## Posts Tagged ‘G’

### How Big is the Earth?

Wednesday, October 8th, 2014
 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. _______________________________________

### Newton’s Law of Gravitation and the Universal Gravitational Constant

Monday, September 29th, 2014
 Last time we introduced the term acceleration of gravity, a physical phenomenon posited by Sir Isaac Newton in his book Philosophia Naturalis Principia Mathematica.   Newton’s Law of Gravitation is also presented in this book.   It provides the basis for his mathematical formula to calculate the acceleration of gravity, g, for any heavenly body in the universe.       Newton’s formula to compute the acceleration of gravity is, g = (G × M) ÷ R2 where, g is the acceleration of gravity, M the mass of the heavenly body, R the radius, and G the universal gravitational constant.       As for the values of the variables in his equation, Newton theorized that G would be a constant, holding the same numerical value throughout the universe.   This universal gravitational constant would be the glue that bound together M, the mass of the object being measured, and R, its radius, and render Newton’s formula a workable equation.   Without these three values, scientists would be unable to determine the acceleration of gravity rate, g, for the heavenly body under study, and Newton’s equation would be useless, relegated to the depths of pure mathematical theory.       In fact, the value for G wasn’t determined until 1796.   At that time Henry Cavendish derived its value as an adjunct to calculating the mass of Earth.   In the end he was able to arrive at values for Earth’s mass, M, as well as its radius, R.   He also came up with the much needed value for G, the universal gravitational constant.   He was able to accomplish so much by building upon the work of other scientists before him.       We’ll see who those earlier scientists were and how they contributed to the world’s discoveries concerning gravity next time. _______________________________________