at conclusions concerning gravity.   He theorized the existence of a universal gravitational constant, G, a set numerical value for all heavenly bodies in our universe, and he developed a formula to determine the acceleration of gravity, g.       Newton felt sure that the gravity at play on the surface of any heavenly body, such as stars and planets, could be determined if one knew the value of  G, along with the object’s mass, M, and radius, R, and he developed this equation to do so, g = (G × M) ÷ R2       At this point you may be thinking, Finding the mass and radius of a heavenly body is hard enough, but what is this universal gravitational constant??   Good point.       Back in Newton’s time, the existence of G was purely speculative.   He conceived it to be a numerical value which would act as a fudge factor, enabling his equation for determining g to work.   As a matter of fact, Newton had no clue of how to determine G and was convinced that it would be beyond anyone’s ability to do so.       The mysterious G factor and its numerical value were not actually determined until more than a hundred years later by Henry Cavendish.   In 1796 Cavendish was focused on determining the Earth’s mass, M, by using Newton’s equation.   To arrive at a value for G, Cavendish conducted experiments which measured the gravitational attraction between two lead spheres attached by way of a torsion balance.   After much testing he eventually concluded that he had computed G to a reasonable degree of accuracy and that its value was equal to 3.439 x 10-8 cubic feet per slug per second squared.   In this case a slug is not a slimy creature living in the garden, but rather a unit of measurement used to quantify the mass of an object.       For the full story, see this article on Cavendish’s experiment by The Physics Classroom.       But even after determining G, Cavendish still had to obtain values for g and R in order to calculate M.   This was made possible thanks to the work of two men who came before him.    One of these was the Greek mathematician Eratosthenes, who way back in 230 B.C. discovered that the radius of the earth, R, could be calculated by simply measuring the shadows of objects cast on Earth’s surface.   All he needed was a measuring stick and geometry.       For the full story see this fascinating article on the subject from Bucknell University. As for the value of g, the acceleration of gravity on Earth, Cavendish was aided by the previous efforts of a Dutch mathematician from Newton’s time, Christiaan Huygens.   You may recall that Huygens was first introduced in a previous blog series on spur gear geometry, where we learned that he studied the motion of clock pendulums.    Through observation, Huygens was able to arrive at a mathematical formula capable of predicting the pendulums’ often erratic motion on ships at sea.       Next time we’ll see how Huygens’ insights gained by watching pendulums ultimately made it possible for him to arrive at a numerical value for Earth’s acceleration of gravity, g. _______________________________________