Mechanical System. Seoul National Univ. School of Mechanical and Aerospace Engineering. Spring 2008

Size: px
Start display at page:

Download "Mechanical System. Seoul National Univ. School of Mechanical and Aerospace Engineering. Spring 2008"

Transcription

1 Mechanical Syste

2 Newton s Laws 1)First law : conservation of oentu no external force no oentu change linear oentu : v Jω angular oentu : dv ) Second law : F = a= dt d T T = J α = J ω dt

3 Three Basic Eleents in Modeling Mechanical Systes i) Inertial eleents ( kinetic energy ): asses: M oents of inertial : J ii) Spring eleents ( Potential energy ) T = k Δθ = k( θ ) 1 θ θ ΔX Torsional spring iii) Daper eleents ( energy dissipation ) Δθ T ΔX b θ θ F = bx T θ T = b θ θ b x x x

4 Exaples of Modeling Mechanical Systes t =, ω() = ω ex1) equation of otion : bw J b dω J = b ω dt dω dω b J + bω = + ω = dt dt J λt λt b λt let, ω( t) = ce λe + e = J b λ = J b t J ω() t = ce, t =, ω() = ω = C ω() t = ω e o b t J ω 1 e =.368ω ω() t w

5 Exaples of Modeling Mechanical Systes ex) Spring-Mass x(t) k t =, x() = x, x() = d x = F = kx dt k x && + kx = && x + x = k k xt () = Acos t+ Bsin t x () = A = x o k k k k k x& () = A sin t + B cos t = B = k xt () = x cos t = x cos ω n t π Period T = [sec] k/ 1 1 k Frequency f = = [ Hz] T π Natural frequency k ωn = π f = [ rad /sec]

6 Exaples of Modeling Mechanical Systes Spring-ass-daper x() x& () = = x x& d x = kx bx& dt b k && x + x& x + = k b b let, = ωn, = ζωn, = ζ k && + & + λ + ζωλ + ωx ω = x+ ζω + =, () = λt nx ωnx xt c e n n x ( ) λ = ζω ± ζω ω = ζω ± ζ 1ω n n n n n

7 Exaples of Modeling Mechanical Systes b k && x + x & x + = x() = x x& () = x& = k b b k let, = ωn, = ζωn, = ζ && x+ x& + x= Laplace Transfor ζω n ω n ( sxs ( ) sx x& ) + ζω ( sxs ( ) x ) + ω Xs ( ) = n n ( s + ζωns+ ωn) X( s) s x ζωnx = ( s + ζω n s+ ω n) X ( s ) = s x + ζω nx X() s = s x + ζω x s n + ζωns+ ωn

8 Exaples of Modeling Mechanical Systes s x + ζω x s x ζω x n n () = = + s + ζωns+ ωn s + ζωns+ ωn s + ζωns+ ωn X s s s + ζω s+ ω : charateristic polynoial n n + ζω s + ω = : characteistic i equation n n ( ) n n n n n s = ζω ± ζω ω = ζω ± ω ζ 1 : charateristic roots 1) ζ < 1 underdaped s = ζω ± ω ζ 1= ζω ± jω 1 ζ n n n n ( s+ ζω ) x ζω x X() s = + ( s+ ζω ) + ω (1 ζ ) ( + ζω ) + ω (1 ζ ) n n n n s n n ζω ζ n xt () = e t xcos 1 ζ ωnt+ x sin ζ ωnt 1 ζ

9 Exaples of Modeling Mechanical Systes ) ζ > 1 overdaped s = ζωn ± ωn ζ X() s = 1 ( s+ ζω ) x ( s + ζω 1)( 1) n ωn ζ s+ ζωn + ωn ζ ( ζωn+ ζ 1 ωn) t ( ζωn ζ 1 ωn) xt () = ke + ke 1 n t 3) ζ = 1 critically daped s = ζωn ( s+ ζωn) x X() s = s + ζω ( ) ( ζω ) t ( ζω ) xt ke kte n n () = + n 1 t

10 Exaples of Modeling Mechanical Systes 1) ζ < 1 Underdaped ζω ζ n x() t = e t xcos 1 ζ ωnt+ x sin ζ ωnt 1 ζ ω n : ζ : natural frequency daping ratio ω d = 1 ζ ωn ) ζ > 1 Overdaped ( ζωn+ ζ 1 ωn) t ( ζωn ζ 1 ωn) xt () = ke + ke 1 λ = ζωn ± ζ 1ω n t x () t ζ > 1 ζ = 1 3) ζ = 1 λ = ζωn xt ke kte n ζωn () = + ζω t 1 t ζ < 1 t

11 Dry Friction (no lubricant) Fμs = Static Friction Force Fμk = Kinetic Friction Force Fμs = μs N μs : Static Friction Coefficient Fμk = μk N k N μk k : Kinetic Friction Coefficient x & = x F μ F if F F = F μssgn( F) if F> F μs μs & F = F sgn( x& ) μ μk

