Chapter 10. Rotational Kinematics

Size: px
Start display at page:

Download "Chapter 10. Rotational Kinematics"

Transcription

1 Chape 10. Roaonal Knemacs Up o now we hae only consdeed ponpacles.e. we hae no consdeed he shape o sze only he mass Also we hae only consdeed he moon o pon-pacles sagh-lne ee-all pojecle moon. Bu eal objecs can also umble wl Ths subjec oaon s wha we exploe n hs chape and n Chape 11. Fs we begn by exendng he conceps o ccula moon

2 Insead o a pon-pacle consde a hn dsk o adus spnnng on s axs y θ z Axs o oaon θ s ac lengh s x θ Ths dsk s a eal objec has sucue We call hese knds o objecs Rgd Bodes Rgd Bodes do no bend ws o lex; o example a bllad ball ac lengh adus s Uns o adans (ad)

3 s θ Fo one complee eoluon s π ccumeence Coneson elaon: π ad 360 θ π ad Now consde he oaon o he dsk om some nal angle θ o a nal angle θ dung some me peod o Δθ θ θ θ θ Δθ Δ ω ag Aeage angula elocy (uns o ad/s) Angula dsplacemen (uns o ad ccw s +) z y θ ccw θ x

4 Smla o nsananeous elocy we can dene he Insananeous Angula Velocy A change n he Angula Velocy ges ω ω ω lm Δ 0 Δθ Δ Δω Δ α ag Aeage Angula Acceleaon (ads/s ) Analogous o Insananeous Angula Velocy he Insananeous Angula Acceleaon s α lm Δ 0 Δω Δ

5 Acually he Angula Veloces and Angula Acceleaon ae magnudes o eco quanes Wha s he decon? ω and α They pon along he axs o oaon wh he sgn deemned by he gh-hand ule Example A an akes.00 s o each s opeang angula speed o 10.0 e/s. Wha s he aeage angula acceleaon (ad/s )?

6 Soluon: Gen:.00 s ω 10.0 e/s Recognze: 0 ω 0 and ha ω needs o be coneed o ad/s ω 10.0 e s 6.8 ad s π ads 1 e α ag ω ω 0.0π α ag 31.4 ad s 0.0π ad s 10.0π ad s

7 Equaons o Roaonal Knemacs Jus as we hae deed a se o equaons o descbe ``lnea o ``anslaonal knemacs we can also oban an analogous se o equaons o oaonal moon Consde coelaon o aables Tanslaonal x dsplacemen θ elocy ω a acceleaon α me Roaonal

8 Replacng each o he anslaonal aables n he anslaonal knemac equaons by he oaonal aables ges he se o oaonal knemac equaons (o consan α) θ θ + 1 (ω + ω )( - ) θ θ + ω ( - ) + 1 α( - ) ω ω + α( - ) ω ω + α(θ -θ ) We can use hese equaons n he same ashon we appled he anslaonal knemac equaons

9 Example Poblem A gue skae s spnnng wh an angula elocy o +15 ad/s. She hen comes o a sop oe a be peod o me. Dung hs me he angula dsplacemen s +5.1 ad. Deemne (a) he aeage angula acceleaon and (b) he me dung whch she comes o es. Soluon: Gen: θ +5.1 ad ω +15 ad/s Ine: θ 0 ω 0 0 Fnd: α?

10 (a) Use las knemac equaon ω ω + α(θ θ ) 0 ω + αθ α ω (15 ad/s) θ (5.1 ad) (b) Use s knemac equaon ad s θ θ θ 0 θ ω ( ω ( ω + (5.1ad) 15 + ω 0)( )( ad/s 0) ) 0.68 s

11 O use he hd knemac equaon ω ω + α( ) 0 ω + α ω α Example Poblem 15 ad/s - ad/s 0.68 s A he local swmmng hole a aoe ck s o un hozonally o a cl ha s 8.3 m aboe he wae uck no a ``ball and oae on he way down o he wae. The aeage angula speed o oaon s 1.6 e/s. Ignong a essance

12 deemne he numbe o eoluons whle on he way down. y y ω Soluon: y x Gen: ω ω 1.6 e/s y 8.3 m Also y 0 0 y 0 Recognze: wo knds o moon; D pojecle moon and oaonal moon wh consan angula elocy. Mehod: #eoluons θ ω. Theeoe need o nd he me o he pojecle moon.

