Gravitation Kepler s Laws
|
|
- Eileen Newman
- 5 years ago
- Views:
Transcription
1 Gravitation Kepler s Laws Lana heridan De Anza College Dec 7, 2017
2 Last time one more tatic Equilibrium example Newton s Law of Universal Gravitation
3 Overview gravitational field escape speed Kepler s Laws orbits
4 Acceleration due to Gravity For an object of mass m near the surface of the Earth: and where F g = mg g = GM E R 2 E M E = kg is the mass of the Earth and R E = m is the radius of the Earth. The force F g acts downwards towards the center of the Earth.
5 Acceleration due to Gravity The acceleration due to gravity, g, can vary with height! F G = GM ( ) Em GME r 2 = m r 2 = mg Depends on r the distance from the center of the Earth. uppose an object is at height h above the surface of the Earth, then: g decreases as h increases. g = GM E (R E + h) 2 g is the not just the acceleration due to gravity, but also the magnitude of the gravitational field.
6 Fields field A field is any kind of physical quantity that has values specified at every point in space and time.
7 Fields field A field is any kind of physical quantity that has values specified at every point in space and time. Fields were first introduced as a calculation tool. A force-field can be used to identify the force a particular particle will feel at a certain point in space and time based on the other objects in its environment that it will interact with. We do not need a description of the sources of the field to describe what their effect is on our particle. Gravitational force: Electrostatic force: F G = m( GMˆr r 2 ) = mg F E = qe
8 Fields Gravitational force: Electrostatic force: F G = m( GMˆr r 2 ) = mg F E = qe Gravitational field: g = F G m Electric field: E = F E q The field tells us what force a test particle of mass m (in the gravitational case) or charge q (in the electrostatic case) would feel at that point in space and time.
9 e perpendicular to the electric field lines passing through them. t the end of ection 25.2, the equipotential surfaces associated ectric Examples field consist of of Fields a family of planes perpendicular to the 25.11a shows some representative equipotential surfaces for this ld produced f charge Fields are drawn with lines showing the direction of force that a test particle will feel at that point. The density of the lines at that point in the diagram indicates the approximate magnitude of the A spherically symmetric electric field produced by a point charge force at that point. An electric field produced by an electric dipole tential surfaces (the dashed blue lines are intersections of these surfaces with the page) and elecq E b c
10 Examples of Fields The gravitational field caused by the un-earth system looks something like: 1 Figure from
11 nce of the field and measure oting Gravitational the force exerted Field of on the it. Earth ject (in this case, the Earth) rce that would be present if a a b Near the surface of the Earth: le in a field analysis model. in an area of space in which d a property of the particle, l version of the particle in a ational, and the property of mass m. The mathematical icle in a field model is Equa- (5.5) e particle in a field model. In lts in a force is electric charge: ure 13.4 (a) The gravitational were placed in the field. The magnitude of the field vector at any location is the magnitude of the free-fall acceleration at that location. Farther out from the Earth: a
12 Gravitational Field of the Earth a Uniform g: e particle establishes a gravitad by measuring the force on a e a particle of mass m is placed at it experiences a gravitational b (5.5) gure 13.4 (a) The gravitational s ld theory vectors of gravitation in the in vicinity of a g m F g mg continued A test mass m experiences a force F g = mg, where g is the field vector.
13 Gravitational Potential (Not gravitational potential energy!) We can define a new quantity gravitational potential. Usually written Φ or V. The change in gravitational potential is equal to the integral of the field along a path (with a minus sign). Φ = g ds (1) Notice: this is very similar to what we had for the relation between force and potential energy: U = F ds (2) In fact, eq (2) = m eq (1)
14 Gravitational Potential The gravitation potential can help us figure out what the gravitational potential energy will be if we put a particle of mass m at a particular point where the gravitational potential is Φ. U = mφ
15 Gravitational Potential Φ = g ds and so, the radial component of g can be found by: g r = dφ dr For the gravitational field around a point-like mass M, g = GM r 2 ˆr, Φ = GM r F = GMm r 2 ˆr, U = GMm r
16 Gravitational Potential 1 Figure from
17 Gravitational Potential A uniform field, as near the surface of the Earth. (a) OTENTIAL + (b) Equipotential surface Field line The blue lines represent the gravitational field. The orange dashed lines are surfaces of equal gravitational potential.
18 Gravitational Potential Energy (13.14) center of the Earth particles inside the U is always negative th system, a similar is, the gravitational es m 1 and m 2 sepa- (13.15) for any pair of par- 1/r 2. Furthermore, and we have chosen finite. Because the do positive work to y the external agent Gravitational potential energy of the Earth particle U(r) = Gm system 1m 2 r O GM E m R E U M E Earth The potential energy goes to zero as r approaches infinity. R E Figure Graph of the grav- r
19 Escape peed which is the energy equivalent of 89 gal of gasoline. NAA engi craft as it ejects burned fuel, something we have not done her effect of this changing mass to yield a greater or a lesser amoun How fast does an object need to be projected with to escape Earth s gravity? Escape peed R E v f v i m 0 h r max uppose an object of mass m i with an initial speed v i as illust to find the value of the initial tance away from the center of system for any configuration. the Earth, v 5 v i and r 5 r i 5 R v f 5 0 and r 5 r f 5 r max. Beca these values into the isolated- olving for v i 2 gives 1 2 M E Figure An object of mass m projected upward from For a given maximum altitud required initial speed. We are now in a position t speed the object must have a
20 Escape peed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy.
21 Escape peed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy. K i + U i = U f
22 Escape peed The object begins with kinetic energy K = 1 2 mv i 2 energy U = GM E m R E. and potential To escape Earth s gravity well the object needs to reach U = 0. Trade kinetic for potential energy. K i + U i = U f 1 2 mv i 2 GM E m R E = mv i 2 = GM E m v i = R E 2GM E R E The initial velocity of the object must be at least this large.