12 ε Friction (with lubricant) bx& + G N sgn( x& ) x& > ε F & μ = F if x < ε and F ( Fs + G N) ( F + s G N)sgn( F) otherwise i.e., if x& < ε and F > ( F + s G N) b: viscous friction coefficient G: load-dependent factor N: noral force Fs: the axiu static friction ε : a sall bound for zero velocity detection

13 Work, Energy, and Power Mechanical work : W = F x [N ] = [Joule] = Force displaceent Energy : capacity or ability to do work. Electrical, Cheical, Mechanical, etc. Mechanical energy : Potential energy position Kinetic energy velocity

14 Potential Energy ex1) F = kx x1 x1 1 dw = Fdx = kx dx dw = kx dx = kx ex) M Mg E h 1 X= = gx (Potential Energy)

15 Kinetic Energy 1 v v : J ω : 1 Jω

16 Work and Energy Syste + External Work = E Energy E1 W E 1 + W = E F v1 v1 l v v E1+ W = E 1 1 v 1 + Fl= v 1 1 v 1 Fl= v

17 Power Power : tie rate & doing work P dw N = dt = sec [ Watt ] Ex) kg V = 7k/h = /s (in 1sec) V = 1 1 v + W = v 1 ()() [ ] W = = N = kn kj W 4 1 N P = = t 1 sec = 4 1 N / s = 4 1 W = 4W 1 hp = W P = 54 hp Power dissipated in a daper P = Fv = bx& x& = bx&

18 An Energy Method for Deriving Equations of Motion Conservative syste : No energy dissipation E + W = E 1 E E = W 1 Kinetic Energy T Potential Energy U Δ(T+U) = ΔW (the change in the total energy) = (the net work done on the syste by the external force) no external force ; ΔW = Δ(T+U) = T+U = constant

19 Exaples of Energy Method x(t) ex1) 1 1 T = x&, U = kx, T + U = C k 1 1 x& + kx = d ( T + U ) =, xx &&& + kxx & = ( x && + kx ) x & = dt x&, ( x && + kx ) = 1 1 d ( T + U ) = bx &, xx &&& + kxx & = bxx && = dt ( x && + kx + bx& ) x& = ex) x(t) x T = x&, U = kx k x&, ( x && + kx+ bx& ) =

20 ex3) Exaples of Energy Method Spring with ass ξ l + x x dξ s l + x k l Potential E x(t) 1 U = kx x+ kx ω n = = k 1 dξ ξ dt = s ( x& ) l + x l + x l+ x l+ x = 1 1 dt s + x& ξ l x d ξ 3 ( ) = x s & ( 3 l + x ) ( l+ x) Kinetic E 1 T = x 1 1 = ( 3 s ) x& T = x& + ( s ) x& = ( + s ) x& ( + s )& x + kx = ω n = k s

Chapter 2: Introduction to Damping in Free and Forced Vibrations

Chapter 2: Introduction to Damping in Free and Forced Vibrations Chapter 2: Introduction to Daping in Free and Forced Vibrations This chapter ainly deals with the effect of daping in two conditions like free and forced excitation of echanical systes. Daping plays an

More information

Dynamics of structures

Dynamics of structures Dynamics of structures 1.2 Viscous damping Luc St-Pierre October 30, 2017 1 / 22 Summary so far We analysed the spring-mass system and found that its motion is governed by: mẍ(t) + kx(t) = 0 k y m x x

More information

Simple Harmonic Motion

Simple Harmonic Motion Reading: Chapter 15 Siple Haronic Motion Siple Haronic Motion Frequency f Period T T 1. f Siple haronic otion x ( t) x cos( t ). Aplitude x Phase Angular frequency Since the otion returns to its initial

More information

which proves the motion is simple harmonic. Now A = a 2 + b 2 = =

which proves the motion is simple harmonic. Now A = a 2 + b 2 = = Worked out Exaples. The potential energy function for the force between two atos in a diatoic olecules can be expressed as follows: a U(x) = b x / x6 where a and b are positive constants and x is the distance

More information

Definition of Work, The basics

Definition of Work, The basics Physics 07 Lecture 16 Lecture 16 Chapter 11 (Work) v Eploy conservative and non-conservative forces v Relate force to potential energy v Use the concept of power (i.e., energy per tie) Chapter 1 v Define

More information

CHECKLIST. r r. Newton s Second Law. natural frequency ω o (rad.s -1 ) (Eq ) a03/p1/waves/waves doc 9:19 AM 29/03/05 1

CHECKLIST. r r. Newton s Second Law. natural frequency ω o (rad.s -1 ) (Eq ) a03/p1/waves/waves doc 9:19 AM 29/03/05 1 PHYS12 Physics 1 FUNDAMENTALS Module 3 OSCILLATIONS & WAVES Text Physics by Hecht Chapter 1 OSCILLATIONS Sections: 1.5 1.6 Exaples: 1.6 1.7 1.8 1.9 CHECKLIST Haronic otion, periodic otion, siple haronic

More information

SIMPLE HARMONIC MOTION: NEWTON S LAW

SIMPLE HARMONIC MOTION: NEWTON S LAW SIMPLE HARMONIC MOTION: NEWTON S LAW siple not siple PRIOR READING: Main 1.1, 2.1 Taylor 5.1, 5.2 http://www.yoops.org/twocw/it/nr/rdonlyres/physics/8-012fall-2005/7cce46ac-405d-4652-a724-64f831e70388/0/chp_physi_pndul.jpg

More information

Work, Energy and Momentum

Work, Energy and Momentum Work, Energy and Moentu Work: When a body oves a distance d along straight line, while acted on by a constant force of agnitude F in the sae direction as the otion, the work done by the force is tered

More information

Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter.

Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. Chapter 3 Mechanical Systems A. Bazoune 3.1 INRODUCION Mathematical Modeling and response analysis of mechanical systems are the subjects of this chapter. 3. MECHANICAL ELEMENS Any mechanical system consists

More information

Physics 41 HW Set 1 Chapter 15 Serway 7 th Edition

Physics 41 HW Set 1 Chapter 15 Serway 7 th Edition Physics HW Set Chapter 5 Serway 7 th Edition Conceptual Questions:, 3, 5,, 6, 9 Q53 You can take φ = π, or equally well, φ = π At t= 0, the particle is at its turning point on the negative side of equilibriu,

More information

PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2

PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2 PHYSICS 110A : CLASSICAL MECHANICS MIDTERM EXAM #2 [1] Two blocks connected by a spring of spring constant k are free to slide frictionlessly along a horizontal surface, as shown in Fig. 1. The unstretched

More information

FOUNDATION STUDIES EXAMINATIONS January 2016

FOUNDATION STUDIES EXAMINATIONS January 2016 1 FOUNDATION STUDIES EXAMINATIONS January 2016 PHYSICS Seester 2 Exa July Fast Track Tie allowed 2 hours for writing 10 inutes for reading This paper consists of 4 questions printed on 11 pages. PLEASE

More information

Periodic Motion is everywhere

Periodic Motion is everywhere Lecture 19 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand and use energy conservation

More information

Lecture #8-3 Oscillations, Simple Harmonic Motion

Lecture #8-3 Oscillations, Simple Harmonic Motion Lecture #8-3 Oscillations Siple Haronic Motion So far we have considered two basic types of otion: translation and rotation. But these are not the only two types of otion we can observe in every day life.

More information

Work and Kinetic Energy

Work and Kinetic Energy Chapter 6 Work and Kinetic Energy PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Copyright 2008 Pearson Education Inc., publishing

More information

Physics 2210 Fall smartphysics 20 Conservation of Angular Momentum 21 Simple Harmonic Motion 11/23/2015

Physics 2210 Fall smartphysics 20 Conservation of Angular Momentum 21 Simple Harmonic Motion 11/23/2015 Physics 2210 Fall 2015 sartphysics 20 Conservation of Angular Moentu 21 Siple Haronic Motion 11/23/2015 Exa 4: sartphysics units 14-20 Midter Exa 2: Day: Fri Dec. 04, 2015 Tie: regular class tie Section

More information

1 (40) Gravitational Systems Two heavy spherical (radius 0.05R) objects are located at fixed positions along

1 (40) Gravitational Systems Two heavy spherical (radius 0.05R) objects are located at fixed positions along (40) Gravitational Systes Two heavy spherical (radius 0.05) objects are located at fixed positions along 2M 2M 0 an axis in space. The first ass is centered at r = 0 and has a ass of 2M. The second ass

More information

Chapter 2 SDOF Vibration Control 2.1 Transfer Function

Chapter 2 SDOF Vibration Control 2.1 Transfer Function Chapter SDOF Vibration Control.1 Transfer Function mx ɺɺ( t) + cxɺ ( t) + kx( t) = F( t) Defines the transfer function as output over input X ( s) 1 = G( s) = (1.39) F( s) ms + cs + k s is a complex number:

More information

T m. Fapplied. Thur Oct 29. ω = 2πf f = (ω/2π) T = 1/f. k m. ω =

T m. Fapplied. Thur Oct 29. ω = 2πf f = (ω/2π) T = 1/f. k m. ω = Thur Oct 9 Assignent 10 Mass-Spring Kineatics (x, v, a, t) Dynaics (F,, a) Tie dependence Energy Pendulu Daping and Resonances x Acos( ωt) = v = Aω sin( ωt) a = Aω cos( ωt) ω = spring k f spring = 1 k

More information

0J2 - Mechanics Lecture Notes 2

0J2 - 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 information

DRAFT. Memo. Contents. To whom it may concern SVN: Jan Mooiman +31 (0) nl

DRAFT. Memo. Contents. To whom it may concern SVN: Jan Mooiman +31 (0) nl Meo To To who it ay concern Date Reference Nuber of pages 219-1-16 SVN: 5744 22 Fro Direct line E-ail Jan Mooian +31 )88 335 8568 jan.ooian@deltares nl +31 6 4691 4571 Subject PID controller ass-spring-daper

More information

Physics 2101 S c e t c i cti n o 3 n 3 March 31st Announcements: Quiz today about Ch. 14 Class Website:

Physics 2101 S c e t c i cti n o 3 n 3 March 31st Announcements: Quiz today about Ch. 14 Class Website: Physics 2101 Section 3 March 31 st Announcements: Quiz today about Ch. 14 Class Website: http://www.phys.lsu.edu/classes/spring2010/phys2101 3/ http://www.phys.lsu.edu/~jzhang/teaching.html Simple Harmonic

More information

and ζ in 1.1)? 1.2 What is the value of the magnification factor M for system A, (with force frequency ω = ωn

and ζ in 1.1)? 1.2 What is the value of the magnification factor M for system A, (with force frequency ω = ωn EN40: Dynais and Vibrations Hoework 6: Fored Vibrations, Rigid Body Kineatis Due Friday April 7, 017 Shool of Engineering Brown University 1. Syste A in the figure is ritially daped. The aplitude of the

More information

+ 1 2 mv 2. Since no forces act on the system in the x-direction, linear momentum in x-direction is conserved: (2) 0 = mv A2. + Rω 2.

+ 1 2 mv 2. Since no forces act on the system in the x-direction, linear momentum in x-direction is conserved: (2) 0 = mv A2. + Rω 2. ME 74 Spring 018 Final Exaination Proble 1 Given: hoogeneous dis of ass and outer radius R is able to roll without slipping on the curved upper surface of a cart. art (of ass ) is able to ove along a sooth,

More information

VTU-NPTEL-NMEICT Project

VTU-NPTEL-NMEICT Project MODULE-II --- SINGLE DOF FREE S VTU-NPTEL-NMEICT Project Progress Report The Project on Development of Remaining Three Quadrants to NPTEL Phase-I under grant in aid NMEICT, MHRD, New Delhi SME Name : Course

More information

Chapter 5 Oscillatory Motion

Chapter 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 information

Exam 2 Spring 2014

Exam 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 information

In the session you will be divided into groups and perform four separate experiments:

In the session you will be divided into groups and perform four separate experiments: Mechanics Lab (Civil Engineers) Nae (please print): Tutor (please print): Lab group: Date of lab: Experients In the session you will be divided into groups and perfor four separate experients: (1) air-track

More information

Vibrations: Two Degrees of Freedom Systems - Wilberforce Pendulum and Bode Plots

Vibrations: Two Degrees of Freedom Systems - Wilberforce Pendulum and Bode Plots .003J/.053J Dynaics and Control, Spring 007 Professor Peacock 5/4/007 Lecture 3 Vibrations: Two Degrees of Freedo Systes - Wilberforce Pendulu and Bode Plots Wilberforce Pendulu Figure : Wilberforce Pendulu.

More information

Classical Mechanics Small Oscillations

Classical Mechanics Small Oscillations Classical Mechanics Sall Oscillations Dipan Kuar Ghosh UM-DAE Centre for Excellence in Basic Sciences, Kalina Mubai 400098 Septeber 4, 06 Introduction When a conservative syste is displaced slightly fro

More information

Kinetics of Rigid (Planar) Bodies

Kinetics of Rigid (Planar) Bodies Kinetics of Rigi (Planar) Boies Types of otion Rectilinear translation Curvilinear translation Rotation about a fixe point eneral planar otion Kinetics of a Syste of Particles The center of ass for a syste

More information

EN40: Dynamics and Vibrations. Midterm Examination Tuesday March

EN40: Dynamics and Vibrations. Midterm Examination Tuesday March EN4: Dynaics and Vibrations Midter Exaination Tuesday March 8 16 School of Engineering Brown University NAME: General Instructions No collaboration of any kind is peritted on this exaination. You ay bring

More information

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx

Chapter 12. Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx Chapter 1 Lecture Notes Chapter 1 Oscillatory Motion Recall that when a spring is stretched a distance x, it will pull back with a force given by: F = -kx When the mass is released, the spring will pull

More information

Question number 1 to 8 carries 2 marks each, 9 to 16 carries 4 marks each and 17 to 18 carries 6 marks each.

Question number 1 to 8 carries 2 marks each, 9 to 16 carries 4 marks each and 17 to 18 carries 6 marks each. IIT-JEE5-PH-1 FIITJEE Solutions to IITJEE 5 Mains Paper Tie: hours Physics Note: Question nuber 1 to 8 carries arks each, 9 to 16 carries 4 arks each and 17 to 18 carries 6 arks each. Q1. whistling train

More information

Q5 We know that a mass at the end of a spring when displaced will perform simple m harmonic oscillations with a period given by T = 2!

