MECH370: Modelling, Simulation and Analysis of Physical Systems

Size: px
Start display at page:

Download "MECH370: Modelling, Simulation and Analysis of Physical Systems"

Transcription

1 MECH370: Modelling, Siultion nd Anlysis o Physicl Systes Chpter Modeling o Trnsltionl Mechnicl Syste Eleents nd eleent lws o trnsltionl echnicl systes Free body digr (FD Interconnection lws Obtining the syste odel Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes, Youin Zhng (CU

2 Modelling (Ch.,3,4,5,6,9,0,, Siultion (4 Anlysis (7,8 Course Outline. Deinition nd clssiiction o dynic systes (chpter. Trnsltionl echnicl systes (chpter 3. Stndrd ors or syste odels (chpter 3 4. loc digrs nd coputer siultion with Mtlb/Siulin (chpter 4 5. Rottionl echnicl systes (chpter 5 6. Electricl systes (chpter 6 7. Anlysis nd solution techniques or liner systes (chpters 7 nd 8 8. Developing liner odel (chpter 9 9. Electroechnicl systes (chpter 0 0.Therl nd luid systes (chpters, Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes

3 Why Mtheticl Models re Needed? Review Anlogous Systes Cn hve the se theticl odel or dierent types o physicl systes Coon nlysis ethods nd tools cn be used Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3

4 Procedure o Syste Modeling Review Divide the syste into idelized coponents Apply physicl lws to the eleents Apply interconnection lws between eleents Cobine the equtions to obtin the odel Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4

5 Overview o Eleent Models in Physicl Systes Mechnicl Trnsltionl Models M orce/velocity orce/position Mss M dv/dt M d /dt Dper (Viscous riction Spring (Stiness v v dt d/dt Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5

6 Overview o Eleent Models in Physicl Systes Mechnicl Rottionl Models J torque/velocity torque/position T, θ Inerti T J dω/dt T J dθ /dt T, θ Viscous riction T ω T dθ/dt s Stiness T s ω dt T s θ T, θ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6

7 Overview o Eleent Models in Physicl Systes Electricl Coponent Models i + v _ i + v _ voltge/current voltge/chrge Inductnce v L di/dt v L dq /dt Resistnce v R i v R dq/dt i + v _ Cpcitnce v /C i dt v /C q Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7

8 Overview o Eleent Models in Physicl Systes Trnsortion Models i i v N N v, Trnsorer v N i N v N i N, L L Lever L L L L T, θ N Gers T N θ N T N θ N T, θ N Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8

9 Mtheticl Modelling o Mechnicl Systes Eleentry prts A ens or storing inetic energy (ss or inerti A ens or storing potentil energy (spring or elsticity A ens by which energy is grdully dissipted (dper Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9

10 Mtheticl Modelling o Mechnicl Systes Motion in echnicl systes cn be Trnsltionl Rottionl, or Cobintion o bove Mechnicl systes cn be o two types Trnsltionl systes Rottionl systes Vribles tht describe otion Displceent, Velocity, v Accelertion, Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 0

11 Modeling o trnsltionl echnicl systes : displceent ( v: velocity (/sec Vribles : ccelertion (/sec : orce (N P: power (N/sec, Wtt W: wor (energy (N, J v & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes d dt dv d d ( dt dt dt P v & W( t & dw dt All these vribles re unctions o tie, t. Power is deined to be the rte t which energy is supplied or dissipted. W( t 0 + t t 0 P( t dt M

12 Modeling o trnsltionl echnicl systes Vribles (cont d The energy supplied between tie t o nd t is t t 0 P( t dt The totl energy supplied ro tie t 0 up to ny lter tie is W( t W( t 0 + t t 0 P( t dt N (Newton-Meter J (Joule Wtt Joule per Second Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes

13 Modeling o trnsltionl echnicl systes Eleents Three priry eleents o interest Mss (inerti (or M Stiness (spring Friction - Dissiption (dper Usully we del with equivlent,, Distributed ss -> luped ss Luped preters Mss intins otion Stiness restores otion Dping eliintes otion Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3

14 Eleents Mss Newton s second lw: su o orces cting on body tie rte o chnge o oentu v d d dv ( v, i 0 dt dt dt Assuptions:. otions deined with respect to inertil reerence re. sclr quntities ( degree o reedo g Energy v Kinetic EK Potentil E P gh Initil conditions v 0, h 0 Grvity Norl orce N Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4

15 Eleents (cont d Stiness (N/ Stiness is the resistnce o n elstic body to delection or deortion by n pplied orce Most coon: idel spring d 0 + d 0 elongtion cused by d ( t length o spring when no orce is pplied d 0 + ( t ( t : stiness constnt d ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5 d 0

16 Eleents (cont d The liner spring is n pproition o soething lie Liner Non-liner Multiple pplied orces: ( Δ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6

17 Eleents (cont d Dping (N-s/ Also nown s viscous riction or liner riction. Friction is the orce tht opposes the reltive otion or tendency o such otion o two surces in contct Δv, where Δv v -v nd viscosity constnt/coeicient v v v Above let: is proportionl to contct re nd viscosity o oil. is inversely proportionl to the thicness o il. Above right: is sll enough to be neglected (this is lwys n pproition. Dping is used to odel dshpot (dper, e.g. shoc bsorbers on crs. Dping is used to odel dshpot (dper e.g. shoc bsorbers on crs. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7 v

18 Eleent Lws Mss: Newton s nd Lw Stiness (Spring: d dt ( dv dt i i Friction (Dping: ( v v Δv : viscosity constnt, unit: N-s/ Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8 v v v v v No riction Idelized shoc bsorber (dshpot