23 Example Escape peed Calculate the escape speed from the Earth for a 5000 kg spacecraft and determine the kinetic energy it must have at the Earth s surface to move infinitely far away from the Earth.
24 Example Escape peed Calculate the escape speed from the Earth for a 5000 kg spacecraft and determine the kinetic energy it must have at the Earth s surface to move infinitely far away from the Earth. v = = 2GM E R E 2( )( ) = m s 1
25 Example Escape peed Calculate the escape speed from the Earth for a 5000 kg spacecraft and determine the kinetic energy it must have at the Earth s surface to move infinitely far away from the Earth. v = = 2GM E R E 2( )( ) = m s 1 K = 1 2 mv 2 = 1 2 (5000)( ) 2 = J
26 Example Escape peed OR Energy conservation U + K = 0 K i = U f U i K = 0 + GM E m R E = ( )( )(5000) = J
27 Motion of the Planets The planets in our solar system orbit the un. (As planets in other systems orbit their stars.) This is called a heliocentric model.
28 Motion of the Planets The planets in our solar system orbit the un. (As planets in other systems orbit their stars.) This is called a heliocentric model. Nicolaus Copernicus (early 1500s A.D.) is credited with the paradigm since he developed a mathematical model and took seriously the idea that the implication was that the Earth moved around the un, but others had similar thoughts: Aristarchus of amos (c. 270 BCE) Martianus Capella (400s A.D.) Aryabhata (500s A.D.), Nilakantha omayaji (1500s A.D.) Najm al-dīn al-qazwīnī al-kātibī (1200s A.D.)
29 Motion of the Planets After Copernicus s proposal, Tycho Brahe gathered a lot of data about the positions of stars and planets. Johannes Kepler inherited Brahe s data and did the calculations to deduce a complete model. Galileo gathered additional data that supported the heliocentric model and popularized it.
30 Kepler s Laws Kepler s Three laws give simple rules for predicting stable planetary orbits. 1. All planets move in elliptical orbits with the un at one focus. 2. The radius vector drawn from the un to a planet sweeps out equal areas in equal time intervals. 3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.
31 is Kepler s the semicenter of the First Law: axis has Elliptical length b. Orbits Defining an ellipse: un, the un y the general re zero. The ection com-, c increases correspond or an ellipse length a, and the semiminor r 1 r 2 F 1 c a F 2 b x eccentricity er hand, the Figure 13.6a cury s orbit. rom a circle, a is the semimajor Each axis focus is located a 2 at = a b 2 + c 2 b is the semiminordistance axis c from the center. c is the distance from the center of the ellipse to the focus e is the eccentricity of the ellipse e = c/a Figure 13.5 Plot of an ellipse.
32 Kepler s First Law: Elliptical Orbits All planets move in elliptical orbits with the un at one focus. er 13 Universal Gravitation pe of h has ) the e of the shape comet ger The un is located at a focus of the ellipse. There is nothing physical located at the center (the black dot) or the other focus (the blue dot). un Orbit of Comet Halley un Center Orbit of Mercury Comet Halley Center a b The planets orbits are close to circular. (Mercury s is the between the planet and the un is a 1 c. At this point, called the aphelion, the planet is at its maximum distance from the un. (For an object in orbit around the least circular.) Earth, this point is called the apogee.) Conversely, when the planet is at the right end of the Halley s ellipse, the Comet distance hasbetween an orbit the planet with and a high the un eccentricity. is a 2 c. At this point, called the perihelion (for an Earth orbit, the perigee), the planet is at its minimum distance from the un. Kepler s first law is a direct result of the inverse-square nature of the gravita-
33 tional force. Circular M p and elliptical the gravitational force center. Thes K that move repeatedly around the un F are also unbound objects, g v a such as el The radius vector drawn from the the unmun to a once planetand sweeps then out never return to equal areas in equal time intervals. these objects also varies as the inve ta allowed paths for these objects inclu What does it mean? ra a th Kepler s econd Law: Equal Areas in Equal Time M p Kepler s econd Law Kepler s second law can d r be v dt shown a r un F g v angular un momentum. Consider a p elliptical orbit (Fig. 13.7a). Let s con M to be so much more da massive than th tational force exerted by the un Eo radius vector, directed toward the a this central force about an axis thro The area swept out by r in When the planet is closer to the un, it Therefore, must be moving because faster. the external a time interval dt is half the d r v dt an isolated area of system the parallelogram. for angular mom va r un planet is a constant of the motion: th K
34 un angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consi to be so much more massive than the tational beige area, force da? exerted by the un on radius vector, directed toward the un this central da force = 1 about r dr an axis throu 2 Therefore, because the external t an isolated system = 1 r for vangular dt 2 mome planet is a constant of the motion: Kepler s econd Law: Equal Areas in Equal Time M a un b un M a r F g da d r dt Figure 13.7 (a) The gravitational force acting on a planet The area swept out by r in a time interval dt is half the area of the parallelogram. un F g M p da v v d r dt r v v are also unbound objects, such as a met the un once and then never return. T these objects also varies as the inverse allowed paths for these objects include Kepler s econd Law Kepler s second law can be shown to angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consid to be so much more massive than the p tational force exerted by the un on t radius vector, directed toward the un this central force about an axis throug Therefore, because the external to an isolated system for angular mome planet is a constant of the motion: Evaluating L for the planet, D L 5 0 L 5 r 3 p 5 M p r We can relate this result to the follo val dt, the radius vector r in Figure 13 the area 0 r 3 d r 0 of the parallelogr the displacement of the planet in the Evaluating L for the planet, D L 5 0 da r 3 d r r 3 p 5 M p r L
