The Two-Body Problem

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "The Two-Body Problem"

Transcription

1 The Two-Body Problem Abstract In my short essay on Kepler s laws of planetary motion and Newton s law of universal gravitation, the trajectory of one massive object near another was shown to be a conic section. However, the frame of reference was a coordinate system centered at one of the masses (typically the heavier). Unfortunately, since even a heavier object will exhibit some acceleration toward the lighter one, such a frame cannot be regarded as inertial, meaning that Newton s laws are not entirely valid (without the introduction of fictitious forces ). The present treatment of eliminates this objection; however, relativistic effects are still not considered. Two-Body Problem Relative to Inertial Frame We consider the motions of two massive objects which we ll call planets relative to an inertial frame of reference, subject only to the mutual forces of gravitation. For convenience, we ll refer to the bodies as planets. Thus, we have a planet of mass whose position vector is x 1 as well as a planet of mass with position vector x 2. In terms of x 1 and x2, the center of mass is located with position vector ( ) 1 ( ) R = x1 + x2, M where M = +. mass mass center of mass x 1 R Figure 1: Two-Body Diagram The force applied to each of the planets has, by Newton s Law of Universal Gravitation, magnitude F = G, r 2 where r = x 1 x 2 is the distance between the planets, and where G is the universal gravitational constant, whose measured value in SI units is approximately G = N /kg 2. Since these forces are opposite each other, Newton s Second Law immediately implies that ( + ) d2 2 R = m1 2 x1 + 2 x2 = 0. x 2

2 This already says that the center of mass of the two-body system exhibits no acceleration relative to the inertial frame of reference. Therefore the moving center of gravity can itself be taken as an inertial frame of reference. Equations Relative to Center of Mass Since a non-rotating coordinate system having origin at the center of mass of our two-body system is an inertial frame, we shall ignore the movement of the center of mass and concentrate only on the movements of the planets relative to their common center of mass. Given this, we ll re-define the center-of-mass coordinate system mass - R R mass position vector R to be that pointing from the center of mass to Planet 1 (having mass ). Knowing how Figure 2: Center-of-mass frame Planet 1 moves about the center of mass will dictate Planet 2 s motion as well, since Planet 2 s position vector R. This configuration is depicted in Figure 2. The task, then, is is just to compute how the position vector R of Planet 1 evolves with time. Rather than letting r be the distance between the two planets, we let r be the length of the position vector ( R. With) this understanding, the distance between the two m1 + planets is simply r. Using Newton s law of universal gravitation, and denoting by r the unit vector in the direction of R, we immediately obtain 2 G R = ( ) 2 r. + r 2 This gives the acceleration of Planet 1 in this frame of reference: a = 2 R = Gm 3 2 ( + ) 2 r 2 r. Now comes the math! Let s gather together all the constants into a single

3 constant by defining Gm 3 2 K = ( + ) 2, leaving us with the task of solving the second-order differential equation a = 2 To deal with the above differential equation, we let r be the unit vector in the direction of R and let s be the d unit vector in the direction of r. From r r = 1, we get 0 = d ( r ) r = 2 r d r, which implies that r s = 0. d Next, from r = (cos θ, sin θ), get dθ r = ( sin θ, cos θ) = dθ s. R = K r 2. (1) Note: R=(rcos θ, rsin θ) center-of-mass coordinate system mass - R unit vector s in direction of d R R Figure 3: Polar coordinates θ mass unit vector r in direction of R Likewise, it follows that d s = dθ r, from which we obtain the velocity vector for Planet 1: v = d d ( ) R = r r = dr r +r ( d r ) = dr r +r dθ s. (2)

4 The acceleration is the time derivative of Equation (2): ( ) d d dr dθ a = v = r +r s = d2 r dr dθ θ r +2 s +r s r 2 ( 2 ( ) ) 2 r dθ = r 2 ( ) 2 dθ r ) r + (2 dr dθ + θ rd2 2 However, recalling from Equation (1) that a = K r 2 r, we extract two equations of interest: and r 2 r ( ) 2 dθ = K (3) r 2 2 dr dθ + θ rd2 = 0. (4) 2 We can immediate extract useful information from Equation (4), namely that ( d r 2dθ ) = 2r dr dθ + θ r2d2 = 0. 2 This means that the quantity L = r 2dθ Next, define the variable u = L2 Kr. is a constant. and that dθ = L r = K2 u 2. 2 L 3 We have, using the Chain Rule, that dr = d ( L 2 dθ Ku ) dθ = L2 dθ du Ku 2 dθ = K du L dθ. Differentiate again and obtain r = d ( ) dr dθ 2 dθ = K u dθ L dθ 2 = u 2 u K3 L 4 dθ 2 s

5 Substitute into Equation (3) and get K3 u 2 ( ) ( u L 2 K 2 L 4 dθ u 2 ) 2 = K3 u 2. 2 Ku L 3 L 4 Dividing by the common factor of K3 u 2 L 4 results in the inhomogeneous secondorder differential equation: u dθ + u = 1. 2 The general solution of this has the form u = u(θ) = 1 + e cos(θ θ 0 ), where e and θ 0 are constants of integration which can be determined from the initial conditions. In turn, we now know that the radial distance of Planet 1 from the center of mass is given, in terms of θ, by where the constants K and L are r = L2 Ku = L 2 K(1 + e cos(θ θ 0 )), (5) K = Gm 3 2 ( + ) 2, and L = r2dθ. The Center-of-Mass Inertial Frame and the Eccentricity The two-body problem will have in all four initial conditions: the two initial positions of the planets and the two initial velocities. Correspondingly, there are four constants of integration which occur: the initial position and velocity of the center of mass relative to the original inertial frame, and the constants e and θ 0 appearing in Equation (5). The motion of the center of mass is easily determined. If x 1 (0) and x 2 (0) are the initial positions of Planets 1 an, respectively, then the initial position of the center of mass is R (0) = 1 + ( x1 (0) + x2 (0)).

6 Likewise, if v 1 (0) and v 2 (0) are the initial velocities of Planets 1 and 2, then the combined system has total momentum (which is conserved) p = m1 v1 (0) + v2 (0), Configuration at t=0 y initial velocity of Planet 1 velocity of frame (relative to v moving frame) center-of-mass u1 (0) coordinate system mass mass u2 (0) and so the center of mass will move with constant velocity 1 v = p. + This means that relative to the moving frame, the initial velocities of Planets 1 an are u1 (0) = v 1 v and initial velocity of Planet 2 (relative to moving frame) Figure 4: Initial conditions relative to moving frame u 2 (0) = v 2 v, respectively. Finally, we choose the x-axis of our moving frame so as to pass through the center of mass of Planet 1 at time t = 0. All of this is indicated in Figure 4 above. For the remainder of the section we shall be operating exclusively in the moving inertial center-of-mass frame. Referring to Figure 4 we let r 0 be the initial distance of Planet 1 from the center of mass. Next we let α be the angle that u 1 makes with the positive x-axis of the moving frame. Since at time t = 0 we have θ = 0, Equation 5 at time t = 0 now reads r 0 = Equation (6) now can be written as L 2 K(1 + e cos θ 0 ). (6) e cos θ 0 = L2 Kr 0 1. (7) Next we take the time derivative of Equation (5) and get: dr = L 2 e sin(θ θ 0 ) dθ K[1 + e cos(θ θ 0 )] 2 ; at t = 0 this simplifies to x u 0 cos α = Kr2 0e sin θ 0 L 2 L r 2 0 = Ke sin θ 0, L

7 where u 0 is the magnitude of the vector u 1 (0). Therefore, e sin θ 0 = Lu 0 cos α K Squaring and adding Equations (7) and (8) leads to (8) ( ) L e = 1 + L2 u 2 0 cos 2 α (9) Kr 0 K 2 Equation (9) can be written purely in terms of the initial data once we realize that the constant L can be expressed as giving rise to Since K is the constant L = r 2dθ = r 0u 0 sin α, ( e 2 r0 u 2 0 sin 2 α = K ) r2 0u 4 0 cos 2 α. K 2 K = Gm2 1 ( + ) 2, we see that the eccentricity of the orbit of Planet 1 (and hence of Planet 2) about the moving center of mass can be expressed purely in terms of the initial data r 0, u 0, the angle α, the masses and and the gravitational constant G. One Planet s Orbit Relative to the Other In the above, we have shown that each member planet of a two-body system will orbit about the system s center of mass in either an ellipse, a parabola, or a hyperbola, depending on the eccentricity. A question that remains is whether one planet s orbit about the other will likewise be a conic section. If we can show this, then Kepler s First Law will be validated, namely that the planets in the solar system orbit about the sun in elliptical orbits.

