Simple Nature - Light and Matter

at its center. We already know that a gravitational energy thatvaries as −1/r is equivalent to a gravitational field proportional to1/r 2 , so it makes sense that a distance that is greater by a factorof 60 corresponds to a gravitational field that is 60 × 60=3600times weaker. Note that the calculation didn’t require knowledgeof the earth’s mass or the gravitational constant, which Newtondidn’t know.In 1665, shortly after Newton graduated from Cambridge, theGreat Plague forced the college to close for two years, and Newtonreturned to the family farm and worked intensely on scientificproblems. During this productive period, he carried out this calculation,but it came out wrong, causing him to doubt his new theoryof gravity. The problem was that during the plague years, he wasunable to use the university’s library, so he had to use a figure forthe radius of the moon’s orbit that he had memorized, and he forgotthat the memorized value was in units of nautical miles ratherthan statute miles. Once he realized his mistake, he found thatthe calculation came out just right, and became confident that histheory was right after all. 9Weighing the earth example 18⊲ Once Cavendish had found G = 6.67 × 10 −11 J · m/kg 2 (p. 101,example 15), it became possible to determine the mass of theearth. By the shell theorem, the gravitational energy of a massm at a distance r from the center of the earth is U = −GMm/r,where M is the mass of the earth. The gravitational field is relatedto this by mg dr = dU, or g = (1/m) dU/ dr = GM/r 2 . Solving forM, we haveM = gr 2 /G= (9.8 m/s2 )(6.4 × 10 6 m) 26.67 × 10 −11 J · m/kg 2= 6.0 × 10 24 m2 · kg 2J · s 2= 6.0 × 10 24 kgGravity inside the earth example 19⊲ The earth is somewhat more dense at greater depths, but as anapproximation let’s assume it has a constant density throughout.How does its internal gravitational field vary with the distance rfrom the center?⊲ Let’s write b for the radius of the earth. The shell theorem tell usthat at a given location r, we only need to consider the mass M