Rotational Inertia of a Point Mass


 Merryl Turner
 2 years ago
 Views:
Transcription
1 Roaional Ineria of a Poin Mass Saddleback College Physics Deparmen, adaped from PASCO Scienific PURPOSE The purpose of his experimen is o find he roaional ineria of a poin experimenally and o verify ha his value corresponds o he calculaed heoreical value. EQUIPMENT  pulley  paper clips (for es < 1 gram)  calipers  es and weigh hanger se  phoogae imers wih 3 phoogae heads  riple beam balance  meersick g (black, square)  roaing plaform  A base  hread/sring SKETCH 8/06 1
2 Figure.3 The seup shown above, wih he phoogae in Pendulum mode, enables one o deermine wheher he hanging is falling a an approximaely consan rae. Figure.4 When arranging he wo phoogaes as shown o he lef, be sure he sring and hook (a he op of he weigh hanger) do no break he phoogae beam. 8/06
3 THEORY Theoreically, he roaional ineria, I, of a poin is given by and R is he disance he is from he axis of roaion. I = MR, where M is he To find he roaional ineria of a poin experimenally, a known orque is applied o he objec and he resuling angular acceleraion is measured. Since " = I!, rearranges o " I =! (1) where! = a () r and! = rt (3) (where r = radius of cylinder abou which he hread is wound and T is he ension in he hread when he apparaus is roaing.) Subsiue equaions () and (3) ino equaion (1) above and wrie he resul in he box below. (4) Apply Newon s Second Law o he hanging, call i m for now, and hen solve he resuling equaion for he ension in he hread, T. Subsiue his value for T ino equaion (4) above and you have a formula yielding he experimenal roaional ineria in erms of he linear acceleraion of he, m. Call his new equaion (5) and wrie i below. (5) The linear acceleraion, a, of he, m, can be found wih he kinemaic equaion below: 1 a! y = v + o (6) where o v is he iniial velociy of he hanging (zero), and is he (average) ime for he hanging o fall from res hrough a displacemen! y. NOTICE: You will be solving he above equaions for wo differen cases o find he experimenal roaional ineria of he poin. To find I you mus find in I hen subrac ou he roaional ineria of he apparaus alone (i.e. po in & apparaus wih no poin ). The mahemaical relaionship is seen below: I = I in po & apparaus po in  apparaus alone po I (7) 8/06 3
4 PROCEDURE Equipmen Seup 1) Leveling he Roaing Plaform: This may ake 30 minues o do correcly, bu i is ime well spen! Before leveling he apparaus you MUST se enire apparaus in he posiion shown in Figure.4. Noice ha he hanging off he pulley canno hi he able and he sring wound around he cylinder mus pass sraigh over he pulley. ONCE APPARATUS IS LEVEL, YOU CANNOT MOVE IT! ) Se up your apparaus as shown in Figures.1 and. (above), making sure he hanging almos reaches he floor when he sring is compleely unwound. Par I: Measuremens for he Theoreical Roaional Ineria 1) Deermine he Mass of he square black, M (he poin ) and record i in able.1. ) Aach he square black o he rack on he roaing plaform a a large radius. 3) Measure he disance from he axis of roaion o he cener of he square and record his radius, R, in Table.1 8/06 4
5 Theoreical Roaional Ineria Mass, M Radius, R Table.1 Par II: Measuremens for he Experimenal Roaional Ineria ACOUNTING FOR FRICTION Because he heory used o find he roaional ineria experimenally does no include fricion, i will be compensaed for in his experimen by finding ou how much over he pulley i akes o overcome kineic fricion and allow he o drop a a consan speed. Then his fricion will be subraced from he hanging ha was used o accelerae M and he resuling will be m, see calculaions. 1) To find he required o overcome kineic fricion, place a phoogae (se on PENDULUM ) so ha he roaing plaform will break he beam and ime one complee cycle of he roaing plaform. ) Tie he sring hrough he hole in he middle cylinder and wind i evenly abou ha cylinder. 3) Place jus enough over he pulley (using paperclips) so ha he roaing plaform moves a a consan angular speed. You migh need o lighly nudge he hanging or pulley o help he sysem overcome saic fricion. Jus eyