# Orbital Mechanics. Angular Momentum

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely determined by the central force of gravity. Johannes Kepler developed the laws of planetary motion we use today to predict the motion of the planets about the sun or the path of a satellite about the earth, and his theories were confirmed when Newton revealed his universal law of gravitation. These laws provide a good approximation of the path of a body in space mechanics. This module explains angular momentum, central forces including Newton's law of gravitation, properties of elliptic orbits such as eccentricity, and Kepler's laws of planetary motion. The associated equations are presented alongside an example in MapleSim and an interactive plot of the eccentricity of an orbit. It is recommended that the Rotational Kinematics module is reviewed prior to completing this module. Angular Momentum Consider an object moving with linear momentum as described by in the Impulse and Momentum module. The moment of this object about point O is the moment of momentum, or angular momentum.

2 ... Eq. (1) The cross product implies that the angular momentum is a vector perpendicular to both the momentum and the position vectors. The magnitude of the angular momentum vector is given as... Eq. (2) Where phi is the angle between the position vector and momentum vector, and is the tangential speed in a rotating frame of reference. Substituting the tangential velocity, this can be written in the form where is the angular speed. Deriving the angular momentum in vector form gives... Eq. (3)... Eq. (4) The cross product of the first term will be zero. The product is a force, which is crossed with the position, resulting in a moment. The rate of change of angular momentum becomes... Eq. (5) It is often more advantageous to express angular momentum per unit mass, denoted by h.... Eq. (6) Central Forces If an object in undergoing a force directed toward a fixed point and no other forces are acting on this object, it is experiencing a central force. Gravity is a common central force. A satellite in orbit experiencing no other (or negligible) forces except for the gravitational pull towards the center of mass of the earth is undergoing a central force. Therefore, if the force on the object is directed at the center of mass, there is no moment on the object.

3 ... Eq. (7) This must mean that the angular momentum is a constant value for objects under a central force. Newton's Law of Universal Gravitation Newton states his law of gravitation as... Eq. (8) where the force F is related to the product of mass M\$m of the attracting objects, the distance r between them, multiplied by the constant of gravitation. The weight that all objects experience under gravity on the surface of a planet of radius R is... Eq. (9) Trajectory of Particles Under a Central Force Particles moving under a central force follow the trajectory given by the following differential equation.... Eq. (10) Where the force F is assumed as an attractive force (positive towards the origin), and u represents 1/r. Deriving the Trajectory Differential Equation

4 The radial and transverse components describe the motion of a particle in terms of r and theta. Figure 1: Radial components of a trajectory The velocity of the particle is the time derivative of its position. The derivative of the unit vector can be represented with.... Eq. (11)... Eq. (12a, 12b) Substituting this in provides the velocity of a particle in radial and transverse components... Eq. (13) Following the same method, differentiating with respect to time and simplifying provides the accelleration of the particle.... Eq. (14) The force applied to the particle given in terms of the radial and transverse components follows F=ma.

5 ... Eq. (15a, 15b) Under a central force, the force experience by a particle will be along the frame directed towards the origin. Including the force in eq. (15) resolves the equations to... Eq. (16) From eq. (6),... Eq. (17a, 17b) Similarly, velocity can be expressed with angular momentum.... Eq. (18) Acceleration is found in a similar fasion.... Eq. (19) Let 1/r be represented with u. Substituting _ and _ into eq. (17a) shows... Eq. (20) Simplifying this resolves to a differential equation.... Eq. (21)... Eq. (10) If the central force is gravity, using Newton's equation from eq. (8) the differential equation represents the trajectory of an object in orbit.

6 ... Eq. (22) Solve the ODE by adding the particular solution u=gm/h 2 to the general solution u=c\$cos( 0 ) with 0 =0. The ratio of the constants decribe the eccentricity of the trajectory.... Eq. (23) This allows eq. (23) to be written in the following form.... Eq. (24)... Eq. (25) Exploring Eccentricity Observe the effects of vaying the eccentricity of the particles trajectory in the interactive section below.