35 un angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consi to be so much more massive than the tational beige area, force da? exerted by the un on radius vector, directed toward the un this central da force = 1 about r dr an axis throu 2 Therefore, because the external t an isolated system = 1 r for vangular dt 2 mome planet is a constant of the motion: Kepler s econd Law: Equal Areas in Equal Time M a un b un M a r F g da d r dt Figure 13.7 (a) The gravitational force acting on a planet The area swept out by r in a time interval dt is half the area of the parallelogram. un F g M p da v v d r dt r v v are also unbound objects, such as a met the un once and then never return. T these objects also varies as the inverse allowed paths for these objects include Kepler s econd Law Kepler s second law can be shown to angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consid to be so much more massive than the p tational force exerted by the un on t radius vector, directed toward the un this central force about an axis throug Therefore, because the external to an isolated system for angular mome planet is a constant of the motion: D L 5 0 r F g τ Evaluating ext = 0 L = const. L for the planet, L = r p L 5 r 3 p 5 M p r L = M p r v We can relate this result to the follo val dt, the radius vector r in Figure 13 the area 0 r 3 d r 0 of the parallelogr the displacement of the planet in the Evaluating L for the planet, D L 5 0 da r 3 d r r 3 p 5 M p r L
36 un angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consi to be so much more massive than the tational beige area, force da? exerted by the un on radius vector, directed toward the un this central da force = 1 about r dr an axis throu 2 Therefore, because the external t an isolated system = 1 r for vangular dt 2 mome planet is a constant of the motion: Kepler s econd Law: Equal Areas in Equal Time M a un b un M a r F g da d r dt Figure 13.7 (a) The gravitational force acting on a planet The area swept out by r in a time interval dt is half the area of the parallelogram. un F g M p da v v d r dt r v v are also unbound objects, such as a met the un once and then never return. T these objects also varies as the inverse allowed paths for these objects include Kepler s econd Law Kepler s second law can be shown to angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consid to be so much more massive than the p tational force exerted by the un on t radius vector, directed toward the un this central force about an axis throug Therefore, because the external to an isolated system for angular mome planet is a constant of the motion: D L 5 0 r F g τ Evaluating ext = 0 L = const. L for the planet, L = r p L 5 r 3 p 5 M p r L = M p r v We can relate this result to the follo val dt, the radius vector r in Figure 13 the area 0 da r 3 d r 0 of the parallelogr the displacement dt = L of 2M the p planet in the Evaluating L for the planet, D L 5 0 da r 3 d r r 3 p 5 M p r L
37 un angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consi to be so much more massive than the tational beige area, force da? exerted by the un on radius vector, directed toward the un this central da force = 1 about r dr an axis throu 2 Therefore, because the external t an isolated system = 1 r for vangular dt 2 mome planet is a constant of the motion: Kepler s econd Law: Equal Areas in Equal Time M a un b un M a r F g da d r dt Figure 13.7 (a) The gravitational force acting on a planet The area swept out by r in a time interval dt is half the area of the parallelogram. un F g M p da v v d r dt r v v are also unbound objects, such as a met the un once and then never return. T these objects also varies as the inverse allowed paths for these objects include Kepler s econd Law Kepler s second law can be shown to angular momentum. Consider a plan elliptical orbit (Fig. 13.7a). Let s consid to be so much more massive than the p tational force exerted by the un on t radius vector, directed toward the un this central force about an axis throug Therefore, because the external to an isolated system for angular mome planet is a constant of the motion: D L 5 0 r F g τ Evaluating ext = 0 L = const. L for the planet, L = r p L 5 r 3 p 5 M p r L = M p r v We can relate this result to the follo val dt, the radius vector r in Figure 13 the area 0 da r 3 d r 0 of the parallelogr the displacement dt = L of 2M the p planet in the da dt = constant! da r 3 d r 0 5 Evaluating L for the planet, L D L r 3 p 5 M p r
38 Kepler s Third Law: T 2 a 3 The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit. We will only prove this for circular orbits. For circular motion: v = 2πr T. F G = F C GM s M p r 2 = M pv 2 r GM s M p r 2 = M p r ( 4π T 2 2 = ( T 2πr GM s ) 2 ) r 3
39 Kepler s Third Law: T 2 a 3 A full derivation for the elliptical orbit case gives the same expression, but with r replaced with a: ( 4π T 2 2 = GM s ) a 3 ometimes the constant is given a name: K s = 4π2 GM s = s 2 m 3
40 Energy of Orbits For a stable orbit E mech = 0. E mech = K + U = constant
41 Energy of Orbits For a stable orbit E mech = 0. E mech = K + U = constant We can use the expression for gravitational potential: E mech = 1 2 mv 2 GMm r = constant
42 Energy of Orbits For a stable orbit E mech = 0. E mech = K + U = constant We can use the expression for gravitational potential: E mech = 1 2 mv 2 GMm r = constant What is the constant? How can we relate v to r?
