Dynamical Systems. Mechanical Systems
|
|
- Derek Berry
- 5 years ago
- Views:
Transcription
1 EE/ME/AE324: Dynamical Systems Chapter 2: Modeling Translational Mechanical Systems
2 Common Variables Used Assumes 1 DoF per mass, i.e., all motion scalar Displacement: x ()[ t [m] Velocity: dx() t vt () [m/s] dt 2 Acceleration: dv() t d x() t 2 at () [m/s] 2 dt dt dmtvt dvt kg m f () t M Ma()[Nor t ] 2 dt dt s Force: ( ( ) ( )) ( ) t Energy (work): wt () wt ( 0) p( ) d J or N m Power: dw () t pt () f() tvt ()[W or J/s] dt t 0
3 Variable Conventions Position, velocity and force: Supplied power ( pt ( ) 0):
4 Mass: M [kg] Element Laws: Mass Assumes constant, non relativistic motion with 1 DoF w f Ma 1 2 Kinetic Energy: 2 Potential Energy: k w Mv Mgh g=9.81 [m/s 2 surface of the earth p h is height above(below) a specified reference point
5 Element Laws: Viscous Friction f Bv B( v v ) Viscous friction: 2 1
6 Element Laws: Viscous Friction
7 Element Laws: Stiffness (Spring) Stiffness: f Kx K( x x ) 2 1 f is a tensile (stretching) force rather than a compressive force when x2 x1 0 (as drawn) d is the natural length of the spring, e.g., when 0 x=0 Potential ti lenergy stored din spring: 1 2 wp K( x) 2
8 Interconnection Laws D Alembert s Law (restatement of Newton s 2 nd Law): i ( f ) Ma f 0 ext i i i Law of Reaction Forces (Newton s 3 rd Law):
9 Law of Displacements: Interconnection Laws ( xx ) 0, around any closed path i i
10 Free body Diagrams Free body diagrams are used as an intermediate step to obtaining system equations of motion (EoM) Assume all elements at equilibrium (EQ) when position and velocity references equal to zero Equilibrium Net force on body = 0, with all inputs constant (zero) Suggested order applying forces to a free body diagram Applied forces, ie i.e., specified inputs Inertial forces, i.e., opposite position reference Spring forces Viscous friction (damping) forces All others, e.g., eg gears, pulleys, levers, etc.
11 Simple Free body Example The MSD systems below have equivalent dynamics Assume elements at equilibrium (EQ) when x x 0 q ( ) Spring stretched when x 0, as drawn
12 Equations of Motion To obtain EoM from free body diagram is straight forward left (up) pointing forces = Given the prior free body diagram right (down) pointing forces We havethe followingeom Mx Bx Kx f () t a
13 Example 2.2 Equations of Motion: M x K x B ( x x ) K ( x x ) M x B( x x ) K ( x x ) f ( t) a
14 Ex. 2.4: Relative Displacements Assume springs in EQ when x z 0 As drawn: Elongation of spring K 2 is x z Inertial forces proportional to absolute (not relative) lti accelerations, e.g., x z
15 Relative Displacements As drawn x,z>0, which implies K 2 stretched, K 1 compressed
16 Relative Displacements EoM: M x B x B x K x B z M ( x z ) B z K ( x z ) f ( t ) a
17 Problem w/absolute References As drawn y>x 2 > x 1 >0, which implies K 1, K 2 stretched x 2 y 1
18 Problem w/absolute References Free body:
19 Ex. 2.5: Vertical Motion Assume x is position of the system at EQ Gravitational force on mass is downward F Fg Mg
20 EoM: Vertical Motion Mx Bx Kx fa () t Mg At EQ with zero applied force, the constant displacement caused by gravity can be calculated Mg Kx Mg x 0 0 K fa ( t) is non-zero x x z When the mass is moving and we can redefine substituting into the EoM yields () 0 Mz Bz K x z f () t Mg 0 a Mz Bz Kz f () t a