Q5 We know that a mass at the end of a spring when displaced will perform simple m harmonic oscillations with a period given by T = 2! Chapter 4.1 Q1 n oscillation is any otion in which the displaceent of a particle fro a fixed point keeps changing direction and there is a periodicity in the otion i.e. the otion repeats in soe way. In

More information

Chapter 1: Basics of Vibrations for Simple Mechanical Systems

Chapter 1: Basics of Vibrations for Simple Mechanical Systems Chapter 1: Basics of Vibrations for Siple Mechanical Systes Introduction: The fundaentals of Sound and Vibrations are part of the broader field of echanics, with strong connections to classical echanics,

More information

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is

Oscillations. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring of constant k is Dr. Alain Brizard College Physics I (PY 10) Oscillations Textbook Reference: Chapter 14 sections 1-8. Simple Harmonic Motion of a Mass on a Spring The equation of motion for a mass m is attached to a spring

More information

Simple Harmonic Motion

Simple Harmonic Motion Siple Haronic Motion Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and Departent of Physics, HKBU Departent of Physics Siple haronic otion In echanical physics,

More information

Dynamical Systems. Mechanical Systems

Dynamical Systems. Mechanical Systems EE/ME/AE324: Dynamical Systems Chapter 2: Modeling Translational Mechanical Systems Common Variables Used Assumes 1 DoF per mass, i.e., all motion scalar Displacement: x ()[ t [m] Velocity: dx() t vt ()

More information

= T. Oscillations and Waves. Example of an Oscillating System IB 12 IB 12

= T. Oscillations and Waves. Example of an Oscillating System IB 12 IB 12 Oscillation: the vibration of an object Oscillations and Waves Eaple of an Oscillating Syste A ass oscillates on a horizontal spring without friction as shown below. At each position, analyze its displaceent,

More information

EN40: Dynamics and Vibrations. Final Examination Monday May : 2pm-5pm

EN40: Dynamics and Vibrations. Final Examination Monday May : 2pm-5pm EN40: Dynaics and Vibrations Final Exaination Monday May 13 013: p-5p School of Engineering Brown University NAME: General Instructions No collaboration of any kind is peritted on this exaination. You

More information

ME Machine Design I. FINAL EXAM. OPEN BOOK AND CLOSED NOTES. Friday, May 8th, 2009

ME Machine Design I. FINAL EXAM. OPEN BOOK AND CLOSED NOTES. Friday, May 8th, 2009 ME 5 - Machine Design I Spring Seester 009 Nae Lab. Div. FINAL EXAM. OPEN BOOK AND LOSED NOTES. Friday, May 8th, 009 Please use the blank paper for your solutions. Write on one side of the paper only.

More information

Physics 207 Lecture 18. Physics 207, Lecture 18, Nov. 3 Goals: Chapter 14

Physics 207 Lecture 18. Physics 207, Lecture 18, Nov. 3 Goals: Chapter 14 Physics 07, Lecture 18, Nov. 3 Goals: Chapter 14 Interrelate the physics and atheatics of oscillations. Draw and interpret oscillatory graphs. Learn the concepts of phase and phase constant. Understand

More information

Vibrations: Second Order Systems with One Degree of Freedom, Free Response

Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single Degree of Freedom System 1.003J/1.053J Dynamics and Control I, Spring 007 Professor Thomas Peacock 5//007 Lecture 0 Vibrations: Second Order Systems with One Degree of Freedom, Free Response Single

More information

5/09/06 PHYSICS 213 Exam #1 NAME FEYNMAN Please write down your name also on the back side of the last page

5/09/06 PHYSICS 213 Exam #1 NAME FEYNMAN Please write down your name also on the back side of the last page 5/09/06 PHYSICS 13 Exa #1 NAME FEYNMAN Please write down your nae also on the back side of the last page 1 he figure shows a horizontal planks of length =50 c, and ass M= 1 Kg, pivoted at one end. he planks

More information

Actuators & Mechanisms Actuator sizing

Actuators & Mechanisms Actuator sizing Course Code: MDP 454, Course Nae:, Second Seester 2014 Actuators & Mechaniss Actuator sizing Contents - Modelling of Mechanical Syste - Mechaniss and Drives The study of Mechatronics systes can be divided

More information

l1, l2, l3, ln l1 + l2 + l3 + ln

l1, 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 information

Physics 4A Solutions to Chapter 15 Homework

Physics 4A Solutions to Chapter 15 Homework Physics 4A Solutions to Chapter 15 Hoework Chapter 15 Questions:, 8, 1 Exercises & Probles 6, 5, 31, 41, 59, 7, 73, 88, 90 Answers to Questions: Q 15- (a) toward -x (b) toward +x (c) between -x and 0 (d)

More information

Physics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information

Physics 121, April 3, Equilibrium and Simple Harmonic Motion. Physics 121. April 3, Physics 121. April 3, Course Information Physics 121, April 3, 2008. Equilibrium and Simple Harmonic Motion. Physics 121. April 3, 2008. Course Information Topics to be discussed today: Requirements for Equilibrium (a brief review) Stress and

More information

Faculty of Computers and Information. Basic Science Department

Faculty of Computers and Information. Basic Science Department 18--018 FCI 1 Faculty of Computers and Information Basic Science Department 017-018 Prof. Nabila.M.Hassan 18--018 FCI Aims of Course: The graduates have to know the nature of vibration wave motions with

More information

ROTATIONAL MOTION FROM TRANSLATIONAL MOTION

ROTATIONAL MOTION FROM TRANSLATIONAL MOTION ROTATIONAL MOTION FROM TRANSLATIONAL MOTION Velocity Acceleration 1-D otion 3-D otion Linear oentu TO We have shown that, the translational otion of a acroscopic object is equivalent to the translational