7 Anim... Eccentricity = 1.07, Path = Hyperbola Figure 2: Visualizing the eccentricity of orbits The eccentricity of an orbit describes the path of the particle quite well. There are four cases of trajectories that concern the eccentricity therefore the vector r tends to infinity. The particle will travel in a hyperbolic trajectory. parabolic ellipse as there is a radius vector for every 4. circular orbit. Escape Velocity

8 Figure 3: At this point, it has an initial position and velocity and the object is located at the perigee, the closest point to earth in its orbit. The constant C can be found with and... Eq. (26) C. Knowing the constant of the trajectory, the eccentricity can be found.... Eq. (27) Which simplifies to... Eq. (28) The different cases of eccentricity can approximate key velocities from eq. (29). To launch a rocket into a circular orbit of radius... Eq. (29)

9 An elliptical path occurs as the eccentricity is between 0 and Eq. (30)... Eq. (31) The minimum velocity required to escape an orbit happens when the eccentricity is 1 and the object is travelling in a parabola.... Eq. (32) Finally, a hyperbolic path requires a velocity that puts the object in an eccentricity greater than Eq. (33) Properties of an Ellipse An elliptical orbit contains two foci, denoted as F1 and F2 in the below figure. The radius from the center of the ellipse to the farthest edge is the semi major axis, a. The distance from the center to the closest edge of the ellipse is the semi minor axis, b.

10 Figure 4: An object in an elliptical orbit A planet or a large mass lies at a focus O of the ellipse. The eccentricity of this elliptical orbit can be described with the largest and smallest r values.... Eq. (34) These radial values can be calculated using eq. (25) at and ). These values help calculate key characteristics of the ellipse such as the magnitude of the semi major and minor axes, a and b. and The period of an object in an elliptical orbit is given by... Eq. (35a, 35b)... Eq. (36) Kepler's Laws of Planetary Motion The preceding equations assumed that the secondary mass such as a satellite or rocket is negligible in the calculations. However, considering the moon moving about the earth, the second mass is a rather large factor in determining its orbit. The approximations obtained from these equations aren't entirely accurate because of the influence of the moon on

11 Earth's orbit. This problem also arises when estimating the path of the planets, however these equations do provide a reasonably good approximation. Johann Kepler, a German astronomer, developed his 3 laws which govern the motion of the planets Each planet describes an elliptical orbit with the sun at one of its two foci. The radius vector drawn from the sun to a planet sweeps equal areas in equal times. The square of the orbital period of a planet is proportional to the cube of the semimajor axis of their orbit. Proof of Kepler's Third Law Adding the equations cancels the eccentricity. Simplify and isolate for the angular momentum.... Eq. (37) Recall the semi major and semi minor axes from eq. (35). Using these in eq. (38) reduces the form.... Eq. (38)... Eq. (39) Kepler's third law relates to the period of an orbit. Using the squared form of the period eq. (36), and the angular momentum in eq. (39), the equation reduces to the third law.

12 ... Eq. (40) Examples with MapleSim Example 1: Maximum Altitude Problem Statement: A satellite enters an orbit 500km above the earth. It's velocity at this point is km/h perpendicular to the earth. What will the maximum altitude that the satellite will reach? Figure 5: A satellite in an elliptic orbit about Earth Analytical Solution Data: Convert units to SI units.

13 Converted to m/s Solution: This satellite has entered an orbit, so it's motion can be estimated using eq. (23). The initial height of the satellite must include the radius of the earth. The angular accelleration will remain constant throughout the orbit. Use the supplied = Subtracting the radius of the earth reveals the maximum altitude of the satellite above earth, in meters. =

14 Therefore, the satellite reaches a maxmimum height of km. MapleSim Solution Step 1: Insert Components Drag the following components into a new workspace. Component Location Multibody > Bodies and Frames Multibody > Bodies and Frames Multibody > Joints and Motions Multibody > Joints and Motions 1D Mechanical > Rotational > Motion Drivers

15 1D Mechanical > Rotational > Sensors 1D Mechanical > Translational > Motion Drivers Step 2: Connect the components. Connect the components as shown in the diagram below. Figure 6: Model Diagram Step 3: Add Parameters 1. Add a parameters block to the model. Select the parameter icon and click on the workspace. Open the parameters block and enter the following values: r0 = 500E+3; v0 = 36900/3.6; R = 6371E+3.

16 Figure 7: Parameter values for MapleSim model Step 4: Enter Block Values 1. Select the Revolute block. On the 'Inspector' pane, change the axis of rotation parameter to [0,1,0]. Change the initial conditions IC to 'Treat as Guess', and set 0 to. 2. For the Prismatic block, change IC s,v to 'Treat as Guess' and set the initial displacement s 0 to r0. Step 5: Create the Custom Component. 1. Add a new custom component to the MapleSim document. From the 'View' menu, select 'Create Attachment'. Select 'Custom Component' and create the attachment. ang(t) and r as r (t) to specify these as inputs and outputs. Press enter/return to save the equation. 3. Enter the constant values under 'params'. Angular momentum h will be the product of the initial radius and velocity. Enter the constant C from eq. (27). Press enter/return.