19 Eleent Lws (cont d Viscous riction: Δv Δv Coulob (dry riction: Asign( Δv A -A Δv Drg: CΔ Δv Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9

20 Interconnection Lws Deterine how to connect eleents D Alebert s lw Just re-stteent o Newton s nd lw, suing eternlly pplied orces to ss ( or ( 0 i et i I you thin o s n dditionl orce I (the inertil orce, or D Alebert s orce, you cn then consider it long with ll other orces nd write i i 0 Lw o rection orces (Newton s 3 rd lw All orces occur in equl nd opposite pirs (ction/rection et i Force eerted by n eleent is equl nd opposite to the orce on the eleent. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 0

21 Lw o Displceents I the ends o two eleents re connected, these ends re orced to ove with the se displceent, velocity, nd ccelertion. v v v v v v Newton s nd lw t point: The su o the orces t connection between eleents equls zero. v dv n 0 P / P i i 0 dt + 0 no ss Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes y

22 Modeling Steps Understnd syste unction, deine proble, nd identiy input/output vribles. Drw sipliied schetics using bsic (idelized eleents. Develop theticl odel (dierentil equtions Identiy reerence point nd positive direction Write eleentl equtions s well s interconnecting equtions by pplying physicl lws. Drw Free-ody-Digr (FD or ech bsic eleent. Cobine equtions by eliinting interedite vribles. Vlidte odel by copring siultion results with physicl esureents. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes

23 Obtining the Syste Model Eple: ss-spring-dper syste, v Free ody Digr (FD: Force eerted by the spring Newton s nd Lw: dv ( t ( t v dt dv d + v + ( t, v dt dt d d + + ( t dt dt && + & + ( t Force eerted by the dshpot (t I (t the pplied orce the initil orce y Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3

24 Obtining the Syste Model (cont d D Alebert s Lw (use the ide o n inertil orce, I (t i 0 ( t && + & + ( t & & 0 I ( t y I Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 4

25 Obtining the Syste Model (cont d Energy distribution: EOM o the bove siple ss-spring-dper syste && Contibutin o Inerti + & o Contribtin the Dper + Contribuion o the Spring Totl Applied Force We now wnt to loo t the energy distribution o the syste. How should we do it? Multiply the bove eqution by the velocity v: (since P is deined s Pv & & + & & + & ( t 443 & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 5 Power Integrte the nd eqution w.r.t. (with respect to tie: t && dt & + t ΔE chnge o enetic energy t & dt & t t & t dt 0 0 Energy dissipted by dper t + dt & t ΔE P Chnge o potentil energy t ( t & ( t dt t W Totl wor done by the pplied orce (t ro tie t to 0 t

26 Obtining the Syste Model (cont d Eple: (t,v (t Drw the FDs nd write the equtions o the syste in ters o nd. Free body digr (FD: ( & & ( - ( & & ( - (t Newton s nd Lw gives: && && ( & & + ( t ( & ( & ( or && ( & & ( 0 && + ( & & + ( ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 6

27 Obtining the Syste Model (cont d How bout deriving the equtions using D Alebert s Lw? Free body digr (FD: ( & & ( - ( & & ( - (t ( & & ( & & (t & ( - ( - & ( & & + ( t ( & ( & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 7 ( && 0 & 0 or && ( & & ( 0 && + ( & & + ( ( t

28 Obtining the Syste Model (cont d Eple: Displceent ecittion (t Drw the FD nd write the eqution o the syste. Free body digr (FD: Newton s nd Lw gives: & ( & ( t & ( (t- && & & & ( ( t + ( ( t Do not need n eqution or since it is deined unction (t! Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 8

29 Obtining the Syste Model (cont d Eple: Reltive displceent z Free body digr (FD: Newton s nd Lw gives: && & z && z& (&& + && z ( Use bsolute ccelertion ( +z Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 9 & z & && + z z& & && + z + z& + && z

30 Assuption or idel pulley: Idel Pulley Eleent T T No ss, no riction, no slippge between cble nd cylinder. Cble is lwys in tension. Cble cnnot stretch. I the pulley is not idel, its ss nd ny rictionl eects ust be considered. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 30

31 Idel Pulley Eleent (cont d Eple: Find the eqution describing the syste with pulley. Free body digr (FD: Equtions: (t ( - & M: && && M : && && ( t + ( + & + ( ( + & & (t + ( - g + soething new here! g + & ( t & ( g T Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3

32 Idel Pulley Eleent (cont d Copre the bove syste including n idel pulley with ollowing syste: Horizontl y Free body digr (FD: (t ( - & ( - & Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 3

33 Lecture Notes on Lecture Notes on MECH 370 MECH 370 Modelling Modelling, Siultion nd Anlysis o Physicl Systes, Siultion nd Anlysis o Physicl Systes Chpter Chpter 33 Idel Pulley Eleent (cont Idel Pulley Eleent (cont d d ( ( ( t ( v v ( t ( ( ( t M : or M: & && & && & & && & && p.3 (8, ( ( t ( v v v v & && & & Equtions or the syste: Finlly, (t (t ( t && ( t ( t && && y ( t ( + + : M & && ( g : M & &&

34 Prllel Cobintions: Prllel Cobintions Newton s nd lw: && where, Generl cse: FD: eq + or, K & -( eq + - equivlent spring stiness eq y eq + Siilrly, eq + Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 34