13 Consde y-componen o pojecle moon snce we hae no nomaon abou he x- componen..1 e e/s)(1.3 s) s 9.80 m) s m (1.6 ) )( ( 1.3 ( y y g y g y g y ) - ( a ) - ( y y θ ω ω ω θ θ

14 Tangenal Velocy z ω θ Fo one complee eoluon he angula dsplacemen s π ad Fom Unom Ccula Moon he me o a complee eoluon s a peod T Theeoe he angula elocy (equency) can be wen Δθ π ω ω Δ T (ad/s)

15 Also he speed o an objec n a ccula pah s π ω T Tangenal speed T (m/s) The angenal speed coesponds o he speed o a pon on a gd body a dsance om s cene oang a an angula speed ω Each pon on he gd body oaes a he same angula speed bu s angenal speed depends on s locaon T 0

16 I he angula elocy changes (ω s no consan) hen we hae an angula acceleaon α Fo some pon on a dsk o example Fom he denon o anslaonal acceleaon ω ω α ω ω ω ω a a Δ Δ Δ Tangenal acceleaon (uns o m/s ) Snce he speed changes hs s no Unom Ccula Moon. Also he Tangenal Acceleaon s deen om he Cenpeal Acceleaon.

17 Recall a c We can nd a oal esulan acceleaon a snce a and a ae pependcula a a + a φ an 1 (a /a ) Peously o he case o unom ccula moon a 0 and aa c a. The acceleaon eco poned o he cene o he ccle. I a 0 acceleaon pons away om he cene a z a φ a a a a

18 Example A hn gd od s oang wh a consan angula acceleaon abou an axs ha passes pependculaly hough one o s ends. A one nsan he oal acceleaon eco (adal plus angenal) a he ohe end o he od makes a 60.0 angle wh espec o he od and has a magnude o 15.0 m/s. The od has an angula speed o.00 ad/s a hs nsan. Wha s he od s lengh? Gen: a 15.0 m/s ω.00 ad/s (a some me)

19 z y L a L a α Lα ω Lω a (Lω) a acosφ Lω L acosφ ω x L a φ a Sole o L (15.0 m )cos60.0 s (.00 ad s ) a Lω 1.88 m

20 Rollng Moon (Tes Bllads) Demo 8.6. B A C Road axs ca Consde a e aelng on a oad wh con (no skddng) beween he e and oad Fs eew concep o elae elocy. Wha s he elocy o B as seen by he gound? I e s suspended eey pon on edge has same

21 Pon B has a elocy wh espec o he A Now A (he e as a whole) s mong o he gh wh elocy ca ca ca G B ca A G A B G B ca A G A B + + Wha s elocy o C as seen by he gound? ca G C ca A G A C G C ca A G A C

22 Example Poblem A e has a adus o m and s cene moes owad wh a lnea speed o 15.0 m/s. (a) Wha s ω o he wheel? (b) Relae o he axle wha s o a pon locaed m om he axle? 15.0 ω ca 45.5 ad/s ω (0.175)(45.5) 7.96 m/s

PHYSICS 161 EXAM III: Thursday December 04, 2003 11:00 a.m.

PHYSICS 161 EXAM III: Thursday December 04, 2003 11:00 a.m. PHYS 6: Eam III Fall 003 PHYSICS 6 EXAM III: Thusda Decembe 04, 003 :00 a.m. Po. N. S. Chan. Please pn ou name and ene ou sea numbe o den ou and ou eamnaon. Suden s Pned Name: Recaon Secon Numbe: Sea Numbe:.

More information

Valuing Long-Lived Assets

Valuing Long-Lived Assets Valuing Long-Lived Asses Olive Tabalski, 008-09-0 This chape explains how you can calculae he pesen value of cash flow. Some vey useful shocu mehods will be shown. These shocus povide a good oppouniy fo

More information

Homework: 49, 56, 67, 60, 64, 74 (p. 234-237)

Homework: 49, 56, 67, 60, 64, 74 (p. 234-237) Hoework: 49, 56, 67, 60, 64, 74 (p. 34-37) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,

More information

. The torque at the rear sprocket is what actually accelerates the bicycle, and so R R

. The torque at the rear sprocket is what actually accelerates the bicycle, and so R R CHAPTE 8: anal Mn Answes Quesns. The dmee desgned 7-nch wheels nceases s eadng by he ccumeence a 7-nch wheel 7 " eey elun he wheel. a 4-nch wheel s used, he dmee wll sll egse 7 " eey elun, bu nly 4 " lnea

More information

HFCC Math Lab Intermediate Algebra - 13 SOLVING RATE-TIME-DISTANCE PROBLEMS

HFCC Math Lab Intermediate Algebra - 13 SOLVING RATE-TIME-DISTANCE PROBLEMS HFCC Mah Lab Inemeiae Algeba - 3 SOLVING RATE-TIME-DISTANCE PROBLEMS The vaiables involve in a moion poblem ae isance (), ae (), an ime (). These vaiables ae elae by he equaion, which can be solve fo any