21 Ex. 2.6: Vertical Motion Assume springs relaxed when x x Determine the static EQ positions due to gravity Note: x is relative to M 2 1 total displacement of M is x +x As drawn: x, x springs K 1, K 2 are stretched t spring K 3 is compressed friction B can be thought of as a damper acting parallel to spring K 2 damper acting parallel to spring K 2 with forces in the same direction
22 Ex. 2.6: Vertical Motion
23 Ex. 2.6: Vertical Motion EoM: M x K x f () t Bx ( K K ) x M g a M ( x x ) Bx ( K K ) x M g To find displacement due to gravity set applied force and all derivatives to zero to obtain: Kx ( K K) x Mg ( K K ) x M g x x Mg 2 K K 2 3 ( M1 M2 ) g K 1
24 Ex. 2.7: Ideal Pulley An ideal pulley is a mass and friction less fi i element that changes direction of motion, e.g., horizontal to vertical, without t cable slippage or stretch t h( (assume in tension)
25 Ex. 2.7: Ideal Pulley The pulley system is the same as the one bl below, except that is includes the gravitational force M 2 g
26 Ex. 2.8: Parallel Combinations For springs with a common natural llength, the variablex ibl can be used to describe their displacement; otherwise use d(t) as shown in the free body bl below:
27 Ex. 2.8: Parallel Combinations The resulting EoM: Mx Bx ( K K ) x Mx Bx K x f ( t ) where K K K eq eq a
28 Ex. 2.8: Series Combinations
29 Ex. 2.8: Series Combinations EoM at A: K ( x x ) K x x EoM at Mass: Kx 1 1 ( K K ) 1 2 Kx Mx Bx 1 K1( x 1 x2) Mx 1 Bx 1 K1( x 1 ) ( K K ) 1 2 Mx Bx KK x Mx Bx K x f t KK 1 2 where Keq K K eq 1 a () K1 K2 1 2
30 Ex. 2.11: Parallel Series Combinations B eq B B ( BB) ( B ) B B 2 3 eq 1 B B B B B B B BB B B BB B B B B B B B
31 P2.21 Assume system at EQ when all 0, and x x 1 2 0; then, as drawn: K compressed by amount 1 1 K stretched by amount x x x K stretched by amount 3 2 B stretched by amount B stretched by amount x 2 2 x x x x i
32 P2.21 M x K x B2x 2 M 2 B ( x x ) f () a t M1x1 Mg 2 K ( x x ) M 1 f () b t Kx 1 1 M1g
33 P2.21 M 2x2 K3x2 B2x2 B1( x1 x2 ) M 1x1 K 2 ( x1 x2 ) M 2 M 1 B1( x 1 x 2) () a M g ( ) f () b t Kx 1 1 M g 1 EOM: f t 2 K2 x1 x2 M x B x K x B ( x x ) K ( x x ) f ( t ) M g a 2 Mx B( x x ) K( xx) Kx f( t) Mg b 1
34 P2.24 Assume system at EQ when all 0, and x x 1 2 0; then, as drawn: K stretched by amount K stretched by amount 2 2 B stretched by amount 1 1 B stretched by amount x 2 2 x x x x x i
35 P M x Bx 2 2 K 2x2 M 2 K ( x x ) M 2g M1x1 fa () t M 1 K ( x x ) B x 1 1 EOM: M x B x K x K ( x x ) M g M x B x K ( x x ) f ( t ) a
36 P2.14 Displacements x, x are relative to x x i Assume system at EQ when all K compressed by amount x x 0; therefore, as drawn: K,B, and B stretched by amount x, x and x, respectively An absolute ref erence to M 1 is y1 x3 x1 y10 An absolute reference to M is y x x y
37 P2.14 My M x x M 1 Bx 1 1 K ( x x ) K2x2 M 2 Bx 1 1 M 2 2 1( 3 1) 2y2 M ( x x ) Bx K2x Bx 2 2 M x M 3 f () a t 3 3
38 My 1 1 M ( x x ) M 1 Bx 1 1 P2.14 K ( x x ) M 2y2 M ( x x ) K x M Bx 2 2 M 3x M 3 3 Bx K2x Bx 2 2 f () a t EoM: M ( x x ) Bx K ( x x ) M ( x x ) K ( x x ) K x B x a () Mx Bx Kx Bx f t
39 Questions?
In-Class Problems 20-21: Work and Kinetic Energy Solutions
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 In-Class Problems 20-21: Work and Kinetic Energy Solutions In-Class-Problem 20 Calculating Work Integrals a) Work
More informationChapter 6 Work and Energy
Chapter 6 Work and Energy Midterm exams will be available next Thursday. Assignment 6 Textbook (Giancoli, 6 th edition), Chapter 6: Due on Thursday, November 5 1. On page 162 of Giancoli, problem 4. 2.