35 Prllel Cobintions (cont d For spring in prllel, or ( ( + ( 3 K eq + ( For dpers in prllel, or ( & & ( + ( 3 eq + & & ( & & Eleents re in prllel i the irst end o ech is connected to the se body nd the reining ends re connected to coon body. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 35

36 Eple: Equivlent syste: Equtions: Prllel Cobintions (cont d && eq eq where, then, && + ( t ( ( t & eq eq 3 eq Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 36 FD: & + + eq + eq & + + 3, ( t 3 & & ( & & + 3 eq + y & (t & 3

37 Series Cobintions Eple: (t A Drw the FDs nd write n eqution only in ters o. FD: (t ( - ( - (0- y & A Newton's nd lw or Newton's nd lw or : && ( & A: ( + 0 ( t ( ( Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 37

38 Series Cobintions (cont d Substitute ( into ( to get: && && ( where, eq + eq ( & + ( t & ( t Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 38

39 Sury Understnd the syste nd identiy the eleents nd vribles. Divide syste into idelized eleents: ss, stiness, riction, pulley. Find the eleent lws: Use the interconnection lws: && i, Δ, Δv Newton s 3 rd lw Lw o displceent Newton s nd lw i i 0 Apply Newton s nd lw to sses, nodes with unnown v nd use eleent, interconnection lws to deterine orces in ters o, v. Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 39

40 Reding Reding nd Eercise Chpter Assignent # E:.,.,.6,., 3.3, 3.8 (to be hnded in or ring Due: Fri., 6/7/07 t lecture E:.0,.7, 3., 3.5, 3.8 (or your prctice, no need to hnd in Chpter Lecture Notes on MECH 370 Modelling,, Siultion nd Anlysis o Physicl Systes 40

Phys101 Lecture 4,5 Dynamics: Newton s Laws of Motion

Phys101 Lecture 4,5 Dynamics: Newton s Laws of Motion Phys101 Lecture 4,5 Dynics: ewton s Lws of Motion Key points: ewton s second lw is vector eqution ction nd rection re cting on different objects ree-ody Digrs riction Inclines Ref: 4-1,2,3,4,5,6,7,8,9.

More information

The Spring. Consider a spring, which we apply a force F A to either stretch it or compress it

The Spring. Consider a spring, which we apply a force F A to either stretch it or compress it The Spring Consider spring, which we pply force F A to either stretch it or copress it F A - unstretched -F A 0 F A k k is the spring constnt, units of N/, different for different terils, nuber of coils

More information

CHAPTER 5 Newton s Laws of Motion

CHAPTER 5 Newton s Laws of Motion CHAPTER 5 Newton s Lws of Motion We ve been lerning kinetics; describing otion without understnding wht the cuse of the otion ws. Now we re going to lern dynics!! Nno otor 103 PHYS - 1 Isc Newton (1642-1727)

More information

SOLUTIONS TO CONCEPTS CHAPTER 6

SOLUTIONS TO CONCEPTS CHAPTER 6 SOLUIONS O CONCEPS CHAPE 6 1. Let ss of the block ro the freebody digr, 0...(1) velocity Agin 0 (fro (1)) g 4 g 4/g 4/10 0.4 he co-efficient of kinetic friction between the block nd the plne is 0.4. Due

More information

PHYS 601 HW3 Solution

PHYS 601 HW3 Solution 3.1 Norl force using Lgrnge ultiplier Using the center of the hoop s origin, we will describe the position of the prticle with conventionl polr coordintes. The Lgrngin is therefore L = 1 2 ṙ2 + 1 2 r2

More information

Project A: Active Vibration Suppression of Lumped-Parameters Systems using Piezoelectric Inertial Actuators *

Project A: Active Vibration Suppression of Lumped-Parameters Systems using Piezoelectric Inertial Actuators * Project A: Active Vibrtion Suppression of Luped-Preters Systes using Piezoelectric Inertil Actutors * A dynic vibrtion bsorber referred to s ctive resontor bsorber (ARA) is considered here, while exploring

More information

The Atwood Machine OBJECTIVE INTRODUCTION APPARATUS THEORY

The Atwood Machine OBJECTIVE INTRODUCTION APPARATUS THEORY The Atwood Mchine OBJECTIVE To derive the ening of Newton's second lw of otion s it pplies to the Atwood chine. To explin how ss iblnce cn led to the ccelertion of the syste. To deterine the ccelertion

More information

UCSD Phys 4A Intro Mechanics Winter 2016 Ch 4 Solutions

UCSD Phys 4A Intro Mechanics Winter 2016 Ch 4 Solutions USD Phys 4 Intro Mechnics Winter 06 h 4 Solutions 0. () he 0.0 k box restin on the tble hs the free-body dir shown. Its weiht 0.0 k 9.80 s 96 N. Since the box is t rest, the net force on is the box ust

More information

X Fx = F A. If applied force is small, book does not move (static), a x =0, then f=f s

X Fx = F A. If applied force is small, book does not move (static), a x =0, then f=f s A Appl ewton s nd Lw X 0 X A I pplied orce is sll, boo does not ove sttic, 0, then s A Increse pplied orce, boo still does not ove Increse A ore, now boo oves, 0 > A A here is soe iu sttic rictionl orce,

More information

PHY 5246: Theoretical Dynamics, Fall Assignment # 5, Solutions. θ = l mr 2 = l

PHY 5246: Theoretical Dynamics, Fall Assignment # 5, Solutions. θ = l mr 2 = l PHY 546: Theoreticl Dynics, Fll 15 Assignent # 5, Solutions 1 Grded Probles Proble 1 (1.) Using the eqution of the orbit or force lw d ( 1 dθ r)+ 1 r = r F(r), (1) l with r(θ) = ke αθ one finds fro which