8 g( x) = 1- x 2 4 f( x) = - 1- x 2 4 The demonstration is actually easier than one might think! Figure 5 depicts both the center-of-mass coordinate system, with planet 1 s orbit about the center of mass, as well as the moving coordinate system centered at Planet 1. We are to show that, in polar coordinates based at Planet 1, the orbit of Planet 2 is also a conic section with the same eccentricity. If r is the polar distance from the center of mass to Planet 1, then in terms of the radial angle θ, we have already shown that r = r=distance of Planet 1 from center of mass d=distance of Planet 1 + from Planet 2 = ( )r L 2 K(1 + e cos(θ θ 0 )). Since the distance d between Planets 1 an is given by d = ( + )r, moving coordinate system at Planet 1 y r θ θ center of mass L 2 x r= K(1+ecos(θ θ 0 )) Figure 5: Coordinate System at Planet 1 it follows that the polar distance s from Planet 1 to Planet 2 must be (note the minus sign!): L 2 ( + ) s = K (1 + e cos(θ θ 0 )), which also describes a conic section with the same eccentricity e.

Kepler s Laws of Planetary Motion and Newton s Law of Universal Gravitation

Kepler s Laws of Planetary Motion and Newton s Law of Universal Gravitation Kepler s Laws of lanetary Motion and Newton s Law of Universal Gravitation Abstract These notes were written with those students in mind having taken (or are taking) A Calculus and A hysics Newton s law

More information

Orbital Mechanics. Angular Momentum

Orbital Mechanics. Angular Momentum 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

More information

Math 1302, Week 3 Polar coordinates and orbital motion

Math 1302, Week 3 Polar coordinates and orbital motion Math 130, Week 3 Polar coordinates and orbital motion 1 Motion under a central force We start by considering the motion of the earth E around the (fixed) sun (figure 1). The key point here is that the

More information

Penn State University Physics 211 ORBITAL MECHANICS 1

Penn State University Physics 211 ORBITAL MECHANICS 1 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

More information

Lesson 5 Rotational and Projectile Motion

Lesson 5 Rotational and Projectile Motion Lesson 5 Rotational and Projectile Motion Introduction: Connecting Your Learning The previous lesson discussed momentum and energy. This lesson explores rotational and circular motion as well as the particular

More information

Mechanics 1: Conservation of Energy and Momentum

Mechanics 1: Conservation of Energy and Momentum Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation

More information

Binary Stars. Kepler s Laws of Orbital Motion

Binary Stars. Kepler s Laws of Orbital Motion Binary Stars Kepler s Laws of Orbital Motion Kepler s Three Laws of orbital motion result from the solution to the equation of motion for bodies moving under the influence of a central 1/r 2 force gravity.

More information

Lecture L17 - Orbit Transfers and Interplanetary Trajectories

Lecture L17 - Orbit Transfers and Interplanetary Trajectories S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17 - Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to

More information

Chapter 6 Circular Motion

Chapter 6 Circular Motion Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

More information

Orbital Dynamics with Maple (sll --- v1.0, February 2012)

Orbital Dynamics with Maple (sll --- v1.0, February 2012) Orbital Dynamics with Maple (sll --- v1.0, February 2012) Kepler s Laws of Orbital Motion Orbital theory is one of the great triumphs mathematical astronomy. The first understanding of orbits was published

More information

Astromechanics Two-Body Problem (Cont)

Astromechanics Two-Body Problem (Cont) 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

More information

Exemplar Problems Physics

Exemplar Problems Physics Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration

More information

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies

CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at

More information

Building Planetary Orbits (the Feynman way)

Building Planetary Orbits (the Feynman way) Building Planetary Orbits (the Feynman way) Using a method of computing known as finite differences we can quickly calculate the geometry of any planetary orbit. All we need are a starting position and

More information

Planetary Orbit Simulator Student Guide

Planetary Orbit Simulator Student Guide 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.

More information

Today. Laws of Motion Conservation Laws Gravity tides. What is the phase of the moon?

