1 What Would Newton Do? David Tong Adams Society, November 2012
2 This talk is partly about physics, but partly about the history of physics. While the physics in the talk speaks for itself, the history needs a little further explanation. For this reason, I ve added some annotations at the beginning of the talk. Anything I said out loud I ve placed in a box. Like this. I should also stress that I m no historian. What follows is my cartoon version of history in the 17 th century.
3 This is really a talk about the Principia. This is Newton's masterpiece. First published in 1687, it describes the laws of motion, his law of gravity and then goes into great detail deriving various consequences from these laws. Not just the obvious stuff like the laws of planetary motion and friction, but hugely complicated detail of how the Earth gets squashed as it rotates and the precession of the equinoxes and a study of the tides and the first look at the three body problem of the Sun-Earth-Moon using perturbation theory and all sorts of things. It is rightly viewed as the beginning of theoretical physics. Now Newton didn't just come up with this stuff over night. It s the culmination of more than 20 years work. Most of the big breakthroughs were made in 1665 when Cambridge closed due to the plague and Newton returned to his home in Grantham.
4 Newton s Principia (1687)
5 But Newton wasn't the only one thinking about gravity. In London, three extraordinarily smart people were also working on these ideas. This group was Edmund Halley (of comet fame), Christopher Wren (of Christopher Wren fame) and, most importantly, Robert Hooke. You probably know Hooke for that stupid spring law, but he was one of the great intellectual forces of the 1600s. By the late 1670s, it seems likely that all three of these guys knew that there should be an inverse square law --- that is a force that drops off as 1-over-distance-squared --- that is responsible for gravity.
6 In fact, as I'll show you shortly, it's actually quite easy to see that gravity must follow an inverse-square-law. This is because 50 years earlier, Kepler had written down three laws that the describe the motion of the planets. The first of these states that all planets move on an ellipse. Laws 2 and 3 tell you exactly how fast the planets move on an ellipse. And, it turns out that if you think about circular orbits, Kepler's third law is equivalent to the inverse-square law for gravity. All of this will be covered later. But what's hard is to show Kepler's first law: to show that an inverse-square law of gravity means that the planets necessarily move on ellipses. That's the tough thing to show.
7 In 1684, Hooke tells his two friends that he s figured this out. He claims that he can show how the planets move on an ellipse. They, understandably, say "show us". Hooke kind of mumbles something and says he s lost the bit of paper and can't do it or won't do it and they leave thinking that he s full of shit. But this prompted Halley to come up to Cambridge to meet with Newton. Halley tells Newton about Hooke s claims and Newton says "oh, I showed that decades ok". Halley says "show me". Newton kind of mumbles and says he can t find the piece of paper and so on and Halley leaves, probably thinking that Newton is full of shit too. Except the next month, Newton sends Halley the proof. It's a short 9 page paper called "De Motu Corporum in Gyrum", or "The Motions of Bodies in Orbit". Halley says "amazing, let's publish". Newton says "well, it's not quite ready. Let me add a few things". Those few things took 2 more years and the result is the 3 books that make up the Principia.
8 (There's actually an interesting extra story here. When it's finally done, Newton sends it to Halley who had arranged for the Royal Society to publish it. Except, with the delay, they've already spent their publication budget on another book called, wonderfully, "The History of Fishes". And there's no more money to publish Newton. So Halley pays for publication out of his own pocket) Anyway, there's one key discovery missing from Newton's Principia: that's the discovery of calculus. It appears that he developed this back in 1665 as well. But, for some reason, he decided to keep it secret. Instead, all the proofs in the Principia are written using traditional methods of geometry.
9 Meanwhile, around the same time that the Principia is published, Leibniz in Germany publishes calculus. And, unlike Newton, Leibniz is really excited about this and telling everyone and, of course, everyone else gets really excited as well because, after all, it s brilliant. And one of the obvious challenges is to take all the proofs in the Principia and rewrite them using the new methods of calculus. This challenge is taken up eagerly by a number of mathematicians and it was very successful; so much so that by the time the second edition of the Principia was published in 1713, no one is thinking about this in terms of geometry. It's all about calculus. Of course, Newton is kind of pissed about this. So he's jumping up and down shouting that he did it first --- which he probably did, but he should have just told someone. And, in 1715, Newton writes that, of course, he had originally proved everything in the Principia using calculus and later translated it into geometry so that people could understand it better. As far as I can tell, most historians think this is just a lie.