More information

Homework: 5, 9, 19, 25, 31, 34, 39 (p )

Homework: 5, 9, 19, 25, 31, 34, 39 (p ) Hoework: 5, 9, 19, 5, 31, 34, 39 (p 130-134) 5. A 3.0 kg block is initilly t rest on horizontl surfce. A force of gnitude 6.0 nd erticl force P re then pplied to the block. The coefficients of friction

More information

ET 438a Control Systems Technology Laboratory 4 Modeling Control Systems with MATLAB/Simulink Position Control with Disturbances

ET 438a Control Systems Technology Laboratory 4 Modeling Control Systems with MATLAB/Simulink Position Control with Disturbances ET 438 Control Systes Technology bortory 4 Modeling Control Systes with MATAB/Siulink Position Control with Disturbnces bortory erning Objectives After copleting this lbortory you will be ble to:.) Convert

More information

Correct answer: 0 m/s 2. Explanation: 8 N

Correct answer: 0 m/s 2. Explanation: 8 N Version 001 HW#3 - orces rts (00223) 1 his print-out should hve 15 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Angled orce on Block 01 001

More information

Exam 1: Tomorrow 8:20-10:10pm

Exam 1: Tomorrow 8:20-10:10pm x : Toorrow 8:0-0:0p Roo Assignents: Lst Ne Roo A-D CCC 00 -J CS A0 K- PUGH 70 N-Q LI 50 R-S RY 30 T-Z W 00 redown o the 0 Probles teril # o Probles Chpter 4 Chpter 3 Chpter 4 6 Chpter 5 3 Chpter 6 5 Crib

More information

ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law

ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law ME 141 Engineering Mechnics Lecture 10: Kinetics of prticles: Newton s nd Lw Ahmd Shhedi Shkil Lecturer, Dept. of Mechnicl Engg, BUET E-mil: sshkil@me.buet.c.bd, shkil6791@gmil.com Website: techer.buet.c.bd/sshkil

More information

The momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is

The momentum of a body of constant mass m moving with velocity u is, by definition, equal to the product of mass and velocity, that is Newtons Lws 1 Newton s Lws There re three lws which ber Newton s nme nd they re the fundmentls lws upon which the study of dynmics is bsed. The lws re set of sttements tht we believe to be true in most

More information

INTRODUCTION. The three general approaches to the solution of kinetics problems are:

INTRODUCTION. The three general approaches to the solution of kinetics problems are: INTRODUCTION According to Newton s lw, prticle will ccelerte when it is subjected to unblnced forces. Kinetics is the study of the reltions between unblnced forces nd the resulting chnges in motion. The

More information

Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc

Forces from Strings Under Tension A string under tension medites force: the mgnitude of the force from section of string is the tension T nd the direc Physics 170 Summry of Results from Lecture Kinemticl Vribles The position vector ~r(t) cn be resolved into its Crtesin components: ~r(t) =x(t)^i + y(t)^j + z(t)^k. Rtes of Chnge Velocity ~v(t) = d~r(t)=

More information

Physics Honors. Final Exam Review Free Response Problems

Physics Honors. Final Exam Review Free Response Problems Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting

More information

ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 17 FORCES OF ELECTRIC ORIGIN ENERGY APPROACH(1)

ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 17 FORCES OF ELECTRIC ORIGIN ENERGY APPROACH(1) ECE 330 POWER CIRCUITS AND ELECTROMECHANICS LECTURE 17 FORCES OF ELECTRIC ORIGIN ENERGY APPROACH(1) Aknowledgent-These hndouts nd leture notes given in lss re bsed on teril fro Prof. Peter Suer s ECE 330

More information

Types of forces. Types of Forces

Types of forces. Types of Forces pes of orces pes of forces. orce of Grvit: his is often referred to s the weiht of n object. It is the ttrctive force of the erth. And is lws directed towrd the center of the erth. It hs nitude equl to

More information

1.2. Linear Variable Coefficient Equations. y + b "! = a y + b " Remark: The case b = 0 and a non-constant can be solved with the same idea as above.

1.2. Linear Variable Coefficient Equations. y + b ! = a y + b  Remark: The case b = 0 and a non-constant can be solved with the same idea as above. 1 12 Liner Vrible Coefficient Equtions Section Objective(s): Review: Constnt Coefficient Equtions Solving Vrible Coefficient Equtions The Integrting Fctor Method The Bernoulli Eqution 121 Review: Constnt

More information

Applied Physics Introduction to Vibrations and Waves (with a focus on elastic waves) Course Outline

Applied Physics Introduction to Vibrations and Waves (with a focus on elastic waves) Course Outline Applied Physics Introduction to Vibrtions nd Wves (with focus on elstic wves) Course Outline Simple Hrmonic Motion && + ω 0 ω k /m k elstic property of the oscilltor Elstic properties of terils Stretching,

More information

OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved.

