# Gravitation Kepler s Laws

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Gravitation Kepler s Laws Lana heridan De Anza College Mar 15, 2015

2 Overview Newton s Law of Universal Gravitation Gravitational field Kepler s Laws

3 Gravitation The force that massive objects exert on one another. Newton s Law of Universal Gravitation F G = Gm 1m 2 r 2 for two objects, masses m 1 and m 2 at a distance r. G = Nm 2 kg 2.

4 r to generate a value for G., each of mass m, fixed to the er Gravitation or thin metal wire as illusf mass M, are placed near the larger spheres causes the rod brium orientation. The angle eam reflected from a mirror.1 is often referred to as an e varies as the inverse square examples of this type of force in vector form by defining a rected from particle 1 toward is (13.3) 5 k/x, where k is a constant. A direct pro- Figure 13.1 Cavendish apparatus for measuring gravitational forces. F 21 m 1 Consistent with Newton s third law, F 21 F 12. rˆ12 r F 12 m 2 Figure 13.2 The gravitational force between two particles is attractive. The unit vector r^12 is directed from particle 1 toward particle 2. r 2 ˆr 1 2 F G,1 2 = Gm 1m 2 for two objects, masses m 1 and m 2 at a distance r. G = Nm 2 kg 2.

5 Gravitational Potential Energy Remember from Chapter 7, F x = du dx ; U = F (x) dx ince W = F (r) dr, this tells us that the work done by gravity on an object is equal to minus the change in potential energy.

6 Gravitational Potential Energy Remember from Chapter 7, F x = du dx ; U = F (x) dx ince W = F (r) dr, this tells us that the work done by gravity on an object is equal to minus the change in potential energy. rf U = r i rf F(r) dr = Gm 1m 2 r i r 2 dr ( 1 = Gm 1 m 2 1 ) r f r i

7 Gravitational Potential Energy It is useful to pick a reference point to set the scale for gravitational potential energy. What would be a good point?

8 Gravitational Potential Energy It is useful to pick a reference point to set the scale for gravitational potential energy. What would be a good point? Infinite distance! For r i =, U(r i ) = 0. Then we can define: U(r) = Gm 1m 2 r This will always be a negative number.

9 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

10 Acceleration due to Gravity This force in that it gives objects weight, F g. 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.

11 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

12 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.

13 Acceleration due to Gravity This force in that it gives objects weight, F g. 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.

14 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

15 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.

16 Fields field A field is any kind of physical quantity that has values specified at every point in space and time.

17 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

18 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.

19 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

20 Examples of Fields The gravitational field caused by the un-earth system looks something like: 1 Figure from

21 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

22 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.

23 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)

24 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

25 Gravitational Potential 1 Figure from

26 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.

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 ummary Gravitational force Gravitational field Kepler s Laws 4th Collected Homework! due Friday. (Uncollected) Homework erway & Jewett, Ch 13, onward from page 410. Questions: ection Qs 3, 9, 11, 15, 19

### Conceptual 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

### Observational 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

### 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

### Gravity. 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

### Gravitation 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

### KEPLER 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

### CH 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

### PHYS 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

### Copyright 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

### Gravitation. 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

### Physics 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

### Planetary 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

### Lesson 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

### L03: 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

### A = 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

### F = 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

### AP 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

### Johannes 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

### Chapter 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

### Newton 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

### ASTRO 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.

### Chapter 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

### 1. 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

### Newton 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

### Gravitation 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

### Physics 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,

### Section 37 Kepler's Rules

Section 37 Kepler's Rules What is the universe made out of and how do the parts interact? That was our goal in this course While we ve learned that objects do what they do because of forces, energy, linear

### 9/12/2010. The Four Fundamental Forces of Nature. 1. Gravity 2. Electromagnetism 3. The Strong Nuclear Force 4. The Weak Nuclear Force

The Four Fundamental Forces of Nature 1. Gravity 2. Electromagnetism 3. The Strong Nuclear Force 4. The Weak Nuclear Force The Universe is made of matter Gravity the force of attraction between matter