Today. Laws of Motion Conservation Laws Gravity tides. What is the phase of the moon? Today Laws of Motion Conservation Laws Gravity tides What is the phase of the moon? How is mass different from weight? Mass the amount of matter in an object Weight the force that acts upon an object You

More information

Lecture 5: Newton s Laws. Astronomy 111

Lecture 5: Newton s Laws. Astronomy 111 Lecture 5: Newton s Laws Astronomy 111 Isaac Newton (1643-1727): English Discovered: three laws of motion, one law of universal gravitation. Newton s great book: Newton s laws are universal in scope,

More information

Chapter 13 - Gravity. David J. Starling Penn State Hazleton Fall Chapter 13 - Gravity. Objectives (Ch 13) Newton s Law of Gravitation

Chapter 13 - Gravity. David J. Starling Penn State Hazleton Fall Chapter 13 - Gravity. Objectives (Ch 13) Newton s Law of Gravitation The moon is essentially gray, no color. It looks like plaster of Paris, like dirty beach sand with lots of footprints in it. -James A. Lovell (from the Apollo 13 mission) David J. Starling Penn State Hazleton

More information

Introduction to Gravity and Orbits. Isaac Newton. Newton s Laws of Motion

Introduction to Gravity and Orbits. Isaac Newton. Newton s Laws of Motion Introduction to Gravity and Orbits Isaac Newton Born in England in 1642 Invented calculus in early twenties Finally published work in gravity in 1687 The Principia Newton s Laws of Motion 1: An object

More information

First Semester Learning Targets

First Semester Learning Targets First Semester Learning Targets 1.1.Can define major components of the scientific method 1.2.Can accurately carry out conversions using dimensional analysis 1.3.Can utilize and convert metric prefixes

More information

Chapter 13 Newton s Theory of Gravity

Chapter 13 Newton s Theory of Gravity Chapter 13 Newton s Theory of Gravity Chapter Goal: To use Newton s theory of gravity to understand the motion of satellites and planets. Slide 13-2 Chapter 13 Preview Slide 13-3 Chapter 13 Preview Slide

More information

Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

More information

Chapter 6. Work and Energy

Chapter 6. Work and Energy Chapter 6 Work and Energy The concept of forces acting on a mass (one object) is intimately related to the concept of ENERGY production or storage. A mass accelerated to a non-zero speed carries energy

More information

Chapter 13. Newton s Theory of Gravity

Chapter 13. Newton s Theory of Gravity Chapter 13. Newton s Theory of Gravity The beautiful rings of Saturn consist of countless centimeter-sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton

More information

1 Newton s Laws of Motion

1 Newton s Laws of Motion 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

More information

Orbital Mechanics and Space Geometry

Orbital Mechanics and Space Geometry Orbital Mechanics and Space Geometry AERO4701 Space Engineering 3 Week 2 Overview First Hour Co-ordinate Systems and Frames of Reference (Review) Kepler s equations, Orbital Elements Second Hour Orbit

More information

Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4. problems: 5.61, 5.67, 6.63, 13.21 Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

More information

Chapter 9 Circular Motion Dynamics

Chapter 9 Circular Motion Dynamics Chapter 9 Circular Motion Dynamics 9. Introduction Newton s Second Law and Circular Motion... 9. Universal Law of Gravitation and the Circular Orbit of the Moon... 9.. Universal Law of Gravitation... 3

More information

1 Kepler s Laws of Planetary Motion

1 Kepler s Laws of Planetary Motion 1 Kepler s Laws of Planetary Motion 1.1 Introduction Johannes Kepler published three laws of planetary motion, the first two in 1609 and the third in 1619. The laws were made possible by planetary data

More information

Lecture L22-2D Rigid Body Dynamics: Work and Energy

Lecture L22-2D Rigid Body Dynamics: Work and Energy 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

More information

Newton s derivation of Kepler s laws (outline)

Newton s derivation of Kepler s laws (outline) Newton s derivation of Kepler s laws (outline) 1. Brief history. The first known proposal for a heliocentric solar system is due to Aristarchus of Samos (ancient Greece, c. 270 BC). Following a long period

More information

Chapter 4. Forces and Newton s Laws of Motion

Chapter 4. Forces and Newton s Laws of Motion Chapter 4 Forces and Newton s Laws of Motion 4.1 The Concepts of Force and Mass A force is a push or a pull. Contact forces arise from physical contact. Action-at-a-distance forces do not require contact

More information

Halliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton

Halliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 13-1 Newton's Law

More information

Lecture Outline Chapter 5. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.