OXFORD H i g h e r E d u c a t i o n Oxford University Press, All rights reserved. Renshw: Mths for Econoics nswers to dditionl exercises Exercise.. Given: nd B 5 Find: () + B + B 7 8 (b) (c) (d) (e) B B B + B T B (where 8 B 6 B 6 8 B + B T denotes the trnspose of ) T 8 B 5 (f) (g) B

More information

SOLUTIONS TO CONCEPTS CHAPTER 10

SOLUTIONS TO CONCEPTS CHAPTER 10 SOLUTIONS TO CONCEPTS CHPTE 0. 0 0 ; 00 rev/s ; ; 00 rd/s 0 t t (00 )/4 50 rd /s or 5 rev/s 0 t + / t 8 50 400 rd 50 rd/s or 5 rev/s s 400 rd.. 00 ; t 5 sec / t 00 / 5 8 5 40 rd/s 0 rev/s 8 rd/s 4 rev/s

More information

The Laws of Motion. chapter

The Laws of Motion. chapter chpter 5 The Lws of Motion 5.1 The Concept of Force 5.2 Newton s First Lw nd Inertil Fres 5.3 Mss 5.4 Newton s econd Lw 5.5 The Grvittionl Force nd Weight 5.6 Newton s Third Lw 5.7 Anlysis Models Using

More information

E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30

E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30 Vector Mechnics for Engineers: Dynmics nnouncement Reminders Wednesdy s clss will strt t 1:00PM. Summry of the chpter 11 ws posted on website nd ws sent you by emil. For the students, who needs hrdcopy,

More information

Dynamics: Newton s Laws of Motion

Dynamics: Newton s Laws of Motion Lecture 7 Chpter 4 Physics I 09.25.2013 Dynmics: Newton s Lws of Motion Solving Problems using Newton s lws Course website: http://fculty.uml.edu/andriy_dnylov/teching/physicsi Lecture Cpture: http://echo360.uml.edu/dnylov2013/physics1fll.html

More information

SOLUTIONS TO CONCEPTS CHAPTER

SOLUTIONS TO CONCEPTS CHAPTER 1. m = kg S = 10m Let, ccelertion =, Initil velocity u = 0. S= ut + 1/ t 10 = ½ ( ) 10 = = 5 m/s orce: = = 5 = 10N (ns) SOLUIONS O CONCEPS CHPE 5 40000. u = 40 km/hr = = 11.11 m/s. 3600 m = 000 kg ; v

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule

PHYS 705: Classical Mechanics. Small Oscillations: Example A Linear Triatomic Molecule PHYS 75: Clssicl echnics Sll Oscilltions: Exple A Liner Tritoic olecule A Liner Tritoic olecule x b b x x3 x Experientlly, one ight be interested in the rdition resulted fro the intrinsic oscilltion odes

More information

Part I: Basic Concepts of Thermodynamics

Part I: Basic Concepts of Thermodynamics Prt I: Bsic Concepts o Thermodynmics Lecture 4: Kinetic Theory o Gses Kinetic Theory or rel gses 4-1 Kinetic Theory or rel gses Recll tht or rel gses: (i The volume occupied by the molecules under ordinry

More information

Unit 3: Direct current and electric resistance Electric current and movement of charges. Intensity of current and drift speed. Density of current in

Unit 3: Direct current and electric resistance Electric current and movement of charges. Intensity of current and drift speed. Density of current in Unit 3: Direct current nd electric resistnce Electric current nd movement of chrges. ntensity of current nd drift speed. Density of current in homogeneous currents. Ohm s lw. esistnce of homogeneous conductor

More information

Physics. Friction.

Physics. Friction. hysics riction www.testprepkrt.co Tble of Content. Introduction.. Types of friction. 3. Grph of friction. 4. riction is cuse of otion. 5. dvntges nd disdvntges of friction. 6. Methods of chnging friction.

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Lecture 8. Newton s Laws. Applications of the Newton s Laws Problem-Solving Tactics. Physics 105; Fall Inertial Frames: T = mg

Lecture 8. Newton s Laws. Applications of the Newton s Laws Problem-Solving Tactics. Physics 105; Fall Inertial Frames: T = mg Lecture 8 Applictions of the ewton s Lws Problem-Solving ctics http://web.njit.edu/~sireno/ ewton s Lws I. If no net force ocects on body, then the body s velocity cnnot chnge. II. he net force on body

More information

MA 131 Lecture Notes Calculus Sections 1.5 and 1.6 (and other material)

MA 131 Lecture Notes Calculus Sections 1.5 and 1.6 (and other material) MA Lecture Notes Clculus Sections.5 nd.6 (nd other teril) Algebr o Functions Su, Dierence, Product, nd Quotient o Functions Let nd g be two unctions with overlpping doins. Then or ll x coon to both doins,

More information

INTRODUCTION TO LINEAR ALGEBRA

INTRODUCTION TO LINEAR ALGEBRA ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

More information

Physics 202, Lecture 10. Basic Circuit Components

Physics 202, Lecture 10. Basic Circuit Components Physics 202, Lecture 10 Tody s Topics DC Circuits (Chpter 26) Circuit components Kirchhoff s Rules RC Circuits Bsic Circuit Components Component del ttery, emf Resistor Relistic Bttery (del) wire Cpcitor

More information

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation

1.1. Linear Constant Coefficient Equations. Remark: A differential equation is an equation 1 1.1. Liner Constnt Coefficient Equtions Section Objective(s): Overview of Differentil Equtions. Liner Differentil Equtions. Solving Liner Differentil Equtions. The Initil Vlue Problem. 1.1.1. Overview

More information

2/20/ :21 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.