More information

OSCILLATIONS AND WAVES

OSCILLATIONS AND WAVES OSCILLATIONS AND WAVES OSCILLATION IS AN EXAMPLE OF PERIODIC MOTION No stories this tie, we are going to get straight to the topic. We say that an event is Periodic in nature when it repeats itself in

More information

NAME NUMBER SEC. PHYCS 101 SUMMER 2001/2002 FINAL EXAME:24/8/2002. PART(I) 25% PART(II) 15% PART(III)/Lab 8% ( ) 2 Q2 Q3 Total 40%

NAME NUMBER SEC. PHYCS 101 SUMMER 2001/2002 FINAL EXAME:24/8/2002. PART(I) 25% PART(II) 15% PART(III)/Lab 8% ( ) 2 Q2 Q3 Total 40% NAME NUMER SEC. PHYCS 101 SUMMER 2001/2002 FINAL EXAME:24/8/2002 PART(I) 25% PART(II) 15% PART(III)/Lab 8% ( ) 2.5 Q1 ( ) 2 Q2 Q3 Total 40% Use the followings: Magnitude of acceleration due to gravity

More information

Chapter 14 Periodic Motion

Chapter 14 Periodic Motion Chapter 14 Periodic Motion 1 Describing Oscillation First, we want to describe the kinematical and dynamical quantities associated with Simple Harmonic Motion (SHM), for example, x, v x, a x, and F x.

More information

Chapter 7 Work and Kinetic Energy. Copyright 2010 Pearson Education, Inc.

Chapter 7 Work and Kinetic Energy. Copyright 2010 Pearson Education, Inc. Chapter 7 Work and Kinetic Energy Units of Chapter 7 Work Done by a Constant Force Kinetic Energy and the Work-Energy Theorem Work Done by a Variable Force Power 7-1 Work Done by a Constant Force The definition

More information

Physics 201, Lecture 15

Physics 201, Lecture 15 Physics 0, Lecture 5 Today s Topics q More on Linear Moentu And Collisions Elastic and Perfect Inelastic Collision (D) Two Diensional Elastic Collisions Exercise: Billiards Board Explosion q Multi-Particle

More information

Modern Control Systems (ECEG-4601) Instructor: Andinet Negash. Chapter 1 Lecture 3: State Space, II

Modern Control Systems (ECEG-4601) Instructor: Andinet Negash. Chapter 1 Lecture 3: State Space, II Modern Control Systes (ECEG-46) Instructor: Andinet Negash Chapter Lecture 3: State Space, II Eaples Eaple 5: control o liquid levels: in cheical plants, it is oten necessary to aintain the levels o liquids.

More information

A. B. C. D. E. v x. ΣF x

A. 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 information

A body of unknown mass is attached to an ideal spring with force constant 123 N/m. It is found to vibrate with a frequency of

A body of unknown mass is attached to an ideal spring with force constant 123 N/m. It is found to vibrate with a frequency of Chapter 14 [ Edit ] Overview Suary View Diagnostics View Print View with Answers Chapter 14 Due: 11:59p on Sunday, Noveber 27, 2016 To understand how points are awarded, read the Grading Policy for this

More information

PS 11 GeneralPhysics I for the Life Sciences

PS 11 GeneralPhysics I for the Life Sciences PS GeneralPhysics I for the Life Sciences W O R K N D E N E R G Y D R. E N J M I N C H N S S O C I T E P R O F E S S O R P H Y S I C S D E P R T M E N T J N U R Y 0 4 Questions and Probles for Conteplation

More information

Kinetic Energy and Work

Kinetic Energy and Work Kinetic Energy and Work 8.01 W06D1 Today s Readings: Chapter 13 The Concept of Energy and Conservation of Energy, Sections 13.1-13.8 Announcements Problem Set 4 due Week 6 Tuesday at 9 pm in box outside

More information

3. Period Law: Simplified proof for circular orbits Equate gravitational and centripetal forces

3. Period Law: Simplified proof for circular orbits Equate gravitational and centripetal forces Physics 106 Lecture 10 Kepler s Laws and Planetary Motion-continued SJ 7 th ed.: Chap 1., 1.6 Kepler s laws of planetary otion Orbit Law Area Law Period Law Satellite and planetary orbits Orbits, potential,

More information

AP Physics. Harmonic Motion. Multiple Choice. Test E

AP Physics. Harmonic Motion. Multiple Choice. Test E AP Physics Harmonic Motion Multiple Choice Test E A 0.10-Kg block is attached to a spring, initially unstretched, of force constant k = 40 N m as shown below. The block is released from rest at t = 0 sec.

More information

PH 221-1D Spring Oscillations. Lectures Chapter 15 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition)

PH 221-1D Spring Oscillations. Lectures Chapter 15 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) PH 1-1D Spring 013 Oscillations Lectures 35-37 Chapter 15 (Halliday/Resnick/Walker, Fundaentals of Physics 9 th edition) 1 Chapter 15 Oscillations In this chapter we will cover the following topics: Displaceent,

More information

CHAPTER 12 OSCILLATORY MOTION

CHAPTER 12 OSCILLATORY MOTION CHAPTER 1 OSCILLATORY MOTION Before starting the discussion of the chapter s concepts it is worth to define some terms we will use frequently in this chapter: 1. The period of the motion, T, is the time

More information

Lecture Notes for PHY 405 Classical Mechanics

Lecture 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 information

Chapter 7. Work and Kinetic Energy

Chapter 7. Work and Kinetic Energy Chapter 7 Work and Kinetic Energy P. Lam 7_16_2018 Learning Goals for Chapter 7 To understand the concept of kinetic energy (energy of motion) To understand the meaning of work done by a force. To apply

More information

Theoretical Basis of Modal Analysis

Theoretical Basis of Modal Analysis American Journal of Mechanical Engineering, 03, Vol., No. 7, 73-79 Available online at http://pubs.sciepub.com/ajme//7/4 Science and Education Publishing DOI:0.69/ajme--7-4 heoretical Basis of Modal Analysis

More information

WYSE Academic Challenge Sectional Physics 2006 Solution Set

WYSE Academic Challenge Sectional Physics 2006 Solution Set WYSE Acadeic Challenge Sectional Physics 6 Solution Set. Correct answer: d. Using Newton s nd Law: r r F 6.N a.kg 6./s.. Correct answer: c. 6. sin θ 98. 3. Correct answer: b. o 37.8 98. N 6. N Using Newton

More information

EN40: Dynamics and Vibrations. Final Examination Tuesday May 15, 2011

EN40: Dynamics and Vibrations. Final Examination Tuesday May 15, 2011 EN40: ynaics and Vibrations Final Exaination Tuesday May 15, 011 School of Engineering rown University NME: General Instructions No collaboration of any ind is peritted on this exaination. You ay use double

More information

Chapter 4. Energy. Work Power Kinetic Energy Potential Energy Conservation of Energy. W = Fs Work = (force)(distance)

Chapter 4. Energy. Work Power Kinetic Energy Potential Energy Conservation of Energy. W = Fs Work = (force)(distance) Chapter 4 Energy In This Chapter: Work Kinetic Energy Potential Energy Conservation of Energy Work Work is a measure of the amount of change (in a general sense) that a force produces when it acts on a

More information

WORK, ENERGY & POWER Work scalar W = F S Cosθ Unit of work in SI system Work done by a constant force

WORK, 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 information

Kinematics (special case) Dynamics gravity, tension, elastic, normal, friction. Energy: kinetic, potential gravity, spring + work (friction)

Kinematics (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

Dr. Gundersen Phy 205DJ Test 2 22 March 2010

Dr. Gundersen Phy 205DJ Test 2 22 March 2010 Signature: Idnumber: Name: Do only four out of the five problems. The first problem consists of five multiple choice questions. If you do more only your FIRST four answered problems will be graded. Clearly

More information

Modeling and Simulation Revision III D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N

Modeling and Simulation Revision III D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N Modeling and Simulation Revision III D R. T A R E K A. T U T U N J I P H I L A D E L P H I A U N I V E R S I T Y, J O R D A N 0 1 4 Block Diagrams Block diagram models consist of two fundamental objects:

More information

Second Order Systems

Second Order Systems Second Order Systems independent energy storage elements => Resonance: inertance & capacitance trade energy, kinetic to potential Example: Automobile Suspension x z vertical motions suspension spring shock

More information

Physics 1401V October 28, 2016 Prof. James Kakalios Quiz No. 2

Physics 1401V October 28, 2016 Prof. James Kakalios Quiz No. 2 This is a closed book, closed notes, quiz. Only simple (non-programmable, nongraphing) calculators are permitted. Define all symbols and justify all mathematical expressions used. Make sure to state all

More information

Two-Mass, Three-Spring Dynamic System Investigation Case Study

Two-Mass, Three-Spring Dynamic System Investigation Case Study Two-ass, Three-Spring Dynamic System Investigation Case Study easurements, Calculations, anufacturer's Specifications odel Parameter Identification Which Parameters to Identify? What Tests to Perform?

More information

Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be.

Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be. Chapter 4 Energy and Stability 4.1 Energy in 1D Consider a particle in 1D at position x(t), subject to a force F (x), so that mẍ = F (x). Define the kinetic energy to be T = 1 2 mẋ2 and the potential energy

More information

Oscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum

Oscillations. Phys101 Lectures 28, 29. Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Phys101 Lectures 8, 9 Oscillations Key points: Simple Harmonic Motion (SHM) SHM Related to Uniform Circular Motion The Simple Pendulum Ref: 11-1,,3,4. Page 1 Oscillations of a Spring If an object oscillates

More information

Chapter 15 - Oscillations

Chapter 15 - Oscillations The pendulum of the mind oscillates between sense and nonsense, not between right and wrong. -Carl Gustav Jung David J. Starling Penn State Hazleton PHYS 211 Oscillatory motion is motion that is periodic