10 Anyway, this leaves the Principia as something of an enigma. One of the most important scientific books ever written, yet using machinery that was out of date before the second edition was published. There s a quote from William Whewell, an old master of trinity, which sums this up nicely
11 "The ponderous instrument of synthesis, so effective in Newton's hands, has never since been grasped by anyone who could use it for such purpose; and we gaze at it with admiring curiosity, as some gigantic implement of war, which stands idle among the memorials of ancient days, and makes us wonder what manner of man he was who could wield as a weapon what we can hardly lift as a burden" William Whewell on Newton s geometric proofs, 1847
12 So what I want to do here is show you a geometrical proof of Kepler's laws in the Newtonian style. This proof is not quite the same as the one presented in the Principia. But it's fairly close. And the proof itself has an impressive pedigree. It was first published in a book by Maxwell, who gave credit to Hamilton. It was later independently rediscovered by Feynman. He gave a lecture on this which was lost apart from an audio recording and some scribbled notes. But in the 1990s, the lecture was reconstructed by David and Judith Goodstein and published in a book called "Feynman's lost lecture. If you want to learn more about this, then looks at this book. (If you really want to learn more about this, then the great astrophysicist Chandrasekhar has written an annotated version of the Principia. Be warned: it s not easy going).
13 First Some Facts About Ellipses
14 Building an ellipse from a piece of string P F / F F / P + FP = constant
15 There s an interesting historical fact here. The points F and F / are the foci of the ellipse. Each is called a focus. As we will see later, the planets move around the Sun on an ellipse with the Sun sitting at one of the focus. And the name focus for this point of the ellipse was chosen by Kepler himself. Because it is latin for fireplace.
16 Unpinning the string: draw a circle with centre F G P F / F
17 Draw the perpendicular bisector of F / G G P F / F
18 Claim: The perpendicular bisector intersects the line FG at Q=P G Q F / F
19 Proof: congruent triangles FQ = QG G Q F / F So distance F / QF is the same as F / G which is the length of the original string point Q lies on the ellipse
20 Moreover No other point on the perpendicular bisector lies on the ellipse G T P F / F FTG > FPG but TG = F / G FTF / > FPPF /
21 This means the perpendicular bisector is tangent to the ellipse G P θ 2 θ 1 F / F θ 1 =θ 2 (using the fact that there are congruent triangles in F / PG)
22 A simple physics corollary: an elliptic mirror P θ θ F / F All light leaving one focus reaches the other
23 Punchline to remember! G P F / F The perpendicular bisector to F / G is tangent to the ellipse at P
24 Planetary Orbits
25 all fields of planetary motion Kepler s laws S= 2 B 2 d4 x Tr E F rwith the Sun at one focus. 1. The orbits of all planets are ellipses, (1609) 2. A planet sweeps out equal areas in equal times. (1619) 3. The time, T, of an orbit is related to the distance, R, of the planet by (1627) E J T R3/2 References  Gibbons and Rychenkova,  Kapustin and Strassler, Tycho Brahe Johannes Kepler
26 all fields Our modern understanding S = d 4 x Tr E 2 B 2 F r 1. The orbits of all planets are ellipses, with the Sun at one focus. 2. A planet sweeps out equal areas in equal times. E J 3. The time, T, of an orbit is related to the distance, R, of the planet by T R 3/2 eferences Force towards the Sun Kepler 2  Gibbons and Rychenkova, Really just angular momentum conservation.  Kapustin and Strassler, Kepler 3 for circular orbits Inverse square law of gravity. Inverse square law Kepler 1.
27 Force towards the Sun In time Δt, the planet travels in a straight line AB B S A
28 Force towards the Sun In time Δt, the planet travels in a straight line AB It then receives an impulse BV towards the Sun V B S A
29 Force towards the Sun In time Δt, the planet travels in a straight line AB It then receives an impulse BV towards the Sun In the next Δt it travels the vector sum of AB + BV. This takes it to C, instead of C / C C / V B S A Note: C / C is parallel to BV
30 Kepler s Second Law: Area SAB = Area SBC C C / V B S A Note: C / C is parallel to BV
31 First show: Area SAB = Area SBC / C C / B S A Area = ½ base x height ½ SB x same distance
32 Then show: Area SAC = Area SBC / C C / B S A Area = ½ base x height ½ SB x same distance This demonstrates Kepler s second law
33 The inverse square law from circular orbits Prob B = B x B y v all fields B z exp (is/) S = d 4 x Tr E 2 B 2 R v v F r E J v T R 3/2 From Kepler 2, the speed, v, must be constant. The time taken to make an orbit is T = 2πR v
34 E J Circular Acceleration T R 3/2 Claim: the acceleration necessary to move in a circle is T = 2πR v a = v2 R rences ibbons and Rychenkova, We will now provide a geometric proof of this well-known fact. The main idea used in this proof will also help us later apustin and Strassler,
35 The velocity vector also rotates around a circle a a v a a
36 all fields By analogy, the time taken to move around the circle is S = d 4 x Tr v F r E S = 2 B 2 d 4 x Tr F r E 2 B 2 F r E J a T R 3/2 E J R E J T = 2πR v v T R 3/2 T = 2πR v T R 3/2 T = 2πR v a = v2 R T = 2πv a ychenkova,  Gibbons and Rychenkova, Strassler, rences References a = v2 R  Kapustin and Strassler,
37 We can rewrite this as The inverse square R law a = R So, using Newton s second law, the T force must be a 2 F = ma = mr T F = ma = mr 2 E J F 1 Now comparing to Kepler s Fthird R 2 1 law R 2 References Prob a = v2 exp (is/) T = 2πR R a = v2 all fields v T = 2πv S a a T = v2 2πv = d 4 x Tr E 2 B 2 Ra a = R T 2 T = 2πv a = R T 2 F = ma = mr T 2 F 1  Gibbons and Rychenkova, R v 4 T 2  Kapustin and Strassler, F r T R 3/2 This, of course, is Newton s famous law of gravity 4
38 Finding ellipses Finally, we need to show that the inverse square law implies that all orbits lie on an ellipse
39 Take an orbit and divide into equal angles D E S θ θ C B F A Equal areas in equal times Planet goes faster around ABC and slower around DEF
40 a = R T 2 F = ma = mr T 2 How much faster does it go? F = ma D = mr E F 1 R 2 F T 2 F 1 R 2 S SE SB = x C A B Claim: Let SE SB = x. Then Area(SEF) Area(SBC) = x2 References Area(SEF) Area(SBC) = x2  Gibbons and Rychenkova, Proof: Follows because they are similar triangles *. Or, if you prefer, we can flip the triangle SEF and overlay it on SBC  Kapustin and Strassler, Rychenkova, Strassler, S C B F E * This proof is not quite good enough. It assumes that CB is parallel to EF. It s not very hard to show that this is assumption is not needed.
41 References F = a = ma = R T 2 SE T SB So = = 2πv x F a 1 R 2 Area(SEF) a = R Area(SBC) SE = x2 T 2 SB = x F = t ma = R 2mR The area of a triangle is proportional to R 2 So the time taken to traverse a segment is proportional to But the force is proportional to  Gibbons and Rychenkova, T 2 Area(SEF) Area(SBC) F 1 = x2 R 2 Which means that the impulse, or change of velocity, felt t R 2 at each corner is constant  Kapustin and Strassler, v = F t = constant References  Gibbons and Rychenkova,  Kapustin and Strassler,
42 The upshot is that, when divided into equal angles S Δv θ θ θ D Δv v 3 Δv Δv C A B v 2 v 1 the size of the change of velocity, Δv, is the same at each kick
43 This means the associated velocity diagram is a regular polygon The planet is fastest when closest to the Sun. Here the velocity arrows are larger v 3 v 2 v 1 Each side has size Δv O The planet is slower when further from the Sun. Here the velocity arrows are smaller
44 And, in the limit Δt goes to zero, this becomes a circle O The point O is off-centre
45 Now we re almost done
46 Let s look at the real orbit and velocity orbit side by side Real Space Velocity Space S v P φ P B p v P b φ C O Note: the angle φ is the same in both cases. This is simplest to see by returning to the jerky version of planetary motion and the resulting polygon
47 SE Rotate the velocity diagram by 90 o SB = x Real Space v P φ S Area(SEF) Area(SBC) = x2 P t R 2 B v = F t = constant Velocity Space p u p φ O C b Because we rotated by 90 o, we must have that the two vectors are orthogonal u P v P =0
48 Rotate the velocity diagram by 90 o Real Space Velocity Space S v P φ P B O u p C φ p b Goal: Reconstruct the orbit in real space from the motion in velocity space. At every point P, the tangent to the orbit, v P, must be perpendicular to u P.