17 Figure 8: Equations for custom component 4. Under the Component Ports section, click 'Clear All Ports' and add 2 new ports. For the first port, set the port type to 'Signal Input' and on the drop down menu for the value, select ang(t). 5. For the second port, set the port type to 'Signal Output' and select the r(t) value. 6. Generate the MapleSim component. Connect it to the model appropriately. The block icon can be found on the 'Project' tab under 'Definitions > Components'. Step 6: Connect probes 1. Connect a probe to flange_b of the Prismatic block to measure the distance from the center of the earth to the satellite. Step 3: Run Simulation Run the simulation and probe the generated graph. Subtract the radius of the earth from the generated value to obtain the maximum altitude of the satellite. Results The simulation generates the following plot.

18 Figure 9: Simulation results The maximum distance measured from the plot is 6.592\$10 7 m. Subtract the radius of the earth, 6371 km, from this value. = meters References: Beer et al. "Vector Mechanics for Engineers: Dynamics", 10th Edition Avenue of the Americas, New York, NY, 2013, McGraw-Hill, Inc.

### Penn State University Physics 211 ORBITAL MECHANICS 1

ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there

### Astromechanics Two-Body Problem (Cont)

5. Orbit Characteristics Astromechanics Two-Body Problem (Cont) We have shown that the in the two-body problem, the orbit of the satellite about the primary (or vice-versa) is a conic section, with the

### USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

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

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

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

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

2. Orbits Topics Orbit types Kepler and Newton laws Coverage area Influence of Earth 1 Orbit types According to inclination angle Equatorial Polar Inclinational orbit According to shape Circular orbit

### Section 4: The Basics of Satellite Orbits

Section 4: The Basics of Satellite Orbits MOTION IN SPACE VS. MOTION IN THE ATMOSPHERE The motion of objects in the atmosphere differs in three important ways from the motion of objects in space. First,

### Orbital Mechanics and Space Geometry

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

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

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

### Planetary Orbit Simulator Student Guide

Name: Planetary Orbit Simulator Student Guide Background Material Answer the following questions after reviewing the Kepler's Laws and Planetary Motion and Newton and Planetary Motion background pages.

### G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M

G U I D E T O A P P L I E D O R B I T A L M E C H A N I C S F O R K E R B A L S P A C E P R O G R A M CONTENTS Foreword... 2 Forces... 3 Circular Orbits... 8 Energy... 10 Angular Momentum... 13 FOREWORD

### 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

### The Two-Body Problem

The Two-Body Problem Abstract In my short essay on Kepler s laws of planetary motion and Newton s law of universal gravitation, the trajectory of one massive object near another was shown to be a conic

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

Chapter 5. Gravitation Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 13. 5.1 Newton s Law of Gravitation We have already studied the effects of gravity through the

### Physics Midterm Review Packet January 2010

Physics Midterm Review Packet January 2010 This Packet is a Study Guide, not a replacement for studying from your notes, tests, quizzes, and textbook. Midterm Date: Thursday, January 28 th 8:15-10:15 Room:

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,

### PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

### Lecture L17 - Orbit Transfers and Interplanetary Trajectories

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

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

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

### A. 81 2 = 6561 times greater. B. 81 times greater. C. equally strong. D. 1/81 as great. E. (1/81) 2 = 1/6561 as great.

Q12.1 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

### Orbital Dynamics: Formulary

Orbital Dynamics: Formulary 1 Introduction Prof. Dr. D. Stoffer Department of Mathematics, ETH Zurich Newton s law of motion: The net force on an object is equal to the mass of the object multiplied by

### Use the following information to deduce that the gravitational field strength at the surface of the Earth is approximately 10 N kg 1.

IB PHYSICS: Gravitational Forces Review 1. This question is about gravitation and ocean tides. (b) State Newton s law of universal gravitation. Use the following information to deduce that the gravitational

### Newton s Law of Universal Gravitation

Newton s Law of Universal Gravitation The greatest moments in science are when two phenomena that were considered completely separate suddenly are seen as just two different versions of the same thing.