2/20/ :21 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. //15 11:1 M Chpter 11 Kinemtics of Prticles 1 //15 11:1 M Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion of bodies under the ction of forces It is

More information

Lumped-Element Modeling

Lumped-Element Modeling EE55: Principles o MEMS Trnsducers (Fll 003) Instructor: Dr. HuiKi Xie st lecture umpedelement Modeling Oneport elements Generlized Cpcitor Generlized Inertnce Kirchho s ws Exmple Tody: Singleport element

More information

13.4 Work done by Constant Forces

13.4 Work done by Constant Forces 13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push

More information

Page 1. Motion in a Circle... Dynamics of Circular Motion. Motion in a Circle... Motion in a Circle... Discussion Problem 21: Motion in a Circle

Page 1. Motion in a Circle... Dynamics of Circular Motion. Motion in a Circle... Motion in a Circle... Discussion Problem 21: Motion in a Circle Dynics of Circulr Motion A boy ties rock of ss to the end of strin nd twirls it in the erticl plne. he distnce fro his hnd to the rock is. he speed of the rock t the top of its trectory is. Wht is the

More information

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

More information

Discussion Introduction P212, Week 1 The Scientist s Sixth Sense. Knowing what the answer will look like before you start.

Discussion Introduction P212, Week 1 The Scientist s Sixth Sense. Knowing what the answer will look like before you start. Discussion Introduction P1, Week 1 The Scientist s Sith Sense As scientist or engineer, uch of your job will be perforing clcultions, nd using clcultions perfored by others. You ll be doing plenty of tht

More information

C D o F. 30 o F. Wall String. 53 o. F y A B C D E. m 2. m 1. m a. v Merry-go round. Phy 231 Sp 03 Homework #8 Page 1 of 4

C D o F. 30 o F. Wall String. 53 o. F y A B C D E. m 2. m 1. m a. v Merry-go round. Phy 231 Sp 03 Homework #8 Page 1 of 4 Phy 231 Sp 3 Hoework #8 Pge 1 of 4 8-1) rigid squre object of negligible weight is cted upon by the forces 1 nd 2 shown t the right, which pull on its corners The forces re drwn to scle in ters of the

More information

2/2/ :36 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.

2/2/ :36 AM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. //16 1:36 AM Chpter 11 Kinemtics of Prticles 1 //16 1:36 AM First Em Wednesdy 4//16 3 //16 1:36 AM Introduction Mechnics Mechnics = science which describes nd predicts the conditions of rest or motion

More information

1/31/ :33 PM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E.

1/31/ :33 PM. Chapter 11. Kinematics of Particles. Mohammad Suliman Abuhaiba,Ph.D., P.E. 1/31/18 1:33 PM Chpter 11 Kinemtics of Prticles 1 1/31/18 1:33 PM First Em Sturdy 1//18 3 1/31/18 1:33 PM Introduction Mechnics Mechnics = science which describes nd predicts conditions of rest or motion

More information

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the

More information

Ideal Gas behaviour: summary

Ideal Gas behaviour: summary Lecture 4 Rel Gses Idel Gs ehviour: sury We recll the conditions under which the idel gs eqution of stte Pn is vlid: olue of individul gs olecules is neglected No interctions (either ttrctive or repulsive)

More information

Version 001 HW#6 - Circular & Rotational Motion arts (00223) 1

Version 001 HW#6 - Circular & Rotational Motion arts (00223) 1 Version 001 HW#6 - Circulr & ottionl Motion rts (00223) 1 This print-out should hve 14 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Circling

More information

Contact Analysis on Large Negative Clearance Four-point Contact Ball Bearing

Contact Analysis on Large Negative Clearance Four-point Contact Ball Bearing Avilble online t www.sciencedirect.co rocedi ngineering 7 0 74 78 The Second SR Conference on ngineering Modelling nd Siultion CMS 0 Contct Anlysis on Lrge Negtive Clernce Four-point Contct Bll Bering

More information

Hints for Exercise 1 on: Current and Resistance

Hints for Exercise 1 on: Current and Resistance Hints for Exercise 1 on: Current nd Resistnce Review the concepts of: electric current, conventionl current flow direction, current density, crrier drift velocity, crrier numer density, Ohm s lw, electric

More information

Lecture 5. Today: Motion in many dimensions: Circular motion. Uniform Circular Motion

Lecture 5. Today: Motion in many dimensions: Circular motion. Uniform Circular Motion Lecture 5 Physics 2A Olg Dudko UCSD Physics Tody: Motion in mny dimensions: Circulr motion. Newton s Lws of Motion. Lws tht nswer why questions bout motion. Forces. Inerti. Momentum. Uniform Circulr Motion

More information

Classical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011

Classical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011 Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,

More information

KINEMATICS OF RIGID BODIES

KINEMATICS OF RIGID BODIES KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

More information

PHYSICS. (c) m D D m. v vt. (c) 3. Width of the river = d = 800 m m/s. 600m. B 800m v m,r. (b) 1000m. v w

PHYSICS. (c) m D D m. v vt. (c) 3. Width of the river = d = 800 m m/s. 600m. B 800m v m,r. (b) 1000m. v w PHYSICS T. T 8 k/hr = /s, ' Hz, ' 7 Hz T. r D D D,. r D D. D. D.... g/c. Width o the rier = d = 8 w /s B 8,r P O. sin t cos., on the copring this eqution, with sin t cos k =, k =. k. /s w Q. =, 9 9 = 8,

More information

r = cos θ + 1. dt ) dt. (1)

r = cos θ + 1. dt ) dt. (1) MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr

More information

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY

UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY UNIQUENESS THEOREMS FOR ORDINARY DIFFERENTIAL EQUATIONS WITH HÖLDER CONTINUITY YIFEI PAN, MEI WANG, AND YU YAN ABSTRACT We estblish soe uniqueness results ner 0 for ordinry differentil equtions of the