43 Energy of Orbits For a stable orbit E mech = 0. E mech = K + U = constant We can use the expression for gravitational potential: E mech = 1 2 mv 2 GMm r = constant What is the constant? How can we relate v to r? Using F c = F G. (Assuming circular orbit, and M >> m.)
44 Energy of Orbits F G = F C GMm r 2 = mv 2 r mv 2 = GMm r o we can write the total mechanical energy of the orbit at a radius r as: E mech = 1 2 mv 2 GMm r = GMm 2r E = GMm 2r GMm r
45 Energy of Orbits For elliptical orbits where a is the semimajor axis. E = GM E m 2a
46 ummary gravitational field escape speed Kepler s Laws orbits (Uncollected) Homework erway & Jewett, Ch 13, onward from page 410. Probs: 19, 27, 35, 37, 39
Gravitation Kepler s Laws
Gravitation Kepler s Laws Lana heridan De Anza College Mar 15, 2015 Overview Newton s Law of Universal Gravitation Gravitational field Kepler s Laws Gravitation The force that massive objects exert on
More informationMechanics Gravity. Lana Sheridan. Nov 29, De Anza College
Mechanics Gravity Lana Sheridan De Anza College Nov 29, 2018 Last time angular momentum of rigid objects conservation of angular momentum examples Overview Newton s law of gravitation gravitational field
More informationConceptual Physics Projectiles Motion of Planets
Conceptual Physics Projectiles Motion of Planets Lana Sheridan De Anza College July 13, 2015 Last time angular momentum gravity gravitational field black holes Overview projectile motion orbital motion
More informationConceptual Physics Projectiles Motion of Planets
Conceptual Physics Projectiles Motion of Planets Lana Sheridan De Anza College July 19, 2017 Last time angular momentum gravity gravitational field black holes Overview projectile motion orbital motion
More informationChapter 13. Universal Gravitation
Chapter 13 Universal Gravitation Planetary Motion A large amount of data had been collected by 1687. There was no clear understanding of the forces related to these motions. Isaac Newton provided the answer.
More informationUniversal Gravitation
Universal Gravitation Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely
More informationChapter 13. Gravitation
Chapter 13 Gravitation e = c/a A note about eccentricity For a circle c = 0 à e = 0 a Orbit Examples Mercury has the highest eccentricity of any planet (a) e Mercury = 0.21 Halley s comet has an orbit
More informationChapter 13. Gravitation
Chapter 13 Gravitation 13.2 Newton s Law of Gravitation 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 x10 11 Nm 2 /kg 2
More informationStatic Equilibrium Gravitation
Static Equilibrium Gravitation Lana Sheridan De Anza College Dec 6, 2017 Overview One more static equilibrium example Newton s Law of Universal Gravitation gravitational potential energy little g Example
More informationPhysics 12. Unit 5 Circular Motion and Gravitation Part 2
Physics 12 Unit 5 Circular Motion and Gravitation Part 2 1. Newton s law of gravitation We have seen in Physics 11 that the force acting on an object due to gravity is given by a well known formula: F
More informationDynamics Laws of Motion More About Forces
Dynamics Laws of Motion More About Forces Lana heridan De Anza College Oct 10, 2017 Overview Newton s first and second laws Warm Up: Newton s econd Law Implications Question. If an object is not accelerating,
More informationChapter 12 Gravity. Copyright 2010 Pearson Education, Inc.
Chapter 12 Gravity Units of Chapter 12 Newton s Law of Universal Gravitation Gravitational Attraction of Spherical Bodies Kepler s Laws of Orbital Motion Gravitational Potential Energy Energy Conservation
More informationObservational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws
Observational Astronomy - Lecture 4 Orbits, Motions, Kepler s and Newton s Laws Craig Lage New York University - Department of Physics craig.lage@nyu.edu February 24, 2014 1 / 21 Tycho Brahe s Equatorial
More informationVISUAL PHYSICS ONLINE
VISUAL PHYSICS ONLINE EXCEL SIMULATION MOTION OF SATELLITES DOWNLOAD the MS EXCEL program PA50satellite.xlsx and view the worksheet Display as shown in the figure below. One of the most important questions
More informationPHYSICS. Chapter 13 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 13 Lecture RANDALL D. KNIGHT Chapter 13 Newton s Theory of Gravity IN THIS CHAPTER, you will learn to understand the motion of satellites
More informationLecture 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 informationThe Heliocentric Model of Copernicus
Celestial Mechanics The Heliocentric Model of Copernicus Sun at the center and planets (including Earth) orbiting along circles. inferior planets - planets closer to Sun than Earth - Mercury, Venus superior
More informationOutline for Today: Newton s Law of Universal Gravitation The Gravitational Field Orbital Motion Gravitational Potential Energy. Hello!