More informationTranslational Mechanical Systems
Translational Mechanical Systems Basic (Idealized) Modeling Elements Interconnection Relationships -Physical Laws Derive Equation of Motion (EOM) - SDOF Energy Transfer Series and Parallel Connections
More informationChapter 7 Energy of a System
Chapter 7 Energy of a System Course Outline : Work Done by a Constant Force Work Done by avarying Force Kinetic Energy and thework-kinetic EnergyTheorem Power Potential Energy of a System (Will be discussed
More informationA. B. C. D. E. v x. ΣF x
Q4.3 The graph to the right shows the velocity of an object as a function of time. Which of the graphs below best shows the net force versus time for this object? 0 v x t ΣF x ΣF x ΣF x ΣF x ΣF x 0 t 0
More informationChapter 6: Work and Kinetic Energy
Chapter 6: Work and Kinetic Energy Suppose you want to find the final velocity of an object being acted on by a variable force. Newton s 2 nd law gives the differential equation (for 1D motion) dv dt =
More informationLesson 5. Luis Anchordoqui. Physics 168. Tuesday, September 26, 17
Lesson 5 Physics 168 1 C. B.-Champagne Luis Anchordoqui 2 2 Work Done by a Constant Force distance moved times component of force in direction of displacement W = Fd cos 3 Work Done by a Constant Force
More informationChapter 5 Oscillatory Motion
Chapter 5 Oscillatory Motion Simple Harmonic Motion An object moves with simple harmonic motion whenever its acceleration is proportional to its displacement from some equilibrium position and is oppositely
More informationKinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction)
Kinematics (special case) a = constant 1D motion 2D projectile Uniform circular Dynamics gravity, tension, elastic, normal, friction Motion with a = constant Newton s Laws F = m a F 12 = F 21 Time & Position
More information( ) = ( ) W net = ΔKE = KE f KE i W F. F d x. KE = 1 2 mv2. Note: Work is the dot product of F and d. Work-Kinetic Energy Theorem
Work-Kinetic Energy Theorem KE = 1 2 mv2 W F change in the kinetic energy of an object F d x net work done on the particle ( ) = ( ) W net = ΔKE = KE f KE i Note: Work is the dot product of F and d W g
More informationWork Done by a Constant Force
Work and Energy Work Done by a Constant Force In physics, work is described by what is accomplished when a force acts on an object, and the object moves through a distance. The work done by a constant
More informationNewton s Laws of Motion and Gravitation
Newton s Laws of Motion and Gravitation Introduction: In Newton s first law we have discussed the equilibrium condition for a particle and seen that when the resultant force acting on the particle is zero,
More informationRead textbook CHAPTER 1.4, Apps B&D
Lecture 2 Read textbook CHAPTER 1.4, Apps B&D Today: Derive EOMs & Linearization undamental equation of motion for mass-springdamper system (1DO). Linear and nonlinear system. Examples of derivation of
More informationWORK, POWER & ENERGY
WORK, POWER & ENERGY Work An applied force acting over a displacement. The force being applied must be parallel to the displacement for work to be occurring. Work Force displacement Units: Newton meter
More informationRecall: Gravitational Potential Energy
Welcome back to Physics 15 Today s agenda: Work Power Physics 15 Spring 017 Lecture 10-1 1 Recall: Gravitational Potential Energy For an object of mass m near the surface of the earth: U g = mgh h is height
More informationPhysics 1 Second Midterm Exam (AM) 2/25/2010
Physics Second Midterm Eam (AM) /5/00. (This problem is worth 40 points.) A roller coaster car of m travels around a vertical loop of radius R. There is no friction and no air resistance. At the top of
More informationModeling Mechanical Systems
Modeling Mechanical Systems Mechanical systems can be either translational or rotational. Although the fundamental relationships for both types are derived from Newton s law, they are different enough
More informationPhysics 2414 Group Exercise 8. Conservation of Energy
Physics 244 Group Exercise 8 Name : OUID : Name 2: OUID 2: Name 3: OUID 3: Name 4: OUID 4: Section Number: Solutions Solutions Conservation of Energy A mass m moves from point i to point f under the action
More information14.4 Change in Potential Energy and Zero Point for Potential Energy
14.4 Change in Potential Energy and Zero Point for Potential Energy We already calculated the work done by different conservative forces: constant gravity near the surface of the earth, the spring force,
More informationOther Examples of Energy Transfer
Chapter 7 Work and Energy Overview energy. Study work as defined in physics. Relate work to kinetic energy. Consider work done by a variable force. Study potential energy. Understand energy conservation.