More information

Chapter 2 Kinematics in One Dimension

Chapter 2 Kinematics in One Dimension Chaper Kinemaics in One Dimension Chaper DESCRIBING MOTION:KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings moe how far (disance and displacemen), how fas (speed and elociy), and how

More information

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore

Imagine a Source (S) of sound waves that emits waves having frequency f and therefore heoreical Noes: he oppler Eec wih ound Imagine a ource () o sound waes ha emis waes haing requency and hereore period as measured in he res rame o he ource (). his means ha any eecor () ha is no moing

More information

The Fresnel Equations and Brewster's Law

The Fresnel Equations and Brewster's Law The Fesnel Equaons and Bewse's Law Equpmen Opcal bench pvo, wo 1 mee opcal benches, geen lase a 543.5 nm, 10cm damee polazes, ecangula polaze, LX-0 phoo-deeco n opcal moun, hck acylc block, hck glass block,

More information

Electric Potential. otherwise to move the object from initial point i to final point f

Electric Potential. otherwise to move the object from initial point i to final point f PHY2061 Enched Physcs 2 Lectue Notes Electc Potental Electc Potental Dsclame: These lectue notes ae not meant to eplace the couse textbook. The content may be ncomplete. Some topcs may be unclea. These

More information

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur

Module 3 Design for Strength. Version 2 ME, IIT Kharagpur Module 3 Design for Srengh Lesson 2 Sress Concenraion Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress

More information

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer

More information

Answer, Key Homework 6 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 6 David McIntyre 45123 Mar 25, 2004 1 Answe, Key Homewok 6 vid McInye 4513 M 5, 004 1 This pin-ou should hve 0 quesions. Muliple-choice quesions my coninue on he nex column o pge find ll choices befoe mking you selecion. The due ime is Cenl

More information

Lecture 40 Induction. Review Inductors Self-induction RL circuits Energy stored in a Magnetic Field

Lecture 40 Induction. Review Inductors Self-induction RL circuits Energy stored in a Magnetic Field ecure 4 nducon evew nducors Self-nducon crcus nergy sored n a Magnec Feld 1 evew nducon end nergy Transfers mf Bv Mechancal energy ransform n elecrc and hen n hermal energy P Fv B v evew eformulaon of

More information

Signal Rectification

Signal Rectification 9/3/25 Signal Recificaion.doc / Signal Recificaion n imporan applicaion of juncion diodes is signal recificaion. here are wo ypes of signal recifiers, half-wae and fullwae. Le s firs consider he ideal

More information

Acceleration Lab Teacher s Guide

Acceleration Lab Teacher s Guide Acceleraion Lab Teacher s Guide Objecives:. Use graphs of disance vs. ime and velociy vs. ime o find acceleraion of a oy car.. Observe he relaionship beween he angle of an inclined plane and he acceleraion

More information

Experiment #1: Reflection, Refraction, and Dispersion

Experiment #1: Reflection, Refraction, and Dispersion Expeimen #1: Reflecion, Refacion, and Dispesion Pupose: To sudy eflecion and efacion of ligh a plane and cuved sufaces, as well as he phenomenon of dispesion. Equipmen: Ray Box wih Slis Opical Accessoies

More information

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1

Answer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1 Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prin-ou should hae 1 quesions. Muliple-choice quesions may coninue on he ne column or page find all choices before making your selecion. The

More information

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field

Gravitation. Definition of Weight Revisited. Newton s Law of Universal Gravitation. Newton s Law of Universal Gravitation. Gravitational Field Defnton of Weght evsted Gavtaton The weght of an object on o above the eath s the gavtatonal foce that the eath exets on the object. The weght always ponts towad the cente of mass of the eath. On o above

More information

Modeling the Yield Curve Dynamics

Modeling the Yield Curve Dynamics FIXED-INCOME SECURITIES Chape 2 Modeling he Yield Cuve Dynamics Ouline Moivaion Inees Rae Tees Single-Faco Coninuous-Time Models Muli-Faco Coninuous-Time Models Abiage Models Moivaion Why do we Cae? Picing

More information

Ź Ź ł ź Ź ś ź ł ź Ś ę ż ż ł ż ż Ż Ś ę Ż Ż ę ś ź ł Ź ł ł ż ż ź ż ż Ś ę ż ż Ź Ł Ż Ż Ą ż ż ę ź Ń Ź ś ł ź ż ł ś ź ź Ą ć ś ś Ź Ś ę ę ć ż Ź Ą Ń Ą ł ć ć ł ł ź ę Ś ę ś ę ł ś ć ź ś ł ś ł ł ł ł ć ć Ś ł ź Ś ł