More information

CE 102: Engineering Mechanics. Minimum Potential Energy

CE 102: Engineering Mechanics. Minimum Potential Energy CE 10: Engineering Mechanics Minimum Potential Energy Work of a Force During a Finite Displacement Work of a force corresponding to an infinitesimal displacement, Work of a force corresponding to a finite

More information

WileyPLUS Assignment 3. Next Week

WileyPLUS Assignment 3. Next Week WileyPLUS Assignent 3 Chapters 6 & 7 Due Wednesday, Noveber 11 at 11 p Next Wee No labs of tutorials Reebrance Day holiday on Wednesday (no classes) 24 Displaceent, x Mass on a spring ωt = 2π x = A cos

More information

Oscillatory Motion SHM

Oscillatory Motion SHM Chapter 15 Oscillatory Motion SHM Dr. Armen Kocharian 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

More information

Chapter 13: Oscillatory Motions

Chapter 13: Oscillatory Motions Chapter 13: Oscillatory Motions Simple harmonic motion Spring and Hooe s law When a mass hanging from a spring and in equilibrium, the Newton s nd law says: Fy ma Fs Fg 0 Fs Fg This means the force due

More information

2. Which of the following best describes the relationship between force and potential energy?

2. Which of the following best describes the relationship between force and potential energy? Work/Energy with Calculus 1. An object oves according to the function x = t 5/ where x is the distance traveled and t is the tie. Its kinetic energy is proportional to (A) t (B) t 5/ (C) t 3 (D) t 3/ (E)

More information

Chapter 7 Kinetic Energy and Work

Chapter 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 information

19. LC and RLC Oscillators

19. LC and RLC Oscillators University of Rhode Island Digitaloons@URI PHY 204: Eleentary Physics II Physics ourse Materials 2015 19. L and RL Oscillators Gerhard Müller University of Rhode Island, guller@uri.edu reative oons License

More information

Comb Resonator Design (4)

Comb Resonator Design (4) Lecture 8: Comb Resonator Design (4) - Intro. to Fluidic Dynamics (Damping) - School of Electrical Engineering and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory Email:

More information

Chapter 6: Work and Kinetic Energy

Chapter 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 information

Physics 231 Lecture 8

Physics 231 Lecture 8 Shelf-life of ideas at the Bookstore Physics 31 Lecture 8 Mi Main points of today s lecture: Frictional forces: kinetic friction: fk = μk N static friction f < μs N s Work: W = FΔs cos( θ ) = FxΔx Kinetic

More information

Vibrations and Waves MP205, Assignment 4 Solutions

Vibrations and Waves MP205, Assignment 4 Solutions Vibrations and Waves MP205, Assignment Solutions 1. Verify that x = Ae αt cos ωt is a possible solution of the equation and find α and ω in terms of γ and ω 0. [20] dt 2 + γ dx dt + ω2 0x = 0, Given x

More information

Chapter 15. Oscillatory Motion

Chapter 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 information

Contents. Dynamics and control of mechanical systems. Focus on

Contents. Dynamics and control of mechanical systems. Focus on Dynamics and control of mechanical systems Date Day 1 (01/08) Day 2 (03/08) Day 3 (05/08) Day 4 (07/08) Day 5 (09/08) Day 6 (11/08) Content Review of the basics of mechanics. Kinematics of rigid bodies

More information

Physics 140 D100 Midterm Exam 2 Solutions 2017 Nov 10

Physics 140 D100 Midterm Exam 2 Solutions 2017 Nov 10 There are 10 ultiple choice questions. Select the correct answer for each one and ark it on the bubble for on the cover sheet. Each question has only one correct answer. (2 arks each) 1. An inertial reference

More information

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION

EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION 1 EQUIVALENT SINGLE-DEGREE-OF-FREEDOM SYSTEM AND FREE VIBRATION The course on Mechanical Vibration is an important part of the Mechanical Engineering undergraduate curriculum. It is necessary for the development

More information

Mechanics 2 Aide Memoire. Work done = or Fx for a constant force, F over a distance, x. = or Pt for constant power, P over t seconds

Mechanics 2 Aide Memoire. Work done = or Fx for a constant force, F over a distance, x. = or Pt for constant power, P over t seconds Mechanics 2 Aide Memoire Work done measured in Joules Energy lost due to overcoming a force (no work done if force acts perpendicular to the direction of motion) Work done = or Fx for a constant force,

More information

dt dt THE AIR TRACK (II)

dt dt THE AIR TRACK (II) THE AIR TRACK (II) References: [] The Air Track (I) - First Year Physics Laoratory Manual (PHY38Y and PHYY) [] Berkeley Physics Laoratory, nd edition, McGraw-Hill Book Copany [3] E. Hecht: Physics: Calculus,

More information

In this chapter we will start the discussion on wave phenomena. We will study the following topics:

In this chapter we will start the discussion on wave phenomena. We will study the following topics: Chapter 16 Waves I In this chapter we will start the discussion on wave phenoena. We will study the following topics: Types of waves Aplitude, phase, frequency, period, propagation speed of a wave Mechanical

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE 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 information