More informationEngineering Mechanics: Statics. Chapter 7: Virtual Work
Engineering Mechanics: Statics Chapter 7: Virtual Work Introduction Previous chapters-- FBD & zero-force and zero-moment equations -- Suitable when equilibrium position is known For bodies composed of
More informationChapter 3 The Laws of motion. The Laws of motion
Chapter 3 The Laws of motion The Laws of motion The Concept of Force. Newton s First Law. Newton s Second Law. Newton s Third Law. Some Applications of Newton s Laws. 1 5.1 The Concept of Force Force:
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 10. Home Page. Title Page. Page 1 of 37.
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics I Fall 2015 Lecture 10 Page 1 of 37 Midterm I summary 100 90 80 70 60 50 40 30 20 39 43 56 28 11 5 3 0 1 Average: 82.00 Page
More informationLecture Outline Chapter 6. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.
Lecture Outline Chapter 6 Physics, 4 th Edition James S. Walker Chapter 6 Applications of Newton s Laws Units of Chapter 6 Frictional Forces Strings and Springs Translational Equilibrium Connected Objects
More informationProblem 1: Find the Equation of Motion from the static equilibrium position for the following systems: 1) Assumptions
Problem 1: Find the Equation of Motion from the static equilibrium position for the following systems: 1) Assumptions k 2 Wheels roll without friction k 1 Motion will not cause block to hit the supports
More informationl1, l2, l3, ln l1 + l2 + l3 + ln
Work done by a constant force: Consider an object undergoes a displacement S along a straight line while acted on a force F that makes an angle θ with S as shown The work done W by the agent is the product
More information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocwmitedu 200 Dynamics and Control Spring 200 For information about citing these materials or our Terms of Use, visit: http://ocwmitedu/terms Massachusetts nstitute of Technology
More informationConservative vs. Non-conservative forces Gravitational Potential Energy. Conservation of Mechanical energy
Next topic Conservative vs. Non-conservative forces Gravitational Potential Energy Mechanical Energy Conservation of Mechanical energy Work done by non-conservative forces and changes in mechanical energy
More informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued 4.9 Static and Kinetic Frictional Forces When an object is in contact with a surface forces can act on the objects. The component of this force acting
More informationPhysics A - PHY 2048C
Physics A - PHY 2048C Mass & Weight, Force, and Friction 10/04/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapters 6.1-6.6? 2 In your own words: What
More informationPHY 101. Work and Kinetic Energy 7.1 Work Done by a Constant Force
PHY 101 DR M. A. ELERUJA KINETIC ENERGY AND WORK POTENTIAL ENERGY AND CONSERVATION OF ENERGY CENTRE OF MASS AND LINEAR MOMENTUM Work is done by a force acting on an object when the point of application
More informationDynamic equilibrium: object moves with constant velocity in a straight line. = 0, a x = i
Dynamic equilibrium: object moves with constant velocity in a straight line. We note that F net a s are both vector quantities, so in terms of their components, (F net ) x = i (F i ) x = 0, a x = i (a
More informationHSC PHYSICS ONLINE B F BA. repulsion between two negatively charged objects. attraction between a negative charge and a positive charge
HSC PHYSICS ONLINE DYNAMICS TYPES O ORCES Electrostatic force (force mediated by a field - long range: action at a distance) the attractive or repulsion between two stationary charged objects. AB A B BA
More informationPhysics B Newton s Laws AP Review Packet
Force A force is a push or pull on an object. Forces cause an object to accelerate To speed up To slow down To change direction Unit: Newton (SI system) Newton s First Law The Law of Inertia. A body in
More informationWhat is a Force? Free-Body diagrams. Contact vs. At-a-Distance 11/28/2016. Forces and Newton s Laws of Motion
Forces and Newton s Laws of Motion What is a Force? In generic terms: a force is a push or a pull exerted on an object that could cause one of the following to occur: A linear acceleration of the object
More informationChapter 13. Simple Harmonic Motion
Chapter 13 Simple Harmonic Motion Hooke s Law F s = - k x F s is the spring force k is the spring constant It is a measure of the stiffness of the spring A large k indicates a stiff spring and a small
More informationLecture 7. Forces: Newton s Laws. Problem-Solving Tactics: Friction and Centripetal Motion. Physics 105; Summer How do we jump?
ecture 7 Problem-Solving Tactics: Friction and Centripetal Motion (H&W, Chapters 5-6) http://web.njit.edu/~sirenko/ Newton s aws I. If no net force acts on a body, then the body s velocity cannot change.
More informationChapter 12. Static Equilibrium and Elasticity
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
More informationWork and kinetic energy. LA info session today at 5pm in UMC235 CAPA homework due tomorrow night.
Work and kinetic energy LA info session today at 5pm in UMC235 CAPA homework due tomorrow night. 1 Work I apply a force of 2N in the x direction to an object that moves 5m in x. How much work have I done
More informationWork and energy. 15 m. c. Find the work done by the normal force exerted by the incline on the crate.
Work and energy 1. A 10.0-kg crate is pulled 15.0 m up along a frictionless incline as shown in the figure below. The crate starts at rest and has a final speed of 6.00 m/s. motor 15 m 5 a. Draw the free-body
More informationModeling of a Mechanical System
Chapter 3 Modeling of a Mechanical System 3.1 Units Currently, there are two systems of units: One is the international system (SI) metric system, the other is the British engineering system (BES) the
More informationChapter 6 Work and Kinetic Energy
Chapter 6 Work and Kinetic Energy Up until now, we have assumed that the force is constant and thus, the acceleration is constant. Is there a simple technique for dealing with non-constant forces? Fortunately,
More information0J2 - Mechanics Lecture Notes 2
0J2 - Mechanics Lecture Notes 2 Work, Power, Energy Work If a force is applied to a body, which then moves, we say the force does work. In 1D, if the force is constant with magnitude F, and the body moves
More informationPhysics H7A, Fall 2011 Homework 6 Solutions
Physics H7A, Fall 2011 Homework 6 Solutions 1. (Hooke s law) (a) What are the dimensions (in terms of M, L, and T ) of the spring constant k in Hooke s law? (b) If it takes 4.00 J of work to stretch a
More informationPSI AP Physics B Dynamics
PSI AP Physics B Dynamics Multiple-Choice questions 1. After firing a cannon ball, the cannon moves in the opposite direction from the ball. This an example of: A. Newton s First Law B. Newton s Second
More informationPhysics 101 Lecture 5 Newton`s Laws
Physics 101 Lecture 5 Newton`s Laws Dr. Ali ÖVGÜN EMU Physics Department The Laws of Motion q Newton s first law q Force q Mass q Newton s second law q Newton s third law qfrictional forces q Examples
More informationChapter 5. The Laws of Motion
Chapter 5 The Laws of Motion The Laws of Motion The description of an object in There was no consideration of what might influence that motion. Two main factors need to be addressed to answer questions
More informationExam 2 Spring 2014
95.141 Exam 2 Spring 2014 Section number Section instructor Last/First name Last 3 Digits of Student ID Number: Answer all questions, beginning each new question in the space provided. Show all work. Show
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
PH105-007 Exam 2 VERSION A Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A 1.0-kg block and a 2.0-kg block are pressed together on a horizontal
More informationChapter 4. Forces and Newton s Laws of Motion. continued
Chapter 4 Forces and Newton s Laws of Motion continued Quiz 3 4.7 The Gravitational Force Newton s Law of Universal Gravitation Every particle in the universe exerts an attractive force on every other
More informationChapter 5 Work and Energy
Chapter 5 Work and Energy Work and Kinetic Energy Work W in 1D Motion: by a Constant orce by a Varying orce Kinetic Energy, KE: the Work-Energy Theorem Mechanical Energy E and Its Conservation Potential
More information2.003 Engineering Dynamics Problem Set 6 with solution
.00 Engineering Dynamics Problem Set 6 with solution Problem : A slender uniform rod of mass m is attached to a cart of mass m at a frictionless pivot located at point A. The cart is connected to a fixed
More informationW = F x W = Fx cosθ W = Fx. Work
Ch 7 Energy & Work Work Work is a quantity that is useful in describing how objects interact with other objects. Work done by an agent exerting a constant force on an object is the product of the component
More informationDeriving 1 DOF Equations of Motion Worked-Out Examples. MCE371: Vibrations. Prof. Richter. Department of Mechanical Engineering. Handout 3 Fall 2017
MCE371: Vibrations Prof. Richter Department of Mechanical Engineering Handout 3 Fall 2017 Masses with Rectilinear Motion Follow Palm, p.63, 67-72 and Sect.2.6. Refine your skill in drawing correct free
More informationENERGY. Conservative Forces Non-Conservative Forces Conservation of Mechanical Energy Power
ENERGY Conservative Forces Non-Conservative Forces Conservation of Mechanical Energy Power Conservative Forces A force is conservative if the work it does on an object moving between two points is independent
More informationStructural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.
Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear
More informationWORK, ENERGY & POWER Work scalar W = F S Cosθ Unit of work in SI system Work done by a constant force
WORK, ENERGY & POWER Work Let a force be applied on a body so that the body gets displaced. Then work is said to be done. So work is said to be done if the point of application of force gets displaced.
More informationMechanics. Time (s) Distance (m) Velocity (m/s) Acceleration (m/s 2 ) = + displacement/time.
Mechanics Symbols: Equations: Kinematics The Study of Motion s = distance or displacement v = final speed or velocity u = initial speed or velocity a = average acceleration s u+ v v v u v= also v= a =
More informationDescription of the motion using vectorial quantities
Description of the motion using vectorial quantities RECTILINEAR MOTION ARBITRARY MOTION (3D) INERTIAL SYSTEM OF REFERENCE Circular motion Free fall Description of the motion using scalar quantities Let's
More informationPower: Sources of Energy
Chapter 5 Energy Power: Sources of Energy Tidal Power SF Bay Tidal Power Project Main Ideas (Encyclopedia of Physics) Energy is an abstract quantity that an object is said to possess. It is not something
More informationCommon Exam 3, Friday, April 13, :30 9:45 A.M. at KUPF 205 Chaps. 6, 7, 8. HW #8 and HW #9: Due tomorrow, April 6 th (Fri)
Common Exam 3, Friday, April 13, 2007 8:30 9:45 A.M. at KUPF 205 Chaps. 6, 7, 8 Bring calculators (Arrive by 8:15) HW #8 and HW #9: Due tomorrow, April 6 th (Fri) Today. Chapter 8 Hints for HW #9 Quiz
More information1D Mechanical Systems
1D Mechanical Systems In this lecture, we shall deal with 1D mechanical systems that can either ih translate or rotate in a one-dimensional i space. We shall demonstrate the similarities betweeneen the
More informationPHYSICS 1 Simple Harmonic Motion
Advanced Placement PHYSICS 1 Simple Harmonic Motion Student 014-015 What I Absolutely Have to Know to Survive the AP* Exam Whenever the acceleration of an object is proportional to its displacement and
More informationPhysics 111 Lecture 4 Newton`s Laws
Physics 111 Lecture 4 Newton`s Laws Dr. Ali ÖVGÜN EMU Physics Department www.aovgun.com he Laws of Motion q Newton s first law q Force q Mass q Newton s second law q Newton s third law q Examples Isaac
More informationChapter 7 Kinetic Energy and Work
Prof. Dr. I. Nasser Chapter7_I 14/11/017 Chapter 7 Kinetic Energy and Work Energy: Measure of the ability of a body or system to do work or produce a change, expressed usually in joules or kilowatt hours
More informationPhys101 Second Major-162 Zero Version Coordinator: Dr. Kunwar S. Saturday, March 25, 2017 Page: N Ans:
Coordinator: Dr. Kunwar S. Saturday, March 25, 2017 Page: 1 Q1. Only two horizontal forces act on a 3.0 kg body that can move over a frictionless floor. One force is 20 N, acting due east, and the other
More informationPHYS 1114, Lecture 33, April 10 Contents:
PHYS 1114, Lecture 33, April 10 Contents: 1 This class is o cially cancelled, and has been replaced by the common exam Tuesday, April 11, 5:30 PM. A review and Q&A session is scheduled instead during class
More information3. Kinetics of Particles
3. Kinetics of Particles 3.1 Force, Mass and Acceleration 3.3 Impulse and Momentum 3.4 Impact 1 3.1 Force, Mass and Acceleration We draw two important conclusions from the results of the experiments. First,
More informationLecture Notes for PHY 405 Classical Mechanics
Lecture Notes for PHY 405 Classical Mechanics From Thorton & Marion s Classical Mechanics Prepared by Dr. Joseph M. Hahn Saint Mary s University Department of Astronomy & Physics September 1, 2005 Chapter
More informationThe Laws of Motion. Newton s first law Force Mass Newton s second law Gravitational Force Newton s third law Examples
The Laws of Motion Newton s first law Force Mass Newton s second law Gravitational Force Newton s third law Examples Gravitational Force Gravitational force is a vector Expressed by Newton s Law of Universal
More informationPHYSICS 110A : CLASSICAL MECHANICS HW 1 SOLUTIONS. r = R vt