More information

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur

Module 4. Single-phase AC circuits. Version 2 EE IIT, Kharagpur Module 4 Single-phase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,

More information

Oblique incidence: Interface between dielectric media

Oblique incidence: Interface between dielectric media lecrmagnec Felds Oblque ncdence: Inerface beween delecrc meda Cnsder a planar nerface beween w delecrc meda. A plane wave s ncden a an angle frm medum. The nerface plane defnes he bundary beween he meda.

More information

Rotation Kinematics, Moment of Inertia, and Torque

Rotation Kinematics, Moment of Inertia, and Torque Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute

More information

12. Rolling, Torque, and Angular Momentum

12. Rolling, Torque, and Angular Momentum 12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

More information

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004

HUT, TUT, LUT, OU, ÅAU / Engineering departments Entrance examination in mathematics May 25, 2004 HUT, TUT, LUT, OU, ÅAU / Engineeing depamens Enane examinaion in mahemais May 5, 4 Insuions. Reseve a sepaae page fo eah poblem. Give you soluions in a lea fom inluding inemediae seps. Wie a lean opy of

More information

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow.

Lecture 2: Telegrapher Equations For Transmission Lines. Power Flow. Whies, EE 481 Lecure 2 Page 1 of 13 Lecure 2: Telegraher Equaions For Transmission Lines. Power Flow. Microsri is one mehod for making elecrical connecions in a microwae circui. I is consruced wih a ground

More information

A Curriculum Module for AP Calculus BC Curriculum Module

A Curriculum Module for AP Calculus BC Curriculum Module Vecors: A Curriculum Module for AP Calculus BC 00 Curriculum Module The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy.

More information

R e t r o f i t o f t C i r u n i s g e C o n t r o l

R e t r o f i t o f t C i r u n i s g e C o n t r o l R e t r o f i t o f t C i r u n i s g e C o n t r o l VB Sprinter D e s c r i p t i o n T h i s r e t r o f i t c o n s i s t s o f i n s t a l l i n g a c r u i s e c o n t r o l s wi t c h k i t i n

More information

Kinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.

Kinematics in 1-D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard. Chaper 2 Kinemaics in 1-D From Problems and Soluions in Inroducory Mechanics (Draf ersion, Augus 2014) Daid Morin, morin@physics.harard.edu As menioned in he preface, his book should no be hough of as

More information

CHARGE AND DISCHARGE OF A CAPACITOR

CHARGE AND DISCHARGE OF A CAPACITOR REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:

More information

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m

Chapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m

More information

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD

HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Journal of Appled Mahemacs and Compuaonal Mechancs 3, (), 45-5 HEAT CONDUCTION PROBLEM IN A TWO-LAYERED HOLLOW CYLINDER BY USING THE GREEN S FUNCTION METHOD Sansław Kukla, Urszula Sedlecka Insue of Mahemacs,

More information

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.

Appendix A: Area. 1 Find the radius of a circle that has circumference 12 inches. Appendi A: Area worked-ou s o Odd-Numbered Eercises Do no read hese worked-ou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa

More information

Reading assignment: Chapter 4 of Aris

Reading assignment: Chapter 4 of Aris Chaper 4 - The Knemacs of Flud Moon Parcle pahs and maeral dervaves Sreamlnes Sreaklnes Dlaaon Reynolds ranspor heorem Conservaon of mass and he equaon of connuy Deformaon and rae of sran Physcal nerpreaon

More information

Permutations and Combinations

Permutations and Combinations Permuaions and Combinaions Combinaorics Copyrigh Sandards 006, Tes - ANSWERS Barry Mabillard. 0 www.mah0s.com 1. Deermine he middle erm in he expansion of ( a b) To ge he k-value for he middle erm, divide

More information

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.