49 But we know how to do this p O C Draw the perpendicular bisector to OP Place the planet at the point where it intersects CP Do this for all points P on the circle and the bisector will always be tanget to the orbit
Lecture 13 Gravity in the Solar System Guiding Questions 1. How was the heliocentric model established? What are monumental steps in the history of the heliocentric model? 2. How do Kepler s three laws
Newton s Law of Gravity and Kepler s Laws Michael Fowler Phys 142E Lec 9 2/6/09. These notes are partly adapted from my Physics 152 lectures, where more mathematical details can be found. The Universal
Name: Planetary Orbit Simulator Student Guide Background Material Answer the following questions after reviewing the Kepler's Laws and Planetary Motion and Newton and Planetary Motion background pages.
Newton s Law of Universal Gravitation The greatest moments in science are when two phenomena that were considered completely separate suddenly are seen as just two different versions of the same thing.
Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015 Why are celestial motions and forces important? They explain the world around
From Aristotle to Newton The history of the Solar System (and the universe to some extent) from ancient Greek times through to the beginnings of modern physics. The Geocentric Model Ancient Greek astronomers
Unit 8 Lesson 2 Gravity and the Solar System Gravity What is gravity? Gravity is a force of attraction between objects that is due to their masses and the distances between them. Every object in the universe
The Newtonian Synthesis The Mathematical Principles of Natural Philosophy SC/NATS 1730, XVIII 1 The Falling Apple According to Newton, it was while he was in the orchard at Woolsthorpe during the plague
Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely
Elliptical Orbit 1/r 2 Force Jeffrey Prentis, Bryan Fulton, and Carol Hesse, University of Michigan-Dearborn, Dearborn, MI Laura Mazzino, University of Louisiana, Lafayette, LA Newton s proof of the connection
G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD
Vocabulary - Understanding Revolution in Universe Galaxy Solar system Planet Moon Comet Asteroid Meteor(ite) Heliocentric Geocentric Satellite Terrestrial planets Jovian (gas) planets Gravity our Solar
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
Astronomy 1140 Quiz 1 Review Prof. Pradhan September 15, 2015 What is Science? 1. Explain the difference between astronomy and astrology. (a) Astrology: nonscience using zodiac sign to predict the future/personality
1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: firstname.lastname@example.org Abstract: There are many longstanding
Exam 1 Ast 4 - Chapter 2 - Newton s Laws Exam 1 is scheduled for the week of Feb 19th Bring Pencil Scantron 882-E (available in the Bookstore) A scientific calculator (you will not be allowed to use you
AE554 Applied Orbital Mechanics Hafta 1 Egemen Đmre A bit of history the beginning Astronomy: Science of heavens. (Ancient Greeks). Astronomy existed several thousand years BC Perfect universe (like circles
Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,
DEFINITIONS Degree A degree is the 1 th part of a straight angle. 180 Right Angle A 90 angle is called a right angle. Perpendicular Two lines are called perpendicular if they form a right angle. Congruent
Exercises 131 The Falling Apple (page 233) 1 Describe the legend of Newton s discovery that gravity extends throughout the universe According to legend, Newton saw an apple fall from a tree and realized
Big Picture 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 2/7/2007 Lecture 1 Newton s Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle Big Picture First
Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:
Lecture 17 Newton on Gravity Patrick Maher Philosophy 270 Spring 2010 Introduction Outline of Newton s Principia Definitions Axioms, or the Laws of Motion Book 1: The Motion of Bodies Book 2: The Motion
39 Symmetry of Plane Figures In this section, we are interested in the symmetric properties of plane figures. By a symmetry of a plane figure we mean a motion of the plane that moves the figure so that
2.1 Force and Motion Kinematics looks at velocity and acceleration without reference to the cause of the acceleration. Dynamics looks at the cause of acceleration: an unbalanced force. Isaac Newton was
Student Outcomes Students learn to construct a line parallel to a given line through a point not on that line using a rotation by 180. They learn how to prove the alternate interior angles theorem using