### By; Jarrick Serdar, Michael Broberg, Trevor Grey, Cameron Kearl, Claire DeCoste, and Kristian Fors

By; Jarrick Serdar, Michael Broberg, Trevor Grey, Cameron Kearl, Claire DeCoste, and Kristian Fors What is gravity? Gravity is defined as the force of attraction by which terrestrial bodies tend to fall

### The 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

### PHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015

PHYS 106 Fall 2151 Homework 3 Due: Thursday, 8 Oct 2015 When you do a calculation, show all your steps. Do not just give an answer. You may work with others, but the work you submit should be your own.

### AP 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

### Kepler's Laws and Newton's Laws

Kepler's Laws and Newton's Laws Kepler's Laws Johannes Kepler (1571-1630) developed a quantitative description of the motions of the planets in the solar system. The description that he produced is expressed

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

### How big is the Universe and where are we in it?

Announcements Results of clicker questions from Monday are on ICON. First homework is graded on ICON. Next homework due one minute before midnight on Tuesday, September 6. Labs start this week. All lab

### AP 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

### GRAVITATION. 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

### ASTR 150. Planetarium Shows begin Sept 9th. Register your iclicker! Last time: The Night Sky Today: Motion and Gravity. Info on course website

Planetarium Shows begin Sept 9th Info on course website Register your iclicker! Last time: The Night Sky Today: Motion and Gravity ASTR 150 Hang on tight! Most math all semester-- get it over with right

### Basics 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

### Test Bank for Life in the Universe, Third Edition Chapter 2: The Science of Life in the Universe

1. The possibility of extraterrestrial life was first considered A) after the invention of the telescope B) only during the past few decades C) many thousands of years ago during ancient times D) at the

### Celestial Mechanics and Orbital Motions. Kepler s Laws Newton s Laws Tidal Forces

Celestial Mechanics and Orbital Motions Kepler s Laws Newton s Laws Tidal Forces Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg,

### Chapter 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.

### Tycho Brahe ( )

Tycho Brahe (1546-1601) Foremost astronomer after the death of Copernicus. King Frederick II of Denmark set him up at Uraniborg, an observatory on the island of Hveen. With new instruments (quadrant),

### Early Theories. Early astronomers believed that the sun, planets and stars orbited Earth (geocentric model) Developed by Aristotle

Planetary Motion Early Theories Early astronomers believed that the sun, planets and stars orbited Earth (geocentric model) Developed by Aristotle Stars appear to move around Earth Observations showed

### Kepler s Laws of Orbital Motion. Lecture 5 January 30, 2014

Kepler s Laws of Orbital Motion Lecture 5 January 30, 2014 Parallax If distance is measured in parsecs then d = 1 PA Where PA is the parallax angle, in arcsec NOTE: The distance from the Sun to the Earth

### Planetary Orbits: Kepler s Laws 1/18/07

Planetary Orbits: Kepler s Laws Announcements The correct link for the course webpage http://www.lpl.arizona.edu/undergrad/classes/spring2007/giacalone_206-2 The first homework due Jan 25 (available for

### Physics Mechanics. Lecture 29 Gravitation

1 Physics 170 - Mechanics Lecture 29 Gravitation Newton, following an idea suggested by Robert Hooke, hypothesized that the force of gravity acting on the planets is inversely proportional to their distances

### Ay 1 Lecture 2. Starting the Exploration

Ay 1 Lecture 2 Starting the Exploration 2.1 Distances and Scales Some Commonly Used Units Distance: Astronomical unit: the distance from the Earth to the Sun, 1 au = 1.496 10 13 cm ~ 1.5 10 13 cm Light

### General 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

### Today. Planetary Motion. Tycho Brahe s Observations. Kepler s Laws of Planetary Motion. Laws of Motion. in physics

Planetary Motion Today Tycho Brahe s Observations Kepler s Laws of Planetary Motion Laws of Motion in physics Page from 1640 text in the KSL rare book collection That the Earth may be a Planet the seeming

### Kepler s Laws of Orbital Motion. Lecture 5 January 24, 2013

Kepler s Laws of Orbital Motion Lecture 5 January 24, 2013 Team Extra Credit Two teams: Io & Genius Every class (that is not an exam/exam review) will have a question asked to a random member of each team

### Chapter 02 The Rise of Astronomy

Chapter 02 The Rise of Astronomy Multiple Choice Questions 1. The moon appears larger when it rises than when it is high in the sky because A. You are closer to it when it rises (angular-size relation).