Differential Equations. Solving for Impulse Response. Linear systems are often described using differential equations. Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given

More information

Transformations. Computer Graphics. Types of Transformations. 2D Scaling from the origin. 2D Translations. 9/22/2011. Geometric Transformation

Transformations. Computer Graphics. Types of Transformations. 2D Scaling from the origin. 2D Translations. 9/22/2011. Geometric Transformation 9// anfomaion. Compue Gaphic Lecue anfomaion Wha i a anfomaion? Wha oe i o? anfom he cooinae / nomal veco of objec Wh ue hem? Moelling -Moving he objec o he eie locaion in he envionmen -Muliple inance

More information

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses, 3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

More information

Phys 2101 Gabriela González. cos. sin. sin

Phys 2101 Gabriela González. cos. sin. sin 1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe

More information

Cointegration: The Engle and Granger approach

Cointegration: The Engle and Granger approach Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be non-saionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require

More information

Chapter 8: Regression with Lagged Explanatory Variables

Chapter 8: Regression with Lagged Explanatory Variables Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One

More information

Motion Along a Straight Line

Motion Along a Straight Line Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his

More information

PY1052 Problem Set 8 Autumn 2004 Solutions

PY1052 Problem Set 8 Autumn 2004 Solutions PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

More information

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling

Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: -Solving Exponenial Equaions (The Mehod of Common Bases) -Solving Exponenial Equaions (Using Logarihms)

More information

AP Calculus BC 2010 Scoring Guidelines

AP Calculus BC 2010 Scoring Guidelines AP Calculus BC Scoring Guidelines The College Board The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board

More information

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary

Random Walk in 1-D. 3 possible paths x vs n. -5 For our random walk, we assume the probabilities p,q do not depend on time (n) - stationary Random Walk in -D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes

More information

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)

Mathematics in Pharmacokinetics What and Why (A second attempt to make it clearer) Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions

More information

Newton s Laws of Motion

Newton s Laws of Motion Newon s Laws of Moion MS4414 Theoreical Mechanics Firs Law velociy. In he absence of exernal forces, a body moves in a sraigh line wih consan F = 0 = v = cons. Khan Academy Newon I. Second Law body. The

More information

Mr. Kepple. Motion at Constant Acceleration 1D Kinematics HW#5. Name: Date: Period: (b) Distance traveled. (a) Acceleration.

Mr. Kepple. Motion at Constant Acceleration 1D Kinematics HW#5. Name: Date: Period: (b) Distance traveled. (a) Acceleration. Moion Consn Accelerion 1D Kinemics HW#5 Mr. Kepple Nme: De: Period: 1. A cr cceleres from 1 m/s o 1 m/s in 6.0 s. () Wh ws is ccelerion? (b) How fr did i rel in his ime? Assume consn ccelerion. () Accelerion

More information

Generalized Difference Sequence Space On Seminormed Space By Orlicz Function

Generalized Difference Sequence Space On Seminormed Space By Orlicz Function Ieaoa Joa of Scece ad Eee Reeach IJSER Vo Ie Decembe -4 5687 568X Geeazed Dffeece Seece Sace O Semomed Sace B Ocz Fco A.Sahaaa Aa ofeo G Ie of TechooCombaoeIda. Abac I h aewe defe he eece ace o emomed

More information

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED

More information

Lecture 11 Inductance and Capacitance

Lecture 11 Inductance and Capacitance Lecure Inducance and apacance ELETRIAL ENGINEERING: PRINIPLES AND APPLIATIONS, Fourh Edon, by Allan R. Hambley, 8 Pearson Educaon, Inc. Goals. Fnd he curren olage for a capacance or nducance gen he olage

More information

The Derivative of a Constant is Zero

The Derivative of a Constant is Zero Sme Simple Algrihms fr Calculaing Derivaives The Derivaive f a Cnsan is Zer Suppse we are l ha x x where x is a cnsan an x represens he psiin f an bjec n a sraigh line pah, in her wrs, he isance ha he

More information

Circle Geometry (Part 3)

Circle Geometry (Part 3) Eam aer 3 ircle Geomery (ar 3) emen andard:.4.(c) yclic uadrilaeral La week we covered u otheorem 3, he idea of a convere and we alied our heory o ome roblem called IE. Okay, o now ono he ne chunk of heory

More information

Autonomic management of scalable load-balancing for ubiquitous networks

Autonomic management of scalable load-balancing for ubiquitous networks Auonomic managemen of scalable -balancing fo ubiquious newoks Toshio TONOUCHI and Yasuyuki BEPPU Inene Sysems Laboaoies, NEC Copoaion {onouchi@cw, y-beppu@ak}.jp.nec.com Absac. In ubiquious newoks, a lo

More information

University Physics AI No. 11 Kinetic Theory

University Physics AI No. 11 Kinetic Theory Unersty hyscs AI No. 11 Knetc heory Class Number Name I.Choose the Correct Answer 1. Whch type o deal gas wll hae the largest alue or C -C? ( D (A Monatomc (B Datomc (C olyatomc (D he alue wll be the same