Lecture Outline Chapter 5. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc. Lecture Outline Chapter 5 Physics, 4 th Edition James S. Walker Chapter 5 Newton s Laws of Motion Dynamics Force and Mass Units of Chapter 5 Newton s 1 st, 2 nd and 3 rd Laws of Motion The Vector Nature

More information

AST 101 Lecture 7. Newton s Laws and the Nature of Matter

AST 101 Lecture 7. Newton s Laws and the Nature of Matter AST 101 Lecture 7 Newton s Laws and the Nature of Matter The Nature of Matter Democritus (c. 470-380 BCE) posited that matter was composed of atoms Atoms: particles that can not be further subdivided 4

More information

Physics 1A Lecture 10C

Physics 1A Lecture 10C Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

More information

Chapter 13. Gravitation

Chapter 13. Gravitation Chapter 13 Gravitation 13.2 Newton s Law of Gravitation In vector notation: Here m 1 and m 2 are the masses of the particles, r is the distance between them, and G is the gravitational constant. G = 6.67

More information

Chapter 15 Collision Theory

Chapter 15 Collision Theory Chapter 15 Collision Theory 151 Introduction 1 15 Reference Frames Relative and Velocities 1 151 Center of Mass Reference Frame 15 Relative Velocities 3 153 Characterizing Collisions 5 154 One-Dimensional

More information

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

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 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

More information

2. Orbits. FER-Zagreb, Satellite communication systems 2011/12

2. Orbits. FER-Zagreb, Satellite communication systems 2011/12 2. Orbits Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1 Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit

More information

Website: Reading: Homework: Discussion:

Website: Reading: Homework: Discussion: Reminders! Website: http://starsarestellar.blogspot.com/ Lectures 1-5 are available for download as study aids. Reading: You should have Chapters 1-4 read, Chapter 5 by the end of today, and Chapters 6

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 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

More information

Isaac Newton s (1642-1727) Laws of Motion

Isaac Newton s (1642-1727) Laws of Motion 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

More information

Halliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton

Halliday, Resnick & Walker Chapter 13. Gravitation. Physics 1A PHYS1121 Professor Michael Burton Halliday, Resnick & Walker Chapter 13 Gravitation Physics 1A PHYS1121 Professor Michael Burton II_A2: Planetary Orbits in the Solar System + Galaxy Interactions (You Tube) 21 seconds 13-1 Newton's Law

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

More information

Motion in Space: Velocity and Acceleration

Motion in Space: Velocity and Acceleration Study Sheet (13.4) Motion in Space: Velocity and Acceleration Here, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study: The motion of an object, including

More information

Physics 53. Gravity. Nature and Nature's law lay hid in night: God said, "Let Newton be!" and all was light. Alexander Pope

Physics 53. Gravity. Nature and Nature's law lay hid in night: God said, Let Newton be! and all was light. Alexander Pope 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

More information

5. Universal Laws of Motion

5. Universal Laws of Motion 5. Universal Laws of Motion If I have seen farther than others, it is because I have stood on the shoulders of giants. Sir Isaac Newton (1642 1727) Physicist 5.1 Describing Motion: Examples from Daily

More information

4.1 Describing Motion. How do we describe motion? Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity

4.1 Describing Motion. How do we describe motion? Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity 4.1 Describing Motion Our goals for learning:! How do we describe motion?! How is mass different from weight? How do we

More information

Galileo and the physics of motion

Galileo and the physics of motion Galileo and the physics of motion Studies of motion important : planetary orbits, cannonball accuracy, basic physics. Galileo among first to make careful observations Looked at velocity, acceleration,

More information

Concept Review. Physics 1

Concept Review. Physics 1 Concept Review Physics 1 Speed and Velocity Speed is a measure of how much distance is covered divided by the time it takes. Sometimes it is referred to as the rate of motion. Common units for speed or

More information

AP1 Gravity. at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit?