### Unit: Planetary Science

Orbital Motion Kepler s Laws GETTING AN ACCOUNT: 1) go to www.explorelearning.com 2) click on Enroll in a class (top right hand area of screen). 3) Where it says Enter class Code enter the number: MLTWD2YAZH

### Chapter 5 Lecture Notes

Formulas: a C = v 2 /r a = a C + a T F = Gm 1 m 2 /r 2 Chapter 5 Lecture Notes Physics 2414 - Strauss Constants: G = 6.67 10-11 N-m 2 /kg 2. Main Ideas: 1. Uniform circular motion 2. Nonuniform circular

### Universal Gravitation

Universal Gravitation Johannes Kepler Johannes Kepler was a German mathematician, astronomer and astrologer, and key figure in the 17th century Scientific revolution. He is best known for his laws of planetary

### Unit 3 Lesson 2 Gravity and the Solar System. Copyright Houghton Mifflin Harcourt Publishing Company

Florida Benchmarks SC.8.N.1.4 Explain how hypotheses are valuable if they lead to further investigations, even if they turn out not to be supported by the data. SC.8.N.1.5 Analyze the methods used to develop

### 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...

### Satellites and Kepler's Laws: An Argument for Simplicity

OpenStax-CNX module: m444 Satellites and Kepler's Laws: An Argument for Simplicity OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License.0 Abstract

### Astronomy 104: Stellar Astronomy

Astronomy 104: Stellar Astronomy Lecture 5: Observing is the key... Brahe and Kepler Spring Semester 2013 Dr. Matt Craig 1 For next time: Read Slater and Freedman 3-5 and 3-6 if you haven't already. Focus

### History. Geocentric model (Ptolemy) Heliocentric model (Aristarchus of Samos)

Orbital Mechanics History Geocentric model (Ptolemy) Heliocentric model (Aristarchus of Samos) Nicholas Copernicus (1473-1543) In De Revolutionibus Orbium Coelestium ("On the Revolutions of the Celestial

### Name Period Date Earth and Space Science. Solar System Review

Name Period Date Earth and Space Science Solar System Review 1. is the spinning a planetary object on its axis. 2. is the backward motion of planets. 3. The is a unit less number between 0 and 1 that describes

### Introduction to Mechanics Dynamics Forces Applying Newton s Laws

Introduction to Mechanics Dynamics Forces Applying Newton s Laws Lana heridan De Anza College Feb 21, 2018 Last time force diagrams Newton s second law examples Overview Newton s second law examples Newton

### Chapter 2. The Rise of Astronomy. Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Chapter 2 The Rise of Astronomy Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Periods of Western Astronomy Western astronomy divides into 4 periods Prehistoric

### Astronomy- The Original Science

Astronomy- The Original Science Imagine that it is 5,000 years ago. Clocks and modern calendars have not been invented. How would you tell time or know what day it is? One way to tell the time is to study

### 7.4 Universal Gravitation

Circular Motion Velocity is a vector quantity, which means that it involves both speed (magnitude) and direction. Therefore an object traveling at a constant speed can still accelerate if the direction

### This Week. 2/3/14 Physics 214 Fall

This Week Circular motion Going round the bend Riding in a ferris wheel, the vomit comet Gravitation Our solar system, satellites (Direct TV) The tides, Dark matter, Space Elevator 2/3/14 Physics 214 Fall