PHY131H1F - Class 13 Outline for Today: Newton s Law of Universal Gravitation The Gravitational Field Orbital Motion Gravitational Potential Energy Under the Flower of Kent apple tree in the Woolsthorpe
More informationA SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE
DOING PHYSICS WITH MATLAB A SIMULATION OF THE MOTION OF AN EARTH BOUND SATELLITE Download Directory: Matlab mscripts mec_satellite_gui.m The [2D] motion of a satellite around the Earth is computed from
More informationGravitation. Luis Anchordoqui
Gravitation Kepler's law and Newton's Synthesis The nighttime sky with its myriad stars and shinning planets has always fascinated people on Earth. Towards the end of the XVI century the astronomer Tycho
More informationKepler, Newton, and laws of motion
Kepler, Newton, and laws of motion First: A Little History Geocentric vs. heliocentric model for solar system (sec. 2.2-2.4)! The only history in this course is this progression: Aristotle (~350 BC) Ptolemy
More informationChapter 8 - Gravity Tuesday, March 24 th
Chapter 8 - Gravity Tuesday, March 24 th Newton s law of gravitation Gravitational potential energy Escape velocity Kepler s laws Demonstration, iclicker and example problems We are jumping backwards to
More informationGravitation and the Motion of the Planets
Gravitation and the Motion of the Planets 1 Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets go around
More informationWelcome back to Physics 211. Physics 211 Spring 2014 Lecture Gravity
Welcome back to Physics 211 Today s agenda: Newtonian gravity Planetary orbits Gravitational Potential Energy Physics 211 Spring 2014 Lecture 14-1 1 Gravity Before 1687, large amount of data collected
More informationChapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity. Copyright 2009 Pearson Education, Inc.
Chapter 4 Making Sense of the Universe: Understanding Motion, Energy, and Gravity How do we describe motion? Precise definitions to describe motion: Speed: Rate at which object moves speed = distance time
More informationCircular Motion. Gravitation
Circular Motion Gravitation Circular Motion Uniform circular motion is motion in a circle at constant speed. Centripetal force is the force that keeps an object moving in a circle. Centripetal acceleration,
More informationWelcome back to Physics 215
Welcome back to Physics 215 Today s agenda: Gravity 15-2 1 Current assignments HW#15 due Monday, 12/12 Final Exam, Thursday, Dec. 15 th, 3-5pm in 104N. Two sheets of handwritten notes and a calculator
More informationWelcome back to Physics 215
Welcome back to Physics 215 Today s agenda: More rolling without slipping Newtonian gravity Planetary orbits Gravitational Potential Energy Physics 215 Spring 2018 Lecture 13-1 1 Rolling without slipping
More informationGravitation and the Waltz of the Planets
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationGravitation and the Waltz of the Planets. Chapter Four
Gravitation and the Waltz of the Planets Chapter Four Guiding Questions 1. How did ancient astronomers explain the motions of the planets? 2. Why did Copernicus think that the Earth and the other planets
More informationNewton s Gravitational Law
1 Newton s Gravitational Law Gravity exists because bodies have masses. Newton s Gravitational Law states that the force of attraction between two point masses is directly proportional to the product of
More informationCopyright 2010 Pearson Education, Inc. GRAVITY. Chapter 12
GRAVITY Chapter 12 Units of Chapter 12 Newton s Law of Universal Gravitation Gravitational Attraction of Spherical Bodies Kepler s Laws of Orbital Motion Gravitational Potential Energy Energy Conservation
More informationNewton s Laws of Motion and Gravity ASTR 2110 Sarazin. Space Shuttle
Newton s Laws of Motion and Gravity ASTR 2110 Sarazin Space Shuttle Discussion Session This Week Friday, September 8, 3-4 pm Shorter Discussion Session (end 3:40), followed by: Intro to Astronomy Department
More informationLesson 9. Luis Anchordoqui. Physics 168. Tuesday, October 24, 17
Lesson 9 Physics 168 1 Static Equilibrium 2 Conditions for Equilibrium An object with forces acting on it but that is not moving is said to be in equilibrium 3 Conditions for Equilibrium (cont d) First
More informationKEPLER S LAWS OF PLANETARY MOTION
KEPLER S LAWS OF PLANETARY MOTION In the early 1600s, Johannes Kepler culminated his analysis of the extensive data taken by Tycho Brahe and published his three laws of planetary motion, which we know
More informationChapter 9. Gravitation
Chapter 9 Gravitation 9.1 The Gravitational Force For two particles that have masses m 1 and m 2 and are separated by a distance r, the force has a magnitude given by the same magnitude of force acts on
More informationOutline for Today: Newton s Law of Universal Gravitation The Gravitational Field Orbital Motion Gravitational Potential Energy
PHY131H1F - Class 13 Outline for Today: Newton s Law of Universal Gravitation The Gravitational Field Orbital Motion Gravitational Potential Energy Under the Flower of Kent apple tree in the Woolsthorpe
More informationPHYS 155 Introductory Astronomy
PHYS 155 Introductory Astronomy - observing sessions: Sunday Thursday, 9pm, weather permitting http://www.phys.uconn.edu/observatory - Exam - Tuesday March 20, - Review Monday 6:30-9pm, PB 38 Marek Krasnansky
More informationGravitation. chapter 9
chapter 9 Gravitation Circular orbits (Section 9.3) 1, 2, and 3 are simple exercises to supplement the quantitative calculations of Examples 4, 5, and 6 in Section 9.3. 1. Satellite near Earth s surface
More informationLecture Outline. Chapter 13 Gravity Pearson Education, Inc. Slide 13-1
Lecture Outline Chapter 13 Gravity Slide 13-1 The plan Lab this week: exam problems will put problems on mastering for chapters without HW; will also go over exam 2 Final coverage: now posted; some sections/chapters
More information6. Summarize Newton s Law of gravity and the inverse square concept. Write out the equation
HW due Today. 1. Read p. 175 180. 2. Summarize the historical account of Brahe and Kepler 3. Write out Kepler s 3 laws. 1) Planets in orbit follow an elliptical path, the Sun is located at a focus of the
More informationPhysics Lecture 03: FRI 29 AUG
Physics 23 Jonathan Dowling Isaac Newton (642 727) Physics 23 Lecture 03: FRI 29 AUG CH3: Gravitation III Version: 8/28/4 Michael Faraday (79 867) 3.7: Planets and Satellites: Kepler s st Law. THE LAW
More informationGat ew ay T o S pace AS EN / AS TR Class # 19. Colorado S pace Grant Consortium
Gat ew ay T o S pace AS EN / AS TR 2500 Class # 19 Colorado S pace Grant Consortium Announcements: - Launch Readiness Review Cards - 11 days to launch Announcements: - Launch Readiness Review Cards - 11
More informationA = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great Pearson Education, Inc.