AP1 Gravity. at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit? 1. A satellite of mass m S orbits a planet of mass m P at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit? (A) v = Gm P R (C)

More information

The derivation of Kepler's three laws using Newton s law of gravity and the law of force. A topic in Mathematical Physics Seminar at Secondary School

The derivation of Kepler's three laws using Newton s law of gravity and the law of force. A topic in Mathematical Physics Seminar at Secondary School The derivation of Kepler's three laws using Newton s law of gravity and the law of force A topic in Mathematical Physics Seminar at Secondary School Name: Martin Kazda Introduction Ph.D student: Faculty

More information

Kepler, Newton, and laws of motion

Kepler, Newton, and laws of motion Kepler, Newton, and laws of motion !! " The only history in this course:!!!geocentric vs. heliocentric model (sec. 2.2-2.4)" The important historical progression is the following:!! Ptolemy (~140 AD) Copernicus

More information

Summary: The Universe in 1650

Summary: The Universe in 1650 Celestial Mechanics: The Why of Planetary Motions Attempts to Describe How Celestial Objects Move Aristotle, Hipparchus, and Ptolemy: The Ptolemaic System Aristarchus, Copernicus, and Kepler: The Copernican

More information

Chapter 3: Force and Motion

Chapter 3: Force and Motion Force and Motion Cause and Effect Chapter 3 Chapter 3: Force and Motion Homework: All questions on the Multiple- Choice and the odd-numbered questions on Exercises sections at the end of the chapter. In

More information

Newton s Law of Gravity

Newton s Law of Gravity 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

More information

Newton s Law of Gravity

Newton s Law of Gravity Newton s Law of Gravity Example 4: What is this persons weight on Earth? Earth s mass = 5.98 10 24 kg Mar s mass = 6.4191 10 23 kg Mar s radius = 3400 km Earth s radius = 6378 km Newton s Form of Kepler

More information

Kepler s Laws of Planetary Motion

Kepler s Laws of Planetary Motion 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

More information

Universal Law of Gravitation Honors Physics

Universal Law of Gravitation Honors Physics Universal Law of Gravitation Honors Physics 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

More information

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers

More information

Chapter 13 Newton s Theory of Gravity

Chapter 13 Newton s Theory of Gravity Chapter 13 Newton s Theory of Gravity The textbook gives a good brief account of the period leading up to Newton s Theory of Gravity. I am not going to spend much time reviewing the history but will show

More information

DERIVING KEPLER S LAWS OF PLANETARY MOTION. By: Emily Davis

DERIVING KEPLER S LAWS OF PLANETARY MOTION. By: Emily Davis DERIVING KEPLER S LAWS OF PLANETARY MOTION By: Emily Davis WHO IS JOHANNES KEPLER? German mathematician, physicist, and astronomer Worked under Tycho Brahe Observation alone Founder of celestial mechanics

More information

Gravitational Potential Energy

Gravitational Potential Energy Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the

More information

Newton s Law of Gravity and Kepler s Laws

Newton s Law of Gravity and Kepler s 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

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

12/3/10. Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.

12/3/10. Copyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The beautiful rings of Saturn consist of countless centimeter-sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton s theory of gravity to understand the

More information

STATICS. Introduction VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.

STATICS. Introduction VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr. Eighth E CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Introduction Lecture Notes: J. Walt Oler Texas Tech University Contents What is Mechanics? Fundamental

More information

Chapter 4. Dynamics: Newton s Laws of Motion

Chapter 4. Dynamics: Newton s Laws of Motion Chapter 4 Dynamics: Newton s Laws of Motion The Concepts of Force and Mass A force is a push or a pull. Contact forces arise from physical contact. Action-at-a-distance forces do not require contact and

More information

NEWTON S LAWS OF MOTION

NEWTON S LAWS OF MOTION NEWTON S LAWS OF MOTION Background: Aristotle believed that the natural state of motion for objects on the earth was one of rest. In other words, objects needed a force to be kept in motion. Galileo studied

More information

M OTION. Chapter2 OUTLINE GOALS

M OTION. Chapter2 OUTLINE GOALS Chapter2 M OTION OUTLINE Describing Motion 2.1 Speed 2.2 Vectors 2.3 Acceleration 2.4 Distance, Time, and Acceleration Acceleration of Gravity 2.5 Free Fall 2.6 Air Resistence Force and Motion 2.7 First

More information

Making Sense of the Universe: Understanding Motion, Energy, and Gravity

Making Sense of the Universe: Understanding Motion, Energy, and Gravity Making Sense of the Universe: Understanding Motion, Energy, and Gravity 1. Newton s Laws 2. Conservation Laws Energy Angular momentum 3. Gravity Review from last time Ancient Greeks: Ptolemy; the geocentric

More information

Single and Double plane pendulum

Single and Double plane pendulum Single and Double plane pendulum Gabriela González 1 Introduction We will write down equations of motion for a single and a double plane pendulum, following Newton s equations, and using Lagrange s equations.