### Electricity and Magnetism Electric Field

Electricity and Magnetism Electric Field Lana Sheridan De Anza College Jan 11, 2018 Last time Coulomb s Law force from many charges R/2 +8Q Warm Up Question (c) articles. p Fig. 21-19 Question 9. 10 In

### Nm kg. The magnitude of a gravitational field is known as the gravitational field strength, g. This is defined as the GM

Copyright FIST EDUCATION 011 0430 860 810 Nick Zhang Lecture 7 Gravity and satellites Newton's Law of Universal Gravitation Gravitation is a force of attraction that acts between any two masses. The gravitation

### Dynamics of the solar system

Dynamics of the solar system Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Planets: Wanderer Through the Sky Ecliptic The zodiac Geometry of the

### History of Astronomy. PHYS 1411 Introduction to Astronomy. Tycho Brahe and Exploding Stars. Tycho Brahe ( ) Chapter 4. Renaissance Period

PHYS 1411 Introduction to Astronomy History of Astronomy Chapter 4 Renaissance Period Copernicus new (and correct) explanation for retrograde motion of the planets Copernicus new (and correct) explanation

### 10/21/2003 PHY Lecture 14 1

Announcements. Second exam scheduled for Oct. 8 th -- practice exams now available -- http://www.wfu.edu/~natalie/f03phy3/extrapractice/. Thursday review of Chapters 9-4 3. Today s lecture Universal law

### Earth Science Unit 6: Astronomy Period: Date: Elliptical Orbits

Earth Science Name: Unit 6: Astronomy Period: Date: Lab # 5 Elliptical Orbits Objective: To compare the shape of the earth s orbit (eccentricity) with the orbits of and with a circle. other planets Focus

### Introduction to Mechanics Dynamics Forces Newton s Laws

Introduction to Mechanics Dynamics Forces Newton s Laws Lana heridan De Anza College Nov 1, 2017 Last time Newton s second law mass and weight examples free-body diagrams Overview Newton s second law examples

### You should have finished reading Chapter 3, and started on chapter 4 for next week.

Announcements Homework due on Sunday at 11:45pm. Thank your classmate! You should have finished reading Chapter 3, and started on chapter 4 for next week. Don t forget your out of class planetarium show

### If Earth had no tilt, what else would happen?

A more in depth explanation from last week: If Earth had no tilt, what else would happen? The equator would be much hotter due to the direct sunlight which would lead to a lower survival rate and little

### Conceptual Physics Mechanics Units, Motion, and Inertia

Conceptual Physics Mechanics Units, Motion, and Inertia Lana Sheridan De Anza College July 5, 2017 Last time Scientific facts, hypotheses, theories, and laws Measurements Physics as modeling the natural

### Electricity and Magnetism Electric Potential Energy Electric Potential

Electricity and Magnetism Electric Potential Energy Electric Potential Lana Sheridan De Anza College Jan 23, 2018 Last time implications of Gauss s law introduced electric potential energy in which the

### Chapter 5 Part 2. Newton s Law of Universal Gravitation, Satellites, and Weightlessness

Chapter 5 Part 2 Newton s Law of Universal Gravitation, Satellites, and Weightlessness Newton s ideas about gravity Newton knew that a force exerted on an object causes an acceleration. Most forces occurred

### GRAVITY IS AN ATTRACTIVE FORCE

WHAT IS GRAVITY? Gravity: force of attraction between objects due to their mass Gravity is a noncontact force that acts between two objects at any distance apart GRAVITY IS AN ATTRACTIVE FORCE Earth s

### Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due on Tuesday, Jan. 19, 2016

Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due on Tuesday, Jan. 19, 2016 Why are celestial motions and forces important? They explain the world around us.