Q13.1 The mass of the Moon is 1/81 of the mass of the Earth. Compared to the gravitational force that the Earth exerts on the Moon, the gravitational force that the Moon exerts on the Earth is A. 81 2
More informationPlanetary Mechanics:
Planetary Mechanics: Satellites A satellite is an object or a body that revolves around another body due to the gravitational attraction to the greater mass. Ex: The planets are natural satellites of the
More informationChapter 9 Lecture. Pearson Physics. Gravity and Circular Motion. Prepared by Chris Chiaverina Pearson Education, Inc.
Chapter 9 Lecture Pearson Physics Gravity and Circular Motion Prepared by Chris Chiaverina Chapter Contents Newton's Law of Universal Gravity Applications of Gravity Circular Motion Planetary Motion and
More informationASTRO 1050 LAB #3: Planetary Orbits and Kepler s Laws
ASTRO 1050 LAB #3: Planetary Orbits and Kepler s Laws ABSTRACT Johannes Kepler (1571-1630), a German mathematician and astronomer, was a man on a quest to discover order and harmony in the solar system.
More informationGravity. Newton s Law of Gravitation Kepler s Laws of Planetary Motion Gravitational Fields
Gravity Newton s Law of Gravitation Kepler s Laws of Planetary Motion Gravitational Fields Simulation Synchronous Rotation https://www.youtube.com/watch?v=ozib_l eg75q Sun-Earth-Moon System https://vimeo.com/16015937
More information(b) The period T and the angular frequency ω of uniform rotation are related to the cyclic frequency f as. , ω = 2πf =
PHY 302 K. Solutions for problem set #9. Non-textbook problem #1: (a) Rotation frequency of 1 Hz means one revolution per second, or 60 revolutions per minute (RPM). The pre-lp vinyl disks rotated at 78
More information5.1. Accelerated Coordinate Systems:
5.1. Accelerated Coordinate Systems: Recall: Uniformly moving reference frames (e.g. those considered at 'rest' or moving with constant velocity in a straight line) are called inertial reference frames.
More informationCH 8. Universal Gravitation Planetary and Satellite Motion
CH 8 Universal Gravitation Planetary and Satellite Motion Sir Isaac Newton UNIVERSAL GRAVITATION Newton: Universal Gravitation Newton concluded that earthly objects and heavenly objects obey the same physical
More informationChapter 13 Gravity Pearson Education, Inc. Slide 13-1
Chapter 13 Gravity Slide 13-1 The plan Lab this week: there will be time for exam problems Final exam: sections posted today; some left out Final format: all multiple choice, almost all short problems,
More informationAP Physics Multiple Choice Practice Gravitation
AP Physics Multiple Choice Practice Gravitation 1. Each of five satellites makes a circular orbit about an object that is much more massive than any of the satellites. The mass and orbital radius of each
More informationChapter 13: universal gravitation
Chapter 13: universal gravitation Newton s Law of Gravitation Weight Gravitational Potential Energy The Motion of Satellites Kepler s Laws and the Motion of Planets Spherical Mass Distributions Apparent
More informationF = ma. G mm r 2. S center
In the early 17 th century, Kepler discovered the following three laws of planetary motion: 1. The planets orbit around the sun in an ellipse with the sun at one focus. 2. As the planets orbit around the
More informationMidterm 3 Thursday April 13th
Welcome back to Physics 215 Today s agenda: rolling friction & review Newtonian gravity Planetary orbits Gravitational Potential Energy Physics 215 Spring 2017 Lecture 13-1 1 Midterm 3 Thursday April 13th
More informationPhysics 115/242 The Kepler Problem
Physics 115/242 The Kepler Problem Peter Young (Dated: April 21, 23) I. INTRODUCTION We consider motion of a planet around the sun, the Kepler problem, see e.g. Garcia, Numerical Methods for Physics, Sec.
More informationHW Chapter 5 Q 7,8,18,21 P 4,6,8. Chapter 5. The Law of Universal Gravitation Gravity
HW Chapter 5 Q 7,8,18,21 P 4,6,8 Chapter 5 The Law of Universal Gravitation Gravity Newton s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force that
More informationPHYSICS 231 INTRODUCTORY PHYSICS I
PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 11 Last Lecture Angular velocity, acceleration " = #$ #t = $ f %$ i t f % t i! = " f # " i t!" #!x $ 0 # v 0 Rotational/ Linear analogy "s = r"# v t = r" $ f
More informationVISUAL PHYSICS ONLINE
VISUAL PHYSICS ONLINE PRACTICAL ACTIVITY HOW DO THE PANETS MOVE? One of the most important questions historically in Physics was how the planets move. Many historians consider the field of Physics to date
More informationChapter 8. Orbits. 8.1 Conics
Chapter 8 Orbits 8.1 Conics Conic sections first studied in the abstract by the Greeks are the curves formed by the intersection of a plane with a cone. Ignoring degenerate cases (such as a point, or pairs
More informationUniversal gravitation
Universal gravitation Physics 211 Syracuse University, Physics 211 Spring 2015 Walter Freeman February 22, 2017 W. Freeman Universal gravitation February 22, 2017 1 / 14 Announcements Extra homework help
More information18. Kepler as a young man became the assistant to A) Nicolaus Copernicus. B) Ptolemy. C) Tycho Brahe. D) Sir Isaac Newton.
