Newton and the 20 October 2015 PHY 1033C

? Why does the moon orbit the earth? So what was the problem The moon presented? ? ? ? ?

Galileo didn ’ t have this question Galileo ’ s explanation: The moon ’ s motion is natural For Galileo “ natural ” motion is uniform and circular

For Newton straight line motion is “ natural ” (First Law) That means and curved motion is constrained - indicates the presence of an unbalanced force (Second Law)

cosmic string

“ 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....

In other words, whatever pulls an apple toward the ground when it is loosed from the branch might be what is pulling on the moon

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

From Newton ’ s Principia Mathematica Newton’s cannon

the structure of Newton ’ s reasoning? What ’ s

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

Newton suspects that gravity weakens with distance Newton figures this out by combining his own insights with those of his predecessors His genius is to see that we can figure out how much from what we know about apples

What he already knew 1. a = 32 ft/sec 2 (by measuring it) 2. d = 1/2 a t 2 (from Galileo) 3. Inertial motion (his own “ corrected ” version) 5. f of tension in a string 6. The relation between a planet ’ s period and its distance from the sun 4. F α a f  mv2v2 /r (from Kepler T 2  r 3 )

f  mv 2 /r v = d/t = f  m(2πr/T) 2 /r = m4π 2 r 2 T2T2 x 1 r m4π 2 r T2T2 = 2πr/T m(4π 2 r 2 /T 2 )x 1/r Now Newton applies formula for centripetal force to the moon case:

m4π 2 r T2T2 But, from Kepler ’ s Third Law T 2  r 3 m4π 2 r T2T2 = r3r3 = m4π 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

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)

p = The same force that affects apples also affects the moon q = The moon falls 16 feet in one minute A. If p then q

B. q C. Therefore p But his argument is

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 To find BD Newton thinks like an engineer

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

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 I win the lotto my taxes will increase q: My taxes have increased. Therefore I won the lotto.

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. ”