Section Section Title 1.1 Identify Points, Lines, and Planes 1.2 Use Segments and Congruence 1.3 Use Midpoint and Distance Formulas Chapter 1: Essentials of Geometry Learning Targets I Can 1. Identify,
(Refer Slide Time: 00:22) Gas Dynamics Prof. T. M. Muruganandam Department of Aerospace Engineering Indian Institute of Technology, Madras Module No - 12 Lecture No - 25 Prandtl-Meyer Function, Numerical
Kepler s Laws of Planetary Motion Charles Byrne (Charles Byrne@uml.edu) Department of Mathematical Sciences University of Massachusetts Lowell Lowell, MA 01854, USA (The most recent version is available
Gravitational Forces 13.1 Newton s Law of Universal Gravity Newton discovered that gravity is universal. Everything pulls on everything else in the universe in a way that involves only mass and distance.
Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital
Dialog: LESSON 120 - MBA A: What are you doing tomorrow? B: I'm starting my MBA. A: I thought you hated business. What changed your mind? B: I do hate it, but I need to start making more money. A: MBA's
1. A student followed the given steps below to complete a construction. Step 1: Place the compass on one endpoint of the line segment. Step 2: Extend the compass from the chosen endpoint so that the width
Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the
USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney email@example.com Introduction The numerical calculations performed by scientists and engineers
5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the
ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there
GeoGebra in 10 lessons Gerrit Stols Acknowledgements GeoGebra is dynamic mathematics open source (free) software for learning and teaching mathematics in schools. It was developed by Markus Hohenwarter
Newton By William Blake ~1800 Ch 5 pg. 91-95 ++ Lecture 3 Isaac Newton & the Newtonian Age If I have ever made any valuable discoveries, it has been owing more to patient attention, than to any other talent.
Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.8-4.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
Einstein s Theory of Special Relativity Made Relatively Simple! by Christopher P. Benton, PhD Young Einstein Albert Einstein was born in 1879 and died in 1955. He didn't start talking until he was three,
Unit 4 The Solar System Chapter 7 ~ The History of the Solar System o Section 1 ~ The Formation of the Solar System o Section 2 ~ Observing the Solar System Chapter 8 ~ The Parts the Solar System o Section
Catapult Engineering Pilot Workshop LA Tech STEP 2007-2008 Some Background Info Galileo Galilei (1564-1642) did experiments regarding Acceleration. He realized that the change in velocity of balls rolling
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
Chapter 5: Circular Motion, the Planets, and Gravity 1. Earth s gravity attracts a person with a force of 120 lbs. The force with which the Earth is attracted towards the person is A. Zero. B. Small but
Bisections and Reflections: A Geometric Investigation Carrie Carden & Jessie Penley Berry College Mount Berry, GA 30149 Email: firstname.lastname@example.org, email@example.com Abstract In this paper we explore a geometric
Gravitation and Newton s Synthesis Vocabulary law of unviversal Kepler s laws of planetary perturbations casual laws gravitation motion casuality field graviational field inertial mass gravitational mass
Right Triangles If I looked at enough right triangles and experimented a little, I might eventually begin to notice a relationship developing if I were to construct squares formed by the legs of a right
How to Make Sure Your Talk Doesn t Suck David Tong This is an annotated version of a talk I gave at a summer school for first year graduate students. Anything sitting in a box, like this, summarizes what
VELOCITY, ACCELERATION, FORCE velocity Velocity v is a vector, with units of meters per second ( m s ). Velocity indicates the rate of change of the object s position ( r ); i.e., velocity tells you how
SHAPE NAMES Three-Dimensional Figures or Space Figures Rectangular Prism Cylinder Cone Sphere Two-Dimensional Figures or Plane Figures Square Rectangle Triangle Circle Name each shape. [triangle] [cone]
Geometry: Unit 1 Vocabulary 1.1 Undefined terms Cannot be defined by using other figures. Point A specific location. It has no dimension and is represented by a dot. Line Plane A connected straight path.
