Newton and the Spring 2007 His 3464

Did Newton discover gravity? What, then did he do? And what did the have do do with it?

Newton at 83 Portrait by Enoch Seeman

Sir James Thronhill’s portrait of Newton at 67

Newton in 1701 (at 59)

Newton at 46

? Why does the moon orbit the earth? The Problem: ? ? ? ?

Complicating the Problem Galileo’s explanation: inertial motion circular

cosmic string

"After dinner, the weather being warm, we went into the garden and drank tea, under the shade of some apple trees, only he and myself. Amidst other discourse, he told me he was in the same situation as when formerly the notion of gravitation came into his mind. It was occasioned by the fall of an apple as he sat in a contemplative mood." William Stuckley, 1726 The legend of the apple

“As he sat alone in a garden, he fell into a speculation on the power of gravity: that as this power is not found sensibly diminished at the remotest distance from the center of the earth to which we can rise, neither at the tops of the loftiest buildings, nor even on the summits of the highest mountains, it appeared to him reasonable to conclude that this power must extend much farther than was usually thought. Why not as high as the moon, said he to himself? And if so, her motion must be influenced by it; perhaps she is retained in her orbit thereby....

The structure of Newton’s argument is p = The same force that affects apples also affects the moon q = The moon falls 16 feet in one minute Given the following statements: A. If p then q B. q C. Therefore p So he must prove both A and B

Gravity likely weakens, but by how much? Newton figures this out by combining his own insights with those of his predecessors

What he already knew a = 32 ft/sec 2 (by measuring it) d = 1/2 a t 2 (from Galileo) Inertial motion (his own “corrected” version) f of tension in a string The relation between a planet’s period and its distance from the sun

r1r1 r2r2 r If r is the average distance of a planet from the sun And T is the time it takes the planet to circle the sun Then Kepler’s 3rd Law says: T2T2  r3r3

cosmic string To find the tension in the cosmic string Newton examined the case of a rock on a string

f  mv2v2 /r

f  mv 2 /r v = d/t = f  m(2 B r/T) 2 /r = m4 B 2 r 2 T2T2 x 1 r m4 B 2 r T2T2 = 2 B r/T m(4 B 2 r 2 /T 2 ) x 1/r Now Newton applies this result to the moon case:

m4 B 2 r T2T2 But, from Kepler’s Third Law T 2  r 3 m4 B 2 r T2T2 = r3r3 = m4 B 2 r2r2 So f  k Thus f " 2 r2r2

f is  1 r2r2 At 1 earth radius: At 2 earth radii: F  1 (2r) 2 r2 r2 1 = some amount F 1 X F 1 = 1 2 F 1 r 2r

Newton’s next move: To show that the moon (like apples) falling body can be considered a

From Newton’s Principia Mathematica

Two cases of projectile fall Thrown slowlyThrown hard

d = 89.4 mi 1 mi (4001) Throw at 4.92 mi/sec for sec (4001) 2 + (89.4) 2 = 4002 C B CB = D = CD = 5280 ft So CD =4002 mi - 1 mi =4001 mi (89.4) CB - BD BD = ½ x 32 ft/sec 2 x (18.166) 2

Since f  a, a  f f is also  1 r2r2 Therefore a is  1 r2r2 At 1 earth radius: At 2 earth radii: a= 32 ft/sec 2 a= 1/2 2 x 32 = 8 ft/sec 2

The moon is 60 earth radii away Therefore at the distance of the moon x32 or ft/sec 2 a =

In one minute (60 sec) the moon will “fall” d = 1/2 a t 2 = 1/2x x = 16 feet He has thus proven A (if p then q)

To prove B (q: the moon “falls” 16 in 1 min) Newton drew the following construction A F C. E D B AEDADF

A F C. E D B His task is to find BD, the distance the moon “falls” in a given amount of time.

A F C. E D B To find BD Newton thinks like an engineer

A F C. E D B AED ADF AE AD = AF SO AD 2 = AE X AF THUS AF= AD 2 AE

A D represents 1 minute. Therefore arc length AD is 1 min minutes in a month xmoon’s orbit We now know AD Arc AD Earth The whole circleis 1 month

A F C. E D B AF= AD 2 AE Since and since AE is known Newton calculated AF to be ?

Recall the structure of Newton’s argument : If p then q, q: therefore p If the “apple force” is affecting the moon, then the moon “falls” 16 feet in 1 minute. The moon does fall 16 feet in one minute. Therefore the apple force affects the moon.

But, look at the following argument: p: If you have enough money, you go to the movies q: You go to the movies. Therefore you have enough money.

Nature was more complicated than Newton ever dreamed.

"The most beautiful experience we can have is the mysterious. It is the fundamental emotion that stands at the cradle of true art and true science. Whoever does not know it and can no longer wonder, no longer marvel, is as good as dead.”