More information

Spline. Computer Graphics. B-splines. B-Splines (for basis splines) Generating a curve. Basis Functions. Lecture 14 Curves and Surfaces II

Spline. Computer Graphics. B-splines. B-Splines (for basis splines) Generating a curve. Basis Functions. Lecture 14 Curves and Surfaces II Lecure 4 Curves and Surfaces II Splne A long flexble srps of meal used by drafspersons o lay ou he surfaces of arplanes, cars and shps Ducks weghs aached o he splnes were used o pull he splne n dfferen

More information

MANAGEMENT SCIENCE doi 10.1287/mnsc.1070.0804ec pp. ec1 ec17

MANAGEMENT SCIENCE doi 10.1287/mnsc.1070.0804ec pp. ec1 ec17 MAAEMET SCIECE doi 087/mn0700804e e e7 e-omanion OY AAIABE I EECTOIC OM infom 008 IOMS Eleoni Comanion Call Cene Ououing Cona Unde Infomaion Aymmey by Samee aija Edieal J inke and obe A Sumky Managemen

More information

Problem Set V Solutions

Problem Set V Solutions Problem Set V Solutions. Consider masses m, m 2, m 3 at x, x 2, x 3. Find X, the C coordinate by finding X 2, the C of mass of and 2, and combining it with m 3. Show this is gives the same result as 3

More information

Multiple criteria network models for project management

Multiple criteria network models for project management Mulple cea newo model fo pojec managemen Víceeální í oé modely pojeoém øízení T. ŠUBRT Czech Uney of Agculue, Pague, Czech Republc Abac: The am of he pape o peen one pobly of how o model and ole a eouce

More information

Capacity Planning. Operations Planning

Capacity Planning. Operations Planning Operaons Plannng Capacy Plannng Sales and Operaons Plannng Forecasng Capacy plannng Invenory opmzaon How much capacy assgned o each producon un? Realsc capacy esmaes Sraegc level Moderaely long me horzon

More information

SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3.

SOLID MECHANICS TUTORIAL GEAR SYSTEMS. This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 3. SOLI MEHNIS TUTORIL GER SYSTEMS This work covers elemens of he syllabus for he Edexcel module 21722P HN/ Mechanical Principles OUTOME 3. On compleion of his shor uorial you should be able o do he following.

More information

Pedro M. Castro Iiro Harjunkoski Ignacio E. Grossmann. Lisbon, Portugal Ladenburg, Germany Pittsburgh, USA

Pedro M. Castro Iiro Harjunkoski Ignacio E. Grossmann. Lisbon, Portugal Ladenburg, Germany Pittsburgh, USA Pedro M. Casro Iro Harjunkosk Ignaco E. Grossmann Lsbon Porugal Ladenburg Germany Psburgh USA 1 Process operaons are ofen subjec o energy consrans Heang and coolng ules elecrcal power Avalably Prce Challengng

More information

Pulse-Width Modulation Inverters

Pulse-Width Modulation Inverters SECTION 3.6 INVERTERS 189 Pulse-Widh Modulaion Inverers Pulse-widh modulaion is he process of modifying he widh of he pulses in a pulse rain in direc proporion o a small conrol signal; he greaer he conrol

More information

Coordinate Systems L. M. Kalnins, March 2009

Coordinate Systems L. M. Kalnins, March 2009 Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

More information

MORE ON TVM, "SIX FUNCTIONS OF A DOLLAR", FINANCIAL MECHANICS. Copyright 2004, S. Malpezzi

MORE ON TVM, SIX FUNCTIONS OF A DOLLAR, FINANCIAL MECHANICS. Copyright 2004, S. Malpezzi MORE ON VM, "SIX FUNCIONS OF A DOLLAR", FINANCIAL MECHANICS Copyrgh 2004, S. Malpezz I wan everyone o be very clear on boh he "rees" (our basc fnancal funcons) and he "fores" (he dea of he cash flow model).

More information

1.- L a m e j o r o p c ió n e s c l o na r e l d i s co ( s e e x p li c a r á d es p u é s ).

1.- L a m e j o r o p c ió n e s c l o na r e l d i s co ( s e e x p li c a r á d es p u é s ). PROCEDIMIENTO DE RECUPERACION Y COPIAS DE SEGURIDAD DEL CORTAFUEGOS LINUX P ar a p od e r re c u p e ra r nu e s t r o c o rt a f u e go s an t e un d es a s t r e ( r ot u r a d e l di s c o o d e l a

More information

Exam 3: Equation Summary

Exam 3: Equation Summary MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

More information

B I N G O B I N G O. Hf Cd Na Nb Lr. I Fl Fr Mo Si. Ho Bi Ce Eu Ac. Md Co P Pa Tc. Uut Rh K N. Sb At Md H. Bh Cm H Bi Es. Mo Uus Lu P F.