More information

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

Mechanics lecture 7 Moment of a force, torque, equilibrium of a body G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

More information

DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION

DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION 1 DIRECT ORBITAL DYNAMICS: USING INDEPENDENT ORBITAL TERMS TO TREAT BODIES AS ORBITING EACH OTHER DIRECTLY WHILE IN MOTION Daniel S. Orton email: dsorton1@gmail.com Abstract: There are many longstanding

More information

Niraj Sir GRAVITATION CONCEPTS. Kepler's law of planetry motion

Niraj Sir GRAVITATION CONCEPTS. Kepler's law of planetry motion GRAVITATION CONCEPTS Kepler's law of planetry motion (a) Kepler's first law (law of orbit): Every planet revolves around the sun in an elliptical orbit with the sun is situated at one focus of the ellipse.

More information

Lecture 13. Gravity in the Solar System

Lecture 13. Gravity in the Solar System 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

More information

Unification of the laws of the Earth and the Universe Why do planets appear to wander slowly across the sky?

Unification of the laws of the Earth and the Universe Why do planets appear to wander slowly across the sky? October 19, 2015 Unification of the laws of the Earth and the Universe Why do planets appear to wander slowly across the sky? Key Words Newton s Laws of motion, and Newton s law of universal gravitation:

More information

5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity.

5. Forces and Motion-I. Force is an interaction that causes the acceleration of a body. A vector quantity. 5. Forces and Motion-I 1 Force is an interaction that causes the acceleration of a body. A vector quantity. Newton's First Law: Consider a body on which no net force acts. If the body is at rest, it will

More information

University Physics 226N/231N Old Dominion University. Chapter 13: Gravity (and then some)

University Physics 226N/231N Old Dominion University. Chapter 13: Gravity (and then some) University Physics 226N/231N Old Dominion University Chapter 13: Gravity (and then some) Dr. Todd Satogata (ODU/Jefferson Lab) satogata@jlab.org http://www.toddsatogata.net/2016-odu Monday, November 28,

More information

Magnetic Dipoles. Magnetic Field of Current Loop. B r. PHY2061 Enriched Physics 2 Lecture Notes

Magnetic Dipoles. Magnetic Field of Current Loop. B r. PHY2061 Enriched Physics 2 Lecture Notes Disclaimer: These lecture notes are not meant to replace the course textbook. The content may be incomplete. Some topics may be unclear. These notes are only meant to be a study aid and a supplement to

More information

circular motion & gravitation physics 111N

circular motion & gravitation physics 111N 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

More information

The Cosmic Perspective Seventh Edition. Making Sense of the Universe: Understanding Motion, Energy, and Gravity. Chapter 4 Lecture

The Cosmic Perspective Seventh Edition. Making Sense of the Universe: Understanding Motion, Energy, and Gravity. Chapter 4 Lecture Chapter 4 Lecture The Cosmic Perspective Seventh Edition Making Sense of the Universe: Understanding Motion, Energy, and Gravity Making Sense of the Universe: Understanding Motion, Energy, and Gravity

More information

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000) 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

More information

Described by Isaac Newton

Described by Isaac Newton Described by Isaac Newton States observed relationships between motion and forces 3 statements cover aspects of motion for single objects and for objects interacting with another object An object at rest

More information

QUESTION BANK UNIT-6 CHAPTER-8 GRAVITATION

QUESTION BANK UNIT-6 CHAPTER-8 GRAVITATION QUESTION BANK UNIT-6 CHAPTER-8 GRAVITATION I. One mark Questions: 1. State Kepler s law of orbits. 2. State Kepler s law of areas. 3. State Kepler s law of periods. 4. Which physical quantity is conserved