Name: Date: 1. The word planet is derived from a Greek term meaning A) bright nighttime object. B) astrological sign. C) wanderer. D) nontwinkling star. 2. The planets that were known before the telescope
More informationLecture 1a: Satellite Orbits
Lecture 1a: Satellite Orbits Meteorological Satellite Orbits LEO view GEO view Two main orbits of Met Satellites: 1) Geostationary Orbit (GEO) 1) Low Earth Orbit (LEO) or polar orbits Orbits of meteorological
More informationL03: Kepler problem & Hamiltonian dynamics
L03: Kepler problem & Hamiltonian dynamics 18.354 Ptolemy circa.85 (Egypt) -165 (Alexandria) Greek geocentric view of the universe Tycho Brahe 1546 (Denmark) - 1601 (Prague) "geo-heliocentric" system last
More informationChapter 14 Satellite Motion
1 Academic Physics Mechanics Chapter 14 Satellite Motion The Mechanical Universe Kepler's Three Laws (Episode 21) The Kepler Problem (Episode 22) Energy and Eccentricity (Episode 23) Navigating in Space
More informationGeneral Physics I. Lecture 7: The Law of Gravity. Prof. WAN, Xin 万歆.
General Physics I Lecture 7: The Law of Gravity Prof. WAN, Xin 万歆 xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Outline Newton's law of universal gravitation Motion of the planets; Kepler's laws Measuring
More informationChapter 12 Gravity. Copyright 2010 Pearson Education, Inc.
Chapter 12 Gravity Units of Chapter 12 Newton s Law of Universal Gravitation Gravitational Attraction of Spherical Bodies Kepler s Laws of Orbital Motion Gravitational Potential Energy Energy Conservation
More informationLecture 22: Gravitational Orbits
Lecture : Gravitational Orbits Astronomers were observing the motion of planets long before Newton s time Some even developed heliocentric models, in which the planets moved around the sun Analysis of
More information1. Which of the following correctly lists our cosmic address from small to large?
1. Which of the following correctly lists our cosmic address from small to large? (a) Earth, solar system, Milky Way Galaxy, Local Group, Local Super Cluster, universe (b) Earth, solar system, Milky Way
More informationAP Physics C - Mechanics
Slide 1 / 78 Slide 2 / 78 AP Physics C - Mechanics Universal Gravitation 2015-12-04 www.njctl.org Table of Contents Slide 3 / 78 Click on the topic to go to that section Newton's Law of Universal Gravitation
More informationJohannes Kepler ( ) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it!
Johannes Kepler (1571-1630) German Mathematician and Astronomer Passionately convinced of the rightness of the Copernican view. Set out to prove it! Kepler s Life Work Kepler sought a unifying principle
More informationIn so many and such important. ways, then, do the planets bear witness to the earth's mobility. Nicholas Copernicus
In so many and such important ways, then, do the planets bear witness to the earth's mobility Nicholas Copernicus What We Will Learn Today What did it take to revise an age old belief? What is the Copernican
More informationAP Physics C - Mechanics
Slide 1 / 78 Slide 2 / 78 AP Physics C - Mechanics Universal Gravitation 2015-12-04 www.njctl.org Table of Contents Slide 3 / 78 Click on the topic to go to that section Newton's Law of Universal Gravitation
More informationPhysics 111. Tuesday, November 9, Universal Law Potential Energy Kepler s Laws. density hydrostatic equilibrium Pascal s Principle
ics Tuesday, ember 9, 2004 Ch 12: Ch 15: Gravity Universal Law Potential Energy Kepler s Laws Fluids density hydrostatic equilibrium Pascal s Principle Announcements Wednesday, 8-9 pm in NSC 118/119 Sunday,
More informationCHAPTER II UNIVERSAL GRAVITATION
CHAPTER II UNIVERSAL GRAVITATION Outline: Newton s Law of Universal Gravitation Measuring gravitational constant Free-fall acceleration and the gravitational force Energy Considerations in Planetary and
More informationCopyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley.
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 informationGRAVITATION. F = GmM R 2
GRAVITATION Name: Partner: Section: Date: PURPOSE: To explore the gravitational force and Kepler s Laws of Planetary motion. INTRODUCTION: Newton s law of Universal Gravitation tells us that the gravitational
More informationChapter 5 Centripetal Force and Gravity. Copyright 2010 Pearson Education, Inc.
Chapter 5 Centripetal Force and Gravity v Centripetal Acceleration v Velocity is a Vector v It has Magnitude and Direction v If either changes, the velocity vector changes. Tumble Buggy Demo v Centripetal
More informationChapter 13 Newton s s Universal Law of Gravity
Chapter 13 Newton s s Universal Law of Gravity F mm 1 rˆ 1 1 = G r G = 6.67x10 11 Nm kg Sun at Center Orbits are Circular Tycho Brahe 1546-1601 Tycho was the greatest observational astronomer of his time.