PHYSICS 11A : CLASSICAL MECHANICS HW 1 SOLUTIONS 2) Taylor 1.46 a) The equations of motion for the puck are: r = R vt φ = Assuming the puck is launched from the position φ =. Technically with the polar
More informationChapter 5: Energy. Energy is one of the most important concepts in the world of science. Common forms of Energy
Chapter 5: Energy Energy is one of the most important concepts in the world of science. Common forms of Energy Mechanical Chemical Thermal Electromagnetic Nuclear One form of energy can be converted to
More information2009 A-level Maths Tutor All Rights Reserved
2 This book is under copyright to A-level Maths Tutor. However, it may be distributed freely provided it is not sold for profit. Contents Newton s Laws 3 connected particles 9 work & energy 18 power &
More informationPOTENTIAL ENERGY AND ENERGY CONSERVATION
7 POTENTIAL ENERGY AND ENERGY CONSERVATION 7.. IDENTIFY: U grav = mgy so ΔU grav = mg( y y ) SET UP: + y is upward. EXECUTE: (a) ΔU = (75 kg)(9.8 m/s )(4 m 5 m) = +6.6 5 J (b) ΔU = (75 kg)(9.8 m/s )(35
More informationGeneral Physics I Work & Energy
General Physics I Work & Energy Forms of Energy Kinetic: Energy of motion. A car on the highway has kinetic energy. We have to remove this energy to stop it. The brakes of a car get HOT! This is an example
More informationThe Concept of Force Newton s First Law and Inertial Frames Mass Newton s Second Law The Gravitational Force and Weight Newton s Third Law Analysis
The Laws of Motion The Concept of Force Newton s First Law and Inertial Frames Mass Newton s Second Law The Gravitational Force and Weight Newton s Third Law Analysis Models using Newton s Second Law Forces
More informationPeriodic Motion. Periodic motion is motion of an object that. regularly repeats
Periodic Motion Periodic motion is motion of an object that regularly repeats The object returns to a given position after a fixed time interval A special kind of periodic motion occurs in mechanical systems
More informationConcept of Force Challenge Problem Solutions
Concept of Force Challenge Problem Solutions Problem 1: Force Applied to Two Blocks Two blocks sitting on a frictionless table are pushed from the left by a horizontal force F, as shown below. a) Draw
More informationDynamics. Dynamics of mechanical particle and particle systems (many body systems)
Dynamics Dynamics of mechanical particle and particle systems (many body systems) Newton`s first law: If no net force acts on a body, it will move on a straight line at constant velocity or will stay at
More informationRutgers University Department of Physics & Astronomy. 01:750:271 Honors Physics I Fall Lecture 20 JJ II. Home Page. Title Page.
Rutgers University Department of Physics & Astronomy 01:750:271 Honors Physics Fall 2015 Lecture 20 Page 1 of 31 1. No quizzes during Thanksgiving week. There will be recitation according to the regular
More informationKinematics 1D Kinematics 2D Dynamics Work and Energy
Kinematics 1D Kinematics 2D Dynamics Work and Energy Kinematics 1 Dimension Kinematics 1 Dimension All about motion problems Frame of Reference orientation of an object s motion Used to anchor coordinate
More informationPhys101 Second Major-152 Zero Version Coordinator: Dr. W. Basheer Monday, March 07, 2016 Page: 1
Phys101 Second Major-15 Zero Version Coordinator: Dr. W. Basheer Monday, March 07, 016 Page: 1 Q1. Figure 1 shows two masses; m 1 = 4.0 and m = 6.0 which are connected by a massless rope passing over a
More informationCh 5 Work and Energy
Ch 5 Work and Energy Energy Provide a different (scalar) approach to solving some physics problems. Work Links the energy approach to the force (Newton s Laws) approach. Mechanical energy Kinetic energy
More informationPHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009
PHYSICS 221, FALL 2009 EXAM #1 SOLUTIONS WEDNESDAY, SEPTEMBER 30, 2009 Note: The unit vectors in the +x, +y, and +z directions of a right-handed Cartesian coordinate system are î, ĵ, and ˆk, respectively.