B I N G O B I N G O. Hf Cd Na Nb Lr. I Fl Fr Mo Si. Ho Bi Ce Eu Ac. Md Co P Pa Tc. Uut Rh K N. Sb At Md H. Bh Cm H Bi Es. Mo Uus Lu P F. Hf Cd Na Nb Lr Ho Bi Ce u Ac I Fl Fr Mo i Md Co P Pa Tc Uut Rh K N Dy Cl N Am b At Md H Y Bh Cm H Bi s Mo Uus Lu P F Cu Ar Ag Mg K Thomas Jefferson National Accelerator Facility - Office of cience ducation

More information

Perturbation Theory and Celestial Mechanics

Perturbation Theory and Celestial Mechanics Copyght 004 9 Petubaton Theoy and Celestal Mechancs In ths last chapte we shall sketch some aspects of petubaton theoy and descbe a few of ts applcatons to celestal mechancs. Petubaton theoy s a vey boad

More information

10. Collisions. Before During After

10. Collisions. Before During After 10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

More information

Ultraconservative Online Algorithms for Multiclass Problems

Ultraconservative Online Algorithms for Multiclass Problems Jounal of Machine Leaning Reseach 3 (2003) 951-991 Submied 2/02; Published 1/03 Ulaconsevaive Online Algoihms fo Muliclass Poblems Koby Camme Yoam Singe School of Compue Science & Engineeing Hebew Univesiy,

More information

Robotics and Autonomous Systems. Cross-spectral visual simultaneous localization and mapping (SLAM) with sensor handover

Robotics and Autonomous Systems. Cross-spectral visual simultaneous localization and mapping (SLAM) with sensor handover Robocs and Auonomous Sysems 61 (213) 19 28 Conens lss avalable a ScVese ScenceDec Robocs and Auonomous Sysems jounal homepage: www.elseve.com/locae/obo Coss-specal vsual smulaneous localzaon and mappng

More information

Steps for D.C Analysis of MOSFET Circuits

Steps for D.C Analysis of MOSFET Circuits 10/22/2004 Seps for DC Analysis of MOSFET Circuis.doc 1/7 Seps for D.C Analysis of MOSFET Circuis To analyze MOSFET circui wih D.C. sources, we mus follow hese five seps: 1. ASSUME an operaing mode 2.

More information

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27 Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

More information

Carter-Penrose diagrams and black holes

Carter-Penrose diagrams and black holes Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

More information

Chem 115 POGIL Worksheet - Week 4 Moles & Stoichiometry

Chem 115 POGIL Worksheet - Week 4 Moles & Stoichiometry Chem 115 POGIL Worksheet - Week 4 Moles & Stoichiometry Why? Chemists are concerned with mass relationships in chemical reactions, usually run on a macroscopic scale (grams, kilograms, etc.). To deal with

More information

Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

More information

9. Capacitor and Resistor Circuits

9. Capacitor and Resistor Circuits ElecronicsLab9.nb 1 9. Capacior and Resisor Circuis Inroducion hus far we have consider resisors in various combinaions wih a power supply or baery which provide a consan volage source or direc curren

More information

Automatic measurement and detection of GSM interferences

Automatic measurement and detection of GSM interferences Auomaic measuremen and deecion of GSM inerferences Poor speech qualiy and dropped calls in GSM neworks may be caused by inerferences as a resul of high raffic load. The radio nework analyzers from Rohde

More information

Future Trends in Airline Pricing, Yield. March 13, 2013

Future Trends in Airline Pricing, Yield. March 13, 2013 Future Trends in Airline Pricing, Yield Management, &AncillaryFees March 13, 2013 THE OPPORTUNITY IS NOW FOR CORPORATE TRAVEL MANAGEMENT BUT FIRST: YOU HAVE TO KNOCK DOWN BARRIERS! but it won t hurt much!