### 4. Gravitation & Planetary Motion. Mars Motion: 2005 to 2006

4. Gravitation & Planetary Motion Geocentric models of ancient times Heliocentric model of Copernicus Telescopic observations of Galileo Galilei Systematic observations of Tycho Brahe Three planetary laws

### Gravitational Potential Energy. The Gravitational Field. Grav. Potential Energy Work. Grav. Potential Energy Work

The Gravitational Field Exists at every point in space The gravitational force experienced by a test particle placed at that point divided by the mass of the test particle magnitude of the freefall acceleration

### PSI AP Physics C Universal Gravity Multiple Choice Questions

PSI AP Physics C Universal Gravity Multiple Choice Questions 1. Who determined the value of the gravitational constant (G)? (A) Newton (B) Galileo (C) Einstein (D) Schrödinger (E) Cavendish 2. Who came

### Space Notes Covers Objectives 1 & 2

Space Notes Covers Objectives 1 & 2 Space Introduction Space Introduction Video Celestial Bodies Refers to a natural object out in space 1) Stars 2) Comets 3) Moons 4) Planets 5) Asteroids Constellations

### Celestial Mechanics Lecture 10

Celestial Mechanics Lecture 10 ˆ This is the first of two topics which I have added to the curriculum for this term. ˆ We have a surprizing amount of firepower at our disposal to analyze some basic problems

### = o + t = ot + ½ t 2 = o + 2

Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

### Rotation Moment of Inertia and Applications

Rotation Moment of Inertia and Applications Lana Sheridan De Anza College Nov 20, 2016 Last time net torque Newton s second law for rotation moments of inertia calculating moments of inertia Overview calculating

### Spacecraft Dynamics and Control

Spacecraft Dynamics and Control Matthew M. Peet Arizona State University Lecture 1: In the Beginning Introduction to Spacecraft Dynamics Overview of Course Objectives Determining Orbital Elements Know

### Chapter 11 Gravity Lecture 2. Measuring G

Chapter 11 Gravity Lecture 2 Physics 201 Fall 2009 The Cavendish experiment (second try) Gravitational potential energy Escape velocity Gravitational Field of a point mass Gravitational Field for mass

### APS 1030 Astronomy Lab 79 Kepler's Laws KEPLER'S LAWS

APS 1030 Astronomy Lab 79 Kepler's Laws KEPLER'S LAWS SYNOPSIS: Johannes Kepler formulated three laws that described how the planets orbit around the Sun. His work paved the way for Isaac Newton, who derived

### Tangent and Normal Vectors

Tangent and Normal Vectors MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Navigation When an observer is traveling along with a moving point, for example the passengers in

### Gravity and the Laws of Motion

Gravity and the Laws of Motion Mass Mass is the amount of stuff (matter) in an object. Measured in grams (kg, mg, cg, etc.) Mass will not change unless matter is added or taken away. Weight Weight is the

### TOPIC 2.1: EXPLORATION OF SPACE

TOPIC 2.1: EXPLORATION OF SPACE S4P-2-1 S4P-2-2 S4P-2-3 S4P-2-4 S4P-2-5 Identify and analyze issues pertaining to space exploration. Examples: scale of the universe, technological advancement, promotion

### Static Equilibrium. Lana Sheridan. Dec 5, De Anza College

tatic Equilibrium Lana heridan De Anza College Dec 5, 2016 Last time simple harmonic motion Overview Introducing static equilibrium center of gravity tatic Equilibrium: ystem in Equilibrium Knowing that

### Study Guide Solutions

Study Guide Solutions Table of Contents Chapter 1 A Physics Toolkit... 3 Vocabulary Review... 3 Section 1.1: Mathematics and Physics... 3 Section 1.2: Measurement... 3 Section 1.3: Graphing Data... 4 Chapter

### Electric Potential II

Electric Potential II Physics 2415 Lecture 7 Michael Fowler, UVa Today s Topics Field lines and equipotentials Partial derivatives Potential along a line from two charges Electric breakdown of air Potential

### Lecture 4: Kepler and Galileo. Astronomy 111 Wednesday September 6, 2017

Lecture 4: Kepler and Galileo Astronomy 111 Wednesday September 6, 2017 Reminders Online homework #2 due Monday at 3pm Johannes Kepler (1571-1630): German Was Tycho s assistant Used Tycho s data to discover