More information"The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were
"The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets
More informationPhysics Unit 7: Circular Motion, Universal Gravitation, and Satellite Orbits. Planetary Motion
Physics Unit 7: Circular Motion, Universal Gravitation, and Satellite Orbits Planetary Motion Geocentric Models --Many people prior to the 1500 s viewed the! Earth and the solar system using a! geocentric
More informationAstr 2320 Tues. Jan. 24, 2017 Today s Topics Review of Celestial Mechanics (Ch. 3)
Astr 2320 Tues. Jan. 24, 2017 Today s Topics Review of Celestial Mechanics (Ch. 3) Copernicus (empirical observations) Kepler (mathematical concepts) Galileo (application to Jupiter s moons) Newton (Gravity
More informationPhys 2101 Gabriela González
Phys 2101 Gabriela González Newton s law : F = Gm 1 m 2 /r 2 Explains why apples fall, why the planets move around the Sun, sciencebulletins.amnh.org And in YouTube! Explains just as well as Newtons why
More informationLecture 15 - Orbit Problems
Lecture 15 - Orbit Problems A Puzzle... The ellipse shown below has one focus at the origin and its major axis lies along the x-axis. The ellipse has a semimajor axis of length a and a semi-minor axis
More informationEnergy Potential Energy and Force Conservation Laws Isolated and Nonisolated Systems
Energy Potential Energy and Force Conservation Laws Isolated and Nonisolated ystems Lana heridan De Anza College Oct 27, 2017 Last time gravitational and spring potential energies conservative and nonconservative
More informationLecture 9 Chapter 13 Gravitation. Gravitation
Lecture 9 Chapter 13 Gravitation Gravitation UNIVERSAL GRAVITATION For any two masses in the universe: F = Gm 1m 2 r 2 G = a constant evaluated by Henry Cavendish +F -F m 1 m 2 r Two people pass in a hall.
More informationGravitation & Kepler s Laws
Gravitation & Kepler s Laws What causes YOU to be pulled down to the surface of the earth? THE EARTH.or more specifically the EARTH S MASS. Anything that has MASS has a gravitational pull towards it. F
More informationBasics of Kepler and Newton. Orbits of the planets, moons,
Basics of Kepler and Newton Orbits of the planets, moons, Kepler s Laws, as derived by Newton. Kepler s Laws Universal Law of Gravity Three Laws of Motion Deriving Kepler s Laws Recall: The Copernican
More informationThe Revolution of the Moons of Jupiter
The Revolution of the Moons of Jupiter Overview: During this lab session you will make use of a CLEA (Contemporary Laboratory Experiences in Astronomy) computer program generously developed and supplied
More information7 Study Guide. Gravitation Vocabulary Review
Date Period Name CHAPTER 7 Study Guide Gravitation Vocabulary Review Write the term that correctly completes the statement. Use each term once. Kepler s second law Newton s law of universal gravitation
More informationPhysics 161 Lecture 10: Universal Gravitation. October 4, /6/20 15
Physics 161 Lecture 10: Universal Gravitation October 4, 2018 1 Midterm announcements 1) The first midterm exam will be Thursday October 18 from 10:00 pm to 11:20 pm in ARC- 103. 2) The exam will be multiple
More information5. 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 (164 177) Physicist Image courtesy of NASA/JPL Sir Isaac Newton (164-177)
More informationIntroduction To Modern Astronomy I
ASTR 111 003 Fall 2006 Lecture 03 Sep. 18, 2006 Introduction To Modern Astronomy I Introducing Astronomy (chap. 1-6) Planets and Moons (chap. 7-17) Ch1: Astronomy and the Universe Ch2: Knowing the Heavens
More information14.1 Earth Satellites. The path of an Earth satellite follows the curvature of the Earth.
The path of an Earth satellite follows the curvature of the Earth. A stone thrown fast enough to go a horizontal distance of 8 kilometers during the time (1 second) it takes to fall 5 meters, will orbit
More information[05] Historical Perspectives (9/12/17)
1 [05] Historical Perspectives (9/12/17) Upcoming Items 1. Homework #2 due now. 2. Read Ch. 4.1 4.2 and do self-study quizzes. 3. Homework #3 due in one week. Ptolemaic system http://static.newworldencyclopedia.org/thumb/3/3a/
More informationASTR-1010: Astronomy I Course Notes Section III
ASTR-1010: Astronomy I Course Notes Section III Dr. Donald G. Luttermoser Department of Physics and Astronomy East Tennessee State University Edition 2.0 Abstract These class notes are designed for use
More informationAP Physics-B Universal Gravitation Introduction: Kepler s Laws of Planetary Motion: Newton s Law of Universal Gravitation: Performance Objectives:
AP Physics-B Universal Gravitation Introduction: Astronomy is the oldest science. Practical needs and imagination acted together to give astronomy an early importance. For thousands of years, the motions
More informationGravitation Part I. Ptolemy, Copernicus, Galileo, and Kepler
Gravitation Part I. Ptolemy, Copernicus, Galileo, and Kepler Celestial motions The stars: Uniform daily motion about the celestial poles (rising and setting). The Sun: Daily motion around the celestial
More informationRotational Motion and the Law of Gravity 1
Rotational Motion and the Law of Gravity 1 Linear motion is described by position, velocity, and acceleration. Circular motion repeats itself in circles around the axis of rotation Ex. Planets in orbit,
More information