2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
Kepler s Third Law ID: 8515 By Michele Impedovo Time required 2 hours Activity Overview Johannes Kepler (1571-1630) is remembered for laying firm mathematical foundations for modern astronomy through his
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
Table of Contents Variable Forces and Differential Equations... 2 Differential Equations... 3 Second Order Linear Differential Equations with Constant Coefficients... 6 Reduction of Differential Equations
Chapter 3 The Science of Astronomy Days of the week were named for Sun, Moon, and visible planets. What did ancient civilizations achieve in astronomy? Daily timekeeping Tracking the seasons and calendar
J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for
Solar System 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X? A) Earth B) Sun C) Moon D) Polaris 2. Which object orbits Earth in both the Earth-centered
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
alternate interior angles two non-adjacent angles that lie on the opposite sides of a transversal between two lines that the transversal intersects (a description of the location of the angles); alternate
Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of
1 Pre-Test Directions: This will help you discover what you know about the subject of motion before you begin this lesson. Answer the following true or false. 1. Aristotle believed that all objects fell
GEOMETRY Constructions OBJECTIVE #: G.CO.12 OBJECTIVE Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic
College Prep. Geometry Course Syllabus Mr. Chris Noll Turner Ashby High School - Room 211 Email: firstname.lastname@example.org Website: http://blogs.rockingham.k12.va.us/cnoll/ School Phone: 828-2008 Text:
Teacher s Guide Grade Level: 6 8 Curriculum Focus: Mathematics Lesson Duration: Three class periods Program Description Discovering Math: Exploring Geometry From methods of geometric construction and threedimensional
Grade 9 Math Unit 8 : CIRCLE GEOMETRY NOTES 1 Chapter 8 in textbook (p. 384 420) 5/50 or 10% on 2011 CRT: 5 Multiple Choice WHAT YOU SHOULD ALREADY KNOW: Radius, diameter, circumference, π (Pi), central
Dialog: VIP LESSON 049 - Future of Business A: We really embarrassed ourselves last night at that business function. B: What are you talking about? A: We didn't even have business cards to hand out. We
Geometry Course Summary Department: Math Semester 1 Learning Objective #1 Geometry Basics Targets to Meet Learning Objective #1 Use inductive reasoning to make conclusions about mathematical patterns Give
1 Calculate the vector product of a and b given that a= 2i + j + k and b = i j k (Ans 3 j - 3 k ) 2 Calculate the vector product of i - j and i + j (Ans ) 3 Find the unit vectors that are perpendicular
Lecture 16 Newton s Second Law for Rotation Moment of Inertia Angular momentum Cutnell+Johnson: 9.4, 9.6 Newton s Second Law for Rotation Newton s second law says how a net force causes an acceleration.
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
Physics 53 Gravity Nature and Nature's law lay hid in night: God said, "Let Newton be!" and all was light. Alexander Pope Kepler s laws Explanations of the motion of the celestial bodies sun, moon, planets
Newton s Second Law Objective The Newton s Second Law experiment provides the student a hands on demonstration of forces in motion. A formulated analysis of forces acting on a dynamics cart will be developed
Chapter 6 Notes: Circles IMPORTANT TERMS AND DEFINITIONS A circle is the set of all points in a plane that are at a fixed distance from a given point known as the center of the circle. Any line segment
Motion Graphs It is said that a picture is worth a thousand words. The same can be said for a graph. Once you learn to read the graphs of the motion of objects, you can tell at a glance if the object in