More information

Optimal Investment and Consumption Decision of Family with Life Insurance

Optimal Investment and Consumption Decision of Family with Life Insurance Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker

More information

Stability. Coefficients may change over time. Evolution of the economy Policy changes

Stability. Coefficients may change over time. Evolution of the economy Policy changes Sabiliy Coefficiens may change over ime Evoluion of he economy Policy changes Time Varying Parameers y = α + x β + Coefficiens depend on he ime period If he coefficiens vary randomly and are unpredicable,

More information

How To Calculate Backup From A Backup From An Oal To A Daa

How To Calculate Backup From A Backup From An Oal To A Daa 6 IJCSNS Inernaonal Journal of Compuer Scence and Nework Secury, VOL.4 No.7, July 04 Mahemacal Model of Daa Backup and Recovery Karel Burda The Faculy of Elecrcal Engneerng and Communcaon Brno Unversy

More information

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural

More information

Derivative Securities: Lecture 7 Further applications of Black-Scholes and Arbitrage Pricing Theory. Sources: J. Hull Avellaneda and Laurence

Derivative Securities: Lecture 7 Further applications of Black-Scholes and Arbitrage Pricing Theory. Sources: J. Hull Avellaneda and Laurence Deivaive ecuiies: Lecue 7 uhe applicaios o Black-choles ad Abiage Picig heoy ouces: J. Hull Avellaeda ad Lauece Black s omula omeimes is easie o hik i ems o owad pices. Recallig ha i Black-choles imilaly

More information

Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleration Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

More information

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem

Functions of a Random Variable: Density. Math 425 Intro to Probability Lecture 30. Definition Nice Transformations. Problem Intoduction One Function of Random Vaiables Functions of a Random Vaiable: Density Math 45 Into to Pobability Lectue 30 Let gx) = y be a one-to-one function whose deiatie is nonzeo on some egion A of the

More information

29 March 2006. Application of Annuity Depreciation in the Presence of Competing Technologies II Telecom New Zealand

29 March 2006. Application of Annuity Depreciation in the Presence of Competing Technologies II Telecom New Zealand 29 Mach 2006 Applicaion of Annuiy Depeciaion in he Pesence of Compeing Technologies II Telecom ew Zealand Pojec Team Tom Hid (Ph.D.) Daniel Young EA Economic Consuling Level 6 33 Exhibiion See Melboune

More information

Interface Design for Rationally Clocked GALS Systems

Interface Design for Rationally Clocked GALS Systems Intefae Deign fo Rationally Cloked GALS Sytem Joyee Mekie, Supatik Chakaboty, Giih Venkataamani,.S. Thiagaajan, D.K. Shama Mah 15, 2006 Motivation SoC : integation of pedeigned I oe Rationally loked ytem

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Inductance and Transient Circuits

Inductance and Transient Circuits Chaper H Inducance and Transien Circuis Blinn College - Physics 2426 - Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual

More information

1. Time Value of Money 3 2. Discounted Cash Flow 35 3. Statistics and Market Returns 49 4. Probabilities 81 5. Key Formulas 109

1. Time Value of Money 3 2. Discounted Cash Flow 35 3. Statistics and Market Returns 49 4. Probabilities 81 5. Key Formulas 109 1. Time Value of Money 3 2. Discouned Cash Flow 35 3. Saisics and Make Reuns 49 4. Pobabiliies 81 5. Key Fomulas 109 Candidae Noe: This is a lenghy Sudy Session ha, along wih Sudy Session 3, you should

More information

Analyzing Energy Use with Decomposition Methods

Analyzing Energy Use with Decomposition Methods nalyzng nergy Use wh Decomposon Mehods eve HNN nergy Technology Polcy Dvson eve.henen@ea.org nergy Tranng Week Pars 1 h prl 213 OCD/ 213 Dscusson nergy consumpon and energy effcency? How can energy consumpon

More information

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere. Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

More information

NURBS Drawing Week 5, Lecture 10

NURBS Drawing Week 5, Lecture 10 CS 43/585 Compute Gaphics I NURBS Dawing Week 5, Lectue 1 David Been, William Regli and Maim Pesakhov Geometic and Intelligent Computing Laboato Depatment of Compute Science Deel Univesit http://gicl.cs.deel.edu

More information

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100...

Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100... Normal (Gaussian) Disribuion Probabiliy De ensiy 0.5 0. 0.5 0. 0.05 0. 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0. 0 3.6 5. 6.8 8.4 0.6 3. 4.8 6.4 8 The Black-Scholes Shl Ml Moel... pricing opions an calculaing

More information

Signal Processing and Linear Systems I

Signal Processing and Linear Systems I Sanford Universiy Summer 214-215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 14-15, Gibbons

More information

REVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ),

REVISTA INVESTIGACION OPERACIONAL Vol. 25, No. 1, 2004. k n ), REVISTA INVESTIGACION OPERACIONAL Vol 25, No, 24 RECURRENCE AND DIRECT FORMULAS FOR TE AL & LA NUMBERS Eduardo Pza Volo Cero de Ivesgacó e Maemáca Pura y Aplcada (CIMPA), Uversdad de Cosa Rca ABSTRACT

More information