More information

Consequently, the temperature must be the same at each point in the cross section at x. Let:

Consequently, the temperature must be the same at each point in the cross section at x. Let: HW 2 Comments: L1-3. Derive the het eqution for n inhomogeneous rod where the therml coefficients used in the derivtion of the het eqution for homogeneous rod now become functions of position x in the

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

An Introduction to Trigonometry

An Introduction to Trigonometry n Introduction to Trigonoetry First of ll, let s check out the right ngled tringle below. The LETTERS, B & C indicte the ngles nd the letters, b & c indicte the sides. c b It is iportnt to note tht side

More information

DIRECT CURRENT CIRCUITS

DIRECT CURRENT CIRCUITS DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

More information

CHAPTER 10 LAGRANGIAN MECHANICS

CHAPTER 10 LAGRANGIAN MECHANICS CHAPTER LAGRANGIAN MECHANICS. Solution ( t (, t + αη ( t ( t (, t +αη ( t where (, t sinωt nd (, cos t ω ωt T V k ω t J α ω dt so: ( ( t t + + ( ω cosωt αη ω ( sinωt αη dt t t t α t J ( α ω ( cos ωt sin

More information

Chapter 5 Exercise 5A

Chapter 5 Exercise 5A Chpter Exercise Q. 1. (i) 00 N,00 N F =,00 00 =,000 F = m,000 = 1,000 = m/s (ii) =, u = 0, t = 0, s =? s = ut + 1 t = 0(0) + 1 ()(00) = 00 m Q.. 0 N 100 N F = 100 0 = 60 F = m 60 = 10 = 1 m/s F = m 60

More information

Discussion Question 1A P212, Week 1 P211 Review: 2-D Motion with Uniform Force

Discussion Question 1A P212, Week 1 P211 Review: 2-D Motion with Uniform Force Discussion Question 1A P1, Week 1 P11 Review: -D otion with Unifor Force The thetics nd phsics of the proble below re siilr to probles ou will encounter in P1, where the force is due to the ction of n

More information

Exponents and Powers

Exponents and Powers EXPONENTS AND POWERS 9 Exponents nd Powers CHAPTER. Introduction Do you know? Mss of erth is 5,970,000,000,000, 000, 000, 000, 000 kg. We hve lredy lernt in erlier clss how to write such lrge nubers ore

More information

Physics Dynamics: Atwood Machine

Physics Dynamics: Atwood Machine plce of ind F A C U L Y O F E D U C A I O N Deprtent of Curriculu nd Pedoy Physics Dynics: Atwood Mchine Science nd Mthetics Eduction Reserch Group Supported by UBC echin nd Lernin Enhnceent Fund 0-04

More information

Chapter 4. Additional Variational Concepts

Chapter 4. Additional Variational Concepts Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.

More information

Model Solutions to Assignment 4

Model Solutions to Assignment 4 Oberlin College Physics 110, Fll 2011 Model Solutions to Assignment 4 Additionl problem 56: A girl, sled, nd n ice-covered lke geometry digrm: girl shore rope sled ice free body digrms: force on girl by

More information

Week 10: Line Integrals

Week 10: Line Integrals Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.

More information

Harman Outline 1A1 Integral Calculus CENG 5131

Harman Outline 1A1 Integral Calculus CENG 5131 Hrmn Outline 1A1 Integrl Clculus CENG 5131 September 5, 213 III. Review of Integrtion A.Bsic Definitions Hrmn Ch14,P642 Fundmentl Theorem of Clculus The fundmentl theorem of clculus shows the intimte reltionship

More information

Problem Solving 7: Faraday s Law Solution

Problem Solving 7: Faraday s Law Solution MASSACHUSETTS NSTTUTE OF TECHNOLOGY Deprtment of Physics: 8.02 Prolem Solving 7: Frdy s Lw Solution Ojectives 1. To explore prticulr sitution tht cn led to chnging mgnetic flux through the open surfce

More information

INVESTIGATION OF THERMAL PROPERTIES OF SOIL BY IMPULSE METHOD Juraj Veselský

INVESTIGATION OF THERMAL PROPERTIES OF SOIL BY IMPULSE METHOD Juraj Veselský THERMOPHYSICS 6 October 6 5 INVESTIGATION OF THERMAL PROPERTIES OF SOIL BY IMPULSE METHOD Jurj Veselský Fculty of Civil Engineering STU Brtislv, Rdlinského, 8 68 Brtislv Abstrct The therl properties of

More information

CHAPTER 4a. ROOTS OF EQUATIONS

CHAPTER 4a. ROOTS OF EQUATIONS CHAPTER 4. ROOTS OF EQUATIONS A. J. Clrk School o Engineering Deprtment o Civil nd Environmentl Engineering by Dr. Ibrhim A. Asskk Spring 00 ENCE 03 - Computtion Methods in Civil Engineering II Deprtment

More information

Math 124A October 04, 2011

Math 124A October 04, 2011 Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model

More information

Heat flux and total heat

Heat flux and total heat Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce

More information

Flow in porous media

Flow in porous media Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ

More information

JURONG JUNIOR COLLEGE

JURONG JUNIOR COLLEGE JURONG JUNIOR COLLEGE 2010 JC1 H1 8866 hysics utoril : Dynmics Lerning Outcomes Sub-topic utoril Questions Newton's lws of motion 1 1 st Lw, b, e f 2 nd Lw, including drwing FBDs nd solving problems by