### From Aristotle to Newton

From Aristotle to Newton The history of the Solar System (and the universe to some extent) from ancient Greek times through to the beginnings of modern physics. The Geocentric Model Ancient Greek astronomers

### Newton s Law of Gravity

Gravitational Potential Energy On Earth, depends on: object s mass (m) strength of gravity (g) distance object could potentially fall Gravitational Potential Energy In space, an object or gas cloud has

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

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

### Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015

Name: Earth 110 Exploration of the Solar System Assignment 1: Celestial Motions and Forces Due in class Tuesday, Jan. 20, 2015 Why are celestial motions and forces important? They explain the world around

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

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

### Newton s Law of Gravity and Kepler s Laws

Newton s Law of Gravity and Kepler s Laws Michael Fowler Phys 142E Lec 9 2/6/09. These notes are partly adapted from my Physics 152 lectures, where more mathematical details can be found. The Universal

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### Gravitation and Newton s Synthesis

Gravitation and Newton s Synthesis Vocabulary law of unviversal Kepler s laws of planetary perturbations casual laws gravitation motion casuality field graviational field inertial mass gravitational mass

### Chapter 3.8 & 6 Solutions

Chapter 3.8 & 6 Solutions P3.37. Prepare: We are asked to find period, speed and acceleration. Period and frequency are inverses according to Equation 3.26. To find speed we need to know the distance traveled

### 3600 s 1 h. 24 h 1 day. 1 day

Week 7 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Vocabulary - Understanding Revolution in. our Solar System

Vocabulary - Understanding Revolution in Universe Galaxy Solar system Planet Moon Comet Asteroid Meteor(ite) Heliocentric Geocentric Satellite Terrestrial planets Jovian (gas) planets Gravity our Solar

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

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

### circular motion & gravitation physics 111N

circular motion & gravitation physics 111N uniform circular motion an object moving around a circle at a constant rate must have an acceleration always perpendicular to the velocity (else the speed would

### Binary Stars. Kepler s Laws of Orbital Motion

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

### Solar System. 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X?

Solar System 1. The diagram below represents a simple geocentric model. Which object is represented by the letter X? A) Earth B) Sun C) Moon D) Polaris 2. Which object orbits Earth in both the Earth-centered

### The Gravitational Field

The Gravitational Field The use of multimedia in teaching physics Texts to multimedia presentation Jan Hrnčíř jan.hrncir@gfxs.cz Martin Klejch martin.klejch@gfxs.cz F. X. Šalda Grammar School, Liberec

### Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam

Physics 2A, Sec B00: Mechanics -- Winter 2011 Instructor: B. Grinstein Final Exam INSTRUCTIONS: Use a pencil #2 to fill your scantron. Write your code number and bubble it in under "EXAM NUMBER;" an entry

### Dynamics of Iain M. Banks Orbitals. Richard Kennaway. 12 October 2005

Dynamics of Iain M. Banks Orbitals Richard Kennaway 12 October 2005 Note This is a draft in progress, and as such may contain errors. Please do not cite this without permission. 1 The problem An Orbital

### Version A Page 1. 1. The diagram shows two bowling balls, A and B, each having a mass of 7.00 kilograms, placed 2.00 meters apart.

Physics Unit Exam, Kinematics 1. The diagram shows two bowling balls, A and B, each having a mass of 7.00 kilograms, placed 2.00 meters apart. What is the magnitude of the gravitational force exerted by

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

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

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 28] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### Chapter 6 Circular Motion

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

### Angular Velocity vs. Linear Velocity

MATH 7 Angular Velocity vs. Linear Velocity Dr. Neal, WKU Given an object with a fixed speed that is moving in a circle with a fixed ius, we can define the angular velocity of the object. That is, we can

### Centripetal Force. This result is independent of the size of r. A full circle has 2π rad, and 360 deg = 2π rad.

Centripetal Force 1 Introduction In classical mechanics, the dynamics of a point particle are described by Newton s 2nd law, F = m a, where F is the net force, m is the mass, and a is the acceleration.

### Math 1302, Week 3 Polar coordinates and orbital motion

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

### Presentation of problem T1 (9 points): The Maribo Meteorite

Presentation of problem T1 (9 points): The Maribo Meteorite Definitions Meteoroid. A small particle (typically smaller than 1 m) from a comet or an asteroid. Meteorite: A meteoroid that impacts the ground

### Mechanics 1: Conservation of Energy and Momentum

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

### Lab 7: Rotational Motion

Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125

### Unit 8 Lesson 2 Gravity and the Solar System

Unit 8 Lesson 2 Gravity and the Solar System Gravity What is gravity? Gravity is a force of attraction between objects that is due to their masses and the distances between them. Every object in the universe

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### PHYS 211 FINAL FALL 2004 Form A

1. Two boys with masses of 40 kg and 60 kg are holding onto either end of a 10 m long massless pole which is initially at rest and floating in still water. They pull themselves along the pole toward each

### Astronomy 1140 Quiz 1 Review

Astronomy 1140 Quiz 1 Review Prof. Pradhan September 15, 2015 What is Science? 1. Explain the difference between astronomy and astrology. (a) Astrology: nonscience using zodiac sign to predict the future/personality

### Magnetism. d. gives the direction of the force on a charge moving in a magnetic field. b. results in negative charges moving. clockwise.