More informationChapter 8. Potential Energy & Conservation of Energy
Chapter 8 Potential Energy & Conservation of Energy 8.1 Potential Energy Technically, potential energy is energy that can be associated with the configuration (arrangement) of a system of objects that
More informationWork changes Energy. Do Work Son!
1 Work changes Energy Do Work Son! 2 Do Work Son! 3 Work Energy Relationship 2 types of energy kinetic : energy of an object in motion potential: stored energy due to position or stored in a spring Work
More informationChapter 6 Energy and Oscillations
Chapter 6 Energy and Oscillations Conservation of Energy In this chapter we will discuss one of the most important and fundamental principles in the universe. Energy is conserved. This means that in any
More informationWork and kinetic energy. If a net force is applied on an object, the object may
Work and kinetic energy If a net force is applied on an object, the object may CHAPTER 6 WORK AND ENERGY experience a change in position, i.e., a displacement. When a net force is applied over a distance,
More informationCHAPTER 6 WORK AND ENERGY
CHAPTER 6 WORK AND ENERGY ANSWERS TO FOCUS ON CONCEPTS QUESTIONS (e) When the force is perpendicular to the displacement, as in C, there is no work When the force points in the same direction as the displacement,
More informationSummary for last week: Newton s 2 nd Law + 1 st Law
! F resultant = Summary for last week: Newton s 2 nd Law + 1 st Law F! " i = F! 1 + F! 2 +...+ F! N = m! all forces acting on object due to other objects a Object if we measure acceleration in an inertial
More informationChapter 15. Oscillatory Motion
Chapter 15 Oscillatory Motion Part 2 Oscillations and Mechanical Waves Periodic motion is the repeating motion of an object in which it continues to return to a given position after a fixed time interval.
More information24/06/13 Forces ( F.Robilliard) 1
R Fr F W 24/06/13 Forces ( F.Robilliard) 1 Mass: So far, in our studies of mechanics, we have considered the motion of idealised particles moving geometrically through space. Why a particular particle
More informationEngineering Mechanics: Statics in SI Units, 12e
Engineering Mechanics: Statics in SI Units, 12e 3 Equilibrium of a Particle 1 Chapter Objectives Concept of the free-body diagram for a particle Solve particle equilibrium problems using the equations
More informationPhys101 Lectures 9 and 10 Conservation of Mechanical Energy
Phys101 Lectures 9 and 10 Conservation of Mechanical Energy Key points: Conservative and Nonconservative Forces Potential Energy Generalized work-energy principle Mechanical Energy and Its Conservation
More informationNewton s Laws.
Newton s Laws http://mathsforeurope.digibel.be/images Forces and Equilibrium If the net force on a body is zero, it is in equilibrium. dynamic equilibrium: moving relative to us static equilibrium: appears
More informationDynamics: Forces and Newton s Laws of Motion
Lecture 7 Chapter 5 Dynamics: Forces and Newton s Laws of Motion Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Today we are going to discuss: Chapter 5: Force, Mass: Section 5.1
More informationQuiz Number 3 PHYSICS March 11, 2009
Instructions Write your name, student ID and name of your TA instructor clearly on all sheets and fill your name and student ID on the bubble sheet. Solve all multiple choice questions. No penalty is given
More informationDynamics: Forces and Newton s Laws of Motion
Lecture 7 Chapter 5 Physics I Dynamics: Forces and Newton s Laws of Motion Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Today we are going to discuss: Chapter 5: Force, Mass:
More informationWork and Energy Definition of work Examples. Definition of Mechanical Energy. Conservation of Mechanical Energy, Pg 1
Work and Energy Definition of work Examples Work and Energy Today s Agenda Definition of Mechanical Energy Conservation of Mechanical Energy Conservative forces Conservation of Mechanical Energy, Pg 1
More informationTranslational and Rotational Dynamics!
Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.
More information