Sir Isaac Newton 16431727

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Sir Isaac Newton (1643-1727)

• Truth is ever to be found in the simplicity, and
  not in the multiplicity and confusion of things.

Allison Wong
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The Life of Sir Isaac Newton

• Can be broken easily into three parts
• Childhood- 1643 to roughly 1669.
• 1669-87. He is a professor at Cambridge.
• Here he makes the majority of his mathematical
  discoveries.
• 1687-1727. He becomes a government official and
  largely abandons math.

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Childhood

• Born January 4, 1643 in a manor house, called
  Woolsthorpe, in Lincolnshire, to a wealthy
  family.
• His father owned livestock and land he was a
  quite wealthy man.
• He came from a long line of farmers
• Had no real education was illiterate.
• Died in October of 1642, three months before his
  son was even born.
• His mother remarried when Newton was two.
• Newton was largely neglected as a child.

Woolsthorpe today!
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Childhood cont

• Newtons stepfather died when he was 10.
• From that point on he began attending the Free
  Grammar School in Grantham.
• He did rather poorly (!). He was described as
  idle and inattentive by his teachers.
• Still, he went on to the Trinity College
  Cambridge in 1661, originally as a law major.
• Turning point he bought an astronomy book and
  was frustrated by the mathematics in it, so he
  decided to study geometry and trigonometry

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Newton as a Young Adult

• Over this period he produces the majority of his
  work.
• Short list of things he did
• 1666- Deduces inverse-square law
• Produces Theory of Universal Gravitation, 3 laws
  of motion.
• 1669- Relates differentials and integrals. (en
  route to better and bigger things!)
• 1671- Writes De Methodis Serierum et Fluxionum.
• 1687- Publishes most well-known work, Principia
• 1704- Publishes Opticks sadly, a lot of it is
  wrong, but he does recognize that white light can
  be refracted through prisms.

There are a lot of portraits of Newton.
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Later in Life

• He was very well-respected.
• However, controversies (like the Leibniz one)
  stressed him.
• He suffered a nervous breakdown (not his first,
  but a more severe one), and retired from research
  entirely.
• He left Cambridge in 1701 after becoming Warden
  of the Royal Mint in 1696 and Master in 1699.
• In 1703 was elected President of the Royal
  Society. He was re-elected every year until his
  death.
• He was knighted by Queen Anne in 1705.
• He died March 31, 1727 in London.

Newton on the 1 pound note
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Newtons Proof of Keplers Laws

• Newton showed in his book Principia Mathematica
  that Keplers three laws are the result of the
  Second Law of Motion and the Law of Universal
  Gravitation.
• Keplers Laws
• A planet revolves around the sun in an elliptical
  orbit with the sun at one focus.
• The line joining the sun to a planet sweeps out
  equal areas in equal times.
• The square of the period of revolution of a
  planet is proportional to the cube of the length
  of the major axis of its orbit.

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Newtons Second Law of Motion

• F m a
• Or, Net force on an object mass of the object
  moving times its acceleration.
• So a unit of force equals a unit of mass times a
  unit of acceleration.
• I.e., 1 Newton 1kg1m/s2

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Newtons Law of Gravitation

• .
• In both this and Fm a, F gravitational force on
  the planet.
• Let rr(t) be the position vector of the planet
  and vr(t) be its velocity vector.
• m mass of planet
• M mass of sun
• G gravitational constant
• U r/r (the unit vector in the r direction)
• Note this equation can be applied to any objects
  in orbit

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Proof of Law 1

• Equating the two expressions for f,
• So since a constant times the position vector,
  they must be parallel.
• So r X a must equal 0.
• So the derivative of r X v r X v r X v v X
  v r X a 000!
• So r X v some constant vector h. h/ 0 (r and
  v arent parallel), so vector r(t) is
  perpendicular to h.
• Therefore, the planet always lies in the plane
  through the oragin perpendicular to h.

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Proof 1 cont

• .
• Also,
• But then since u is the unit vector, and u1, u
  u1 and u u0 (u is orthogonal to u for all
  u).
• Therefore a X h GMu

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Law 1 Continued

• Since a X h GMu,
• (v X h)' v' X ha X h GMu.
• (c is a constant vector).
• So let a standard vector k point in the h
  direction, so the planet moves in the xy plane.
• c therefore lies in the xy plane.
• If ? is the angle between c and r, then (r, ?)
  are polar coordinates of the planet.

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Cont

• Remember this?
• Then
• So the magnitude of r r equals
• Let

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Almost done!

• Yet r dot (v cross h) also equals the magnitude
  of h, so
• Substituting in dh2/c, we get

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End of Proof of Law 1

• This is the polar equation of a conic section
  with the focus at the origin and eccentricity e.
• So since the planets orbit is a closed curve,
  the conic is an ellipse.
• Yay, were done!

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Proof of Keplers 2nd Law

• Let r be the vector from the sun to the planet
  (or, the position of the orbiting body.)
• The three vectors r (when t0), ?r, and r (when
  t ?t), form a triangle.
• The area of this triangle approximates the area
  swept out by vector r.
• This triangle is half of the parallelogram
  defined by vectors ro and ?r.
• You know what happens next

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Law II Cont

• Cross product!!!
• Since where l is a constant vector
• (according to Newtons law of angular motion,
  which I will not prove here)
• Then l/mr X v.

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Yay! End!

• So substitute that in
• Well, since the mass of the orbiting body is
  constant, and the angular velocity l is constant,
  the change in area over time is constant also.

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Works Cited

• http//www.alcyone.com/max/physics/kepler/5.html
• http//www.alcyone.com/max/physics/kepler/3.html
• http//www-history.mcs.st-andrews.ac.uk/history/Bi
  ographies/Newton.html
• http//www-history.mcs.st-andrews.ac.uk/history/Pi
  ctDisplay/Newton.html
• http//www-groups.dcs.st-and.ac.uk/history/Mathem
  aticians/Newton.html
• http//www.newton.cam.ac.uk/newtlife.html

Diagram of his proof of Law 3
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More!

• http//www.newtonproject.ic.ac.uk/prism.php?id15
• http//www-groups.dcs.st-and.ac.uk/history/Quotat
  ions/Newton.html
• http//www.blupete.com/Literature/Biographies/Scie
  nce/Newton.htm
• http//scidiv.bcc.ctc.edu/Math/Newton.html
• http//www.blupete.com/Literature/Biographies/Scie
  nce/Newton.htmCalculus
• http//www.maths.tcd.ie/pub/HistMath/People/Newton
  /RouseBall/RB_Newton.html
• http//www.lucidcafe.com/library/95dec/newton.html
  
• http//www.physics.upenn.edu/courses/gladney/mathp
  hys/java/sect5/subsubsection5_1_1_1.html

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Works Cited

• Images
• http//www.graywizard.net/images/Astronomy/solarsy
  s.gif
• http//www.shef.ac.uk/physics/people/vdhillon/teac
  hing/phy105/areas.gif
• http//www-groups.dcs.st-and.ac.uk/history/Poster
  s2/Newton.html
• http//www.mhhe.com/physsci/astronomy/fix/student/
  images/04f15.jpg