Magnetism 1. An electron which moves with a speed of 3.0 10 4 m/s parallel to a uniform magnetic field of 0.40 T experiences a force of what magnitude? (e = 1.6 10 19 C) a. 4.8 10 14 N c. 2.2 10 24 N b.

### Chapter 6. Work and Energy

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

### Exam 1 Review Questions PHY 2425 - Exam 1

Exam 1 Review Questions PHY 2425 - Exam 1 Exam 1H Rev Ques.doc - 1 - Section: 1 7 Topic: General Properties of Vectors Type: Conceptual 1 Given vector A, the vector 3 A A) has a magnitude 3 times that

### Unit 4 Practice Test: Rotational Motion

Unit 4 Practice Test: Rotational Motion Multiple Guess Identify the letter of the choice that best completes the statement or answers the question. 1. How would an angle in radians be converted to an angle

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### 226 Chapter 15: OSCILLATIONS

Chapter 15: OSCILLATIONS 1. In simple harmonic motion, the restoring force must be proportional to the: A. amplitude B. frequency C. velocity D. displacement E. displacement squared 2. An oscillatory motion

### Series and Parallel Resistive Circuits

Series and Parallel Resistive Circuits The configuration of circuit elements clearly affects the behaviour of a circuit. Resistors connected in series or in parallel are very common in a circuit and act

### Satellites and Space Stations

Satellites and Space Stations A satellite is an object or a body that revolves around another object, which is usually much larger in mass. Natural satellites include the planets, which revolve around

### Determination of Acceleration due to Gravity

Experiment 2 24 Kuwait University Physics 105 Physics Department Determination of Acceleration due to Gravity Introduction In this experiment the acceleration due to gravity (g) is determined using two

### Name Class Period. F = G m 1 m 2 d 2. G =6.67 x 10-11 Nm 2 /kg 2

Gravitational Forces 13.1 Newton s Law of Universal Gravity Newton discovered that gravity is universal. Everything pulls on everything else in the universe in a way that involves only mass and distance.

### 11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

### State Newton's second law of motion for a particle, defining carefully each term used.

5 Question 1. [Marks 20] An unmarked police car P is, travelling at the legal speed limit, v P, on a straight section of highway. At time t = 0, the police car is overtaken by a car C, which is speeding

### CATIA V5 Tutorials. Mechanism Design & Animation. Release 18. Nader G. Zamani. University of Windsor. Jonathan M. Weaver. University of Detroit Mercy

CATIA V5 Tutorials Mechanism Design & Animation Release 18 Nader G. Zamani University of Windsor Jonathan M. Weaver University of Detroit Mercy SDC PUBLICATIONS Schroff Development Corporation www.schroff.com

### Astrodynamics (AERO0024)

Astrodynamics (AERO0024) 6. Interplanetary Trajectories Gaëtan Kerschen Space Structures & Systems Lab (S3L) Course Outline THEMATIC UNIT 1: ORBITAL DYNAMICS Lecture 02: The Two-Body Problem Lecture 03:

### Newton s proof of the connection between

Elliptical Orbit 1/r 2 Force Jeffrey Prentis, Bryan Fulton, and Carol Hesse, University of Michigan-Dearborn, Dearborn, MI Laura Mazzino, University of Louisiana, Lafayette, LA Newton s proof of the connection

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

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

### Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

### Solar System Fundamentals. What is a Planet? Planetary orbits Planetary temperatures Planetary Atmospheres Origin of the Solar System

Solar System Fundamentals What is a Planet? Planetary orbits Planetary temperatures Planetary Atmospheres Origin of the Solar System Properties of Planets What is a planet? Defined finally in August 2006!