More information

Chapter 6 Work and Energy

Chapter 6 Work and Energy Chapter 6 WORK AND ENERGY PREVIEW Work is the scalar product of the force acting on an object and the displacement through which it acts. When work is done on or by a system, the energy of that system

More information

There are three different properties associated with the mass of an object:

There are three different properties associated with the mass of an object: Mechanics Notes II Forces, Inertia and Motion The mathematics of calculus, which enables us to work with instantaneous rates of change, provides a language to describe motion. Our perception of force is

More information

THE EQUATIONS OF PLANETARY MOTION AND THEIR SOLUTION. By: Kyriacos Papadatos ABSTRACT

THE EQUATIONS OF PLANETARY MOTION AND THEIR SOLUTION. By: Kyriacos Papadatos ABSTRACT THE EQUATIONS OF PLANETARY MOTION AND THEIR SOLUTION By: Kyriacos Papadatos ABSTRACT Newton's iginal wk on the they of gravitation presented in the Principia, even in its best translation, is difficult

More information

Newton s Laws of Motion

Newton s Laws of Motion Chapter 4 Newton s Laws of Motion PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 7_8_2016 Goals for Chapter 4

More information

Physics Midterm Review Packet January 2010

Physics Midterm Review Packet January 2010 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:

More information

and from (14.23), we find the differential cross section db do R cos (0 / 2) R sin(0/2) R 2

and from (14.23), we find the differential cross section db do R cos (0 / 2) R sin(0/2) R 2 574 Chapter 14 Collision Theory Figure 14.10 A point projectile bouncing off a fixed rigid sphere obeys the law of reflection, that the two adjacent angles labelled a are equal. The impact parameter is

More information

Lecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014

Lecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014 Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,

More information

Chapter 5. Determining Masses of Astronomical Objects

Chapter 5. Determining Masses of Astronomical Objects Chapter 5. Determining Masses of Astronomical Objects One of the most fundamental and important properties of an object is its mass. On Earth we can easily weigh objects, essentially measuring how much

More information

Kepler s Laws, Newton s Laws, and the Search for New Planets

Kepler s Laws, Newton s Laws, and the Search for New Planets Integre Technical Publishing Co., Inc. American Mathematical Monthly 108:9 July 12, 2001 2:22 p.m. osserman.tex page 813 Kepler s Laws, Newton s Laws, and the Search for New Planets Robert Osserman Introduction.

More information

Gravitation. Physics 1425 Lecture 11. Michael Fowler, UVa

Gravitation. Physics 1425 Lecture 11. Michael Fowler, UVa Gravitation Physics 1425 Lecture 11 Michael Fowler, UVa The Inverse Square Law Newton s idea: the centripetal force keeping the Moon circling the Earth is the same gravitational force that pulls us to

More information

PHYSICS 111 HOMEWORK SOLUTION, week 4, chapter 5, sec 1-7. February 13, 2013

PHYSICS 111 HOMEWORK SOLUTION, week 4, chapter 5, sec 1-7. February 13, 2013 PHYSICS 111 HOMEWORK SOLUTION, week 4, chapter 5, sec 1-7 February 13, 2013 0.1 A 2.00-kg object undergoes an acceleration given by a = (6.00î + 4.00ĵ)m/s 2 a) Find the resultatnt force acting on the object

More information

Newton s Laws. Newton s Imaginary Cannon. Michael Fowler Physics 142E Lec 6 Jan 22, 2009

Newton s Laws. Newton s Imaginary Cannon. Michael Fowler Physics 142E Lec 6 Jan 22, 2009 Newton s Laws Michael Fowler Physics 142E Lec 6 Jan 22, 2009 Newton s Imaginary Cannon Newton was familiar with Galileo s analysis of projectile motion, and decided to take it one step further. He imagined

More information

Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION

Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION 1 P a g e Inertia Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION The property of an object by virtue of which it cannot change its state of rest or of uniform motion along a straight line its own, is

More information

Orbits of the Lennard-Jones Potential

Orbits of the Lennard-Jones Potential Orbits of the Lennard-Jones Potential Prashanth S. Venkataram July 28, 2012 1 Introduction The Lennard-Jones potential describes weak interactions between neutral atoms and molecules. Unlike the potentials

More information