More information

Chapter 14. Gas-Vapor Mixtures and Air-Conditioning. Study Guide in PowerPoint

Chapter 14. Gas-Vapor Mixtures and Air-Conditioning. Study Guide in PowerPoint Chpter 14 Gs-Vpor Mixtures nd Air-Conditioning Study Guide in PowerPoint to ccopny Therodynics: An Engineering Approch, 5th edition by Yunus A. Çengel nd Michel A. Boles We will be concerned with the ixture

More information

Kinematics equations, some numbers

Kinematics equations, some numbers Kinemtics equtions, some numbers Kinemtics equtions: x = x 0 + v 0 t + 1 2 t2, v = v 0 + t. They describe motion with constnt ccelertion. Brking exmple, = 1m/s. Initil: x 0 = 10m, v 0 = 10m/s. x(t=1s)

More information

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3

I1 = I2 I1 = I2 + I3 I1 + I2 = I3 + I4 I 3 2 The Prllel Circuit Electric Circuits: Figure 2- elow show ttery nd multiple resistors rrnged in prllel. Ech resistor receives portion of the current from the ttery sed on its resistnce. The split is

More information

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016

HOMEWORK SOLUTIONS MATH 1910 Sections 7.9, 8.1 Fall 2016 HOMEWORK SOLUTIONS MATH 9 Sections 7.9, 8. Fll 6 Problem 7.9.33 Show tht for ny constnts M,, nd, the function yt) = )) t ) M + tnh stisfies the logistic eqution: y SOLUTION. Let Then nd Finlly, y = y M

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

The University of New South Wales FINAL EXAMINATION. Session ELEC4613 ELECTRIC DRIVE SYSTEMS. 1. Time allowed 3 hours

The University of New South Wales FINAL EXAMINATION. Session ELEC4613 ELECTRIC DRIVE SYSTEMS. 1. Time allowed 3 hours The University of New South Wles FINAL EXAMINATION Session 010 ELEC4613 ELECTRIC DRIVE SYSTEMS 1. Tie llowed 3 hours. Reding tie: 10 inutes 3. Totl nuber of questions in this pper = SIX 4. Answer ny FOUR

More information

Math 0230 Calculus 2 Lectures

Math 0230 Calculus 2 Lectures Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two

More information

FULL MECHANICS SOLUTION

FULL MECHANICS SOLUTION FULL MECHANICS SOLUION. m 3 3 3 f For long the tngentil direction m 3g cos 3 sin 3 f N m 3g sin 3 cos3 from soling 3. ( N 4) ( N 8) N gsin 3. = ut + t = ut g sin cos t u t = gsin cos = 4 5 5 = s] 3 4 o

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com 1. A uniform circulr disc hs mss m, centre O nd rdius. It is free to rotte bout fixed smooth horizontl xis L which lies in the sme plne s the disc nd which is tngentil to the disc t the point A. The disc

More information

Physics 212. Faraday s Law

Physics 212. Faraday s Law Phsics 1 Lecture 17 Frd s Lw Phsics 1 Lecture 17, Slide 1 Motionl EMF Chnge Are of loop Chnge mgnetic field through loop Chnge orienttion of loop reltive to In ech cse the flu of the mgnetic field through

More information

a) mass inversely proportional b) force directly proportional

a) mass inversely proportional b) force directly proportional 1. Wht produces ccelertion? A orce 2. Wht is the reltionship between ccelertion nd ) mss inersely proportionl b) orce directly proportionl 3. I you he orce o riction, 30N, on n object, how much orce is

More information

Hung problem # 3 April 10, 2011 () [4 pts.] The electric field points rdilly inwrd [1 pt.]. Since the chrge distribution is cylindriclly symmetric, we pick cylinder of rdius r for our Gussin surfce S.

More information

Numerical Problems With Solutions(STD:-XI)

Numerical Problems With Solutions(STD:-XI) Numericl Problems With Solutions(STD:-XI) Topic:-Uniform Circulr Motion. An irplne executes horizontl loop of rdius 000m with stedy speed of 900kmh -. Wht is its centripetl ccelertion? Ans:- Centripetl

More information

Answers to the Conceptual Questions

Answers to the Conceptual Questions Chpter 3 Explining Motion 41 Physics on Your Own If the clss is not too lrge, tke them into freight elevtor to perform this exercise. This simple exercise is importnt if you re going to cover inertil forces

More information

Eunil Won Dept. of Physics, Korea University 1. Ch 03 Force. Movement of massive object. Velocity, acceleration. Force. Source of the move

Eunil Won Dept. of Physics, Korea University 1. Ch 03 Force. Movement of massive object. Velocity, acceleration. Force. Source of the move Eunil Won Dept. of Phsics, Kore Uniersit 1 Ch 03 orce Moement of mssie object orce Source of the moe Velocit, ccelertion Eunil Won Dept. of Phsics, Kore Uniersit m ~ 3.305 m ~ 1.8 m 1.8 m Eunil Won Dept.

More information

PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3.

PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3. PROLEM 15.105 A 5-m steel bem is lowered by mens of two cbles unwinding t the sme speed from overhed crnes. As the bem pproches the ground, the crne opertors pply brkes to slow the unwinding motion. At

More information

Verification Analysis of the Redi Rock Wall

Verification Analysis of the Redi Rock Wall Verifiction Mnul no. Updte 06/06 Verifiction Anlysis of the Redi Rock Wll Progr File Redi Rock Wll Deo_v_etric_en_0.grr In this verifiction nul you will find hnd-de verifiction nlysis of the Redi Rock

More information