### Satellite Posi+oning. Lecture 5: Satellite Orbits. Jan Johansson jan.johansson@chalmers.se Chalmers University of Technology, 2013

Lecture 5: Satellite Orbits Jan Johansson jan.johansson@chalmers.se Chalmers University of Technology, 2013 Geometry Satellite Plasma Posi+oning physics Antenna theory Geophysics Time and Frequency GNSS

### EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1

Instructor: L. M. Khandro EDMONDS COMMUNITY COLLEGE ASTRONOMY 100 Winter Quarter 2007 Sample Test # 1 1. An arc second is a measure of a. time interval between oscillations of a standard clock b. time

### Chapter 9 Circular Motion Dynamics

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

### ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION

ENGINEERING COUNCIL DYNAMICS OF MECHANICAL SYSTEMS D225 TUTORIAL 1 LINEAR AND ANGULAR DISPLACEMENT, VELOCITY AND ACCELERATION This tutorial covers pre-requisite material and should be skipped if you are

### 8.012 Physics I: Classical Mechanics Fall 2008

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

### The orbit of Halley s Comet

The orbit of Halley s Comet Given this information Orbital period = 76 yrs Aphelion distance = 35.3 AU Observed comet in 1682 and predicted return 1758 Questions: How close does HC approach the Sun? What

### 11. Describing Angular or Circular Motion

11. Describing Angular or Circular Motion Introduction Examples of angular motion occur frequently. Examples include the rotation of a bicycle tire, a merry-go-round, a toy top, a food processor, a laboratory

### AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017

AP PHYSICS C Mechanics - SUMMER ASSIGNMENT FOR 2016-2017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material

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

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### AP Physics C: Mechanics: Syllabus 1

AP Physics C: Mechanics: Syllabus 1 Scoring Components Page(s) SC1 The course covers instruction in kinematics. 3, 5 SC2 The course covers instruction in Newton s laws of 3 4 motion. SC3 The course covers

### AP Physics C. Oscillations/SHM Review Packet

AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

### AE554 Applied Orbital Mechanics. Hafta 1 Egemen Đmre

AE554 Applied Orbital Mechanics Hafta 1 Egemen Đmre A bit of history the beginning Astronomy: Science of heavens. (Ancient Greeks). Astronomy existed several thousand years BC Perfect universe (like circles

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

Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving

### Candidate Number. General Certificate of Education Advanced Level Examination June 2010

entre Number andidate Number Surname Other Names andidate Signature General ertificate of Education dvanced Level Examination June 1 Physics PHY4/1 Unit 4 Fields and Further Mechanics Section Friday 18

### Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces. Copyright 2009 Pearson Education, Inc.

Chapter 5 Using Newton s Laws: Friction, Circular Motion, Drag Forces Units of Chapter 5 Applications of Newton s Laws Involving Friction Uniform Circular Motion Kinematics Dynamics of Uniform Circular

### Tennessee State University

Tennessee State University Dept. of Physics & Mathematics PHYS 2010 CF SU 2009 Name 30% Time is 2 hours. Cheating will give you an F-grade. Other instructions will be given in the Hall. MULTIPLE CHOICE.

### Chapter 2. Mission Analysis. 2.1 Mission Geometry

Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude

### Lectures on Gravity Michael Fowler, University of Virginia, Physics 152 Notes, May, 2007

Lectures on Gravity Michael Fowler, University of Virginia, Physics 15 Notes, May, 007 DISCOVERING GRAVITY...3 Terrestrial Gravity: Galileo Analyzes a Cannonball Trajectory...3 Moving Up: Newton Puts the

### Lecture Presentation Chapter 7 Rotational Motion

Lecture Presentation Chapter 7 Rotational Motion Suggested Videos for Chapter 7 Prelecture Videos Describing Rotational Motion Moment of Inertia and Center of Gravity Newton s Second Law for Rotation Class

### Newton s derivation of Kepler s laws (outline)

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

### Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE

1 P a g e Motion Physics Notes Class 11 CHAPTER 3 MOTION IN A STRAIGHT LINE If an object changes its position with respect to its surroundings with time, then it is called in motion. Rest If an object

### Chapter 5: Circular Motion, the Planets, and Gravity

Chapter 5: Circular Motion, the Planets, and Gravity 1. Earth s gravity attracts a person with a force of 120 lbs. The force with which the Earth is attracted towards the person is A. Zero. B. Small but

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### Lecture PowerPoints. Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli

Lecture PowerPoints Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the

### Artificial Satellites Earth & Sky

Artificial Satellites Earth & Sky Name: Introduction In this lab, you will have the opportunity to find out when satellites may be visible from the RPI campus, and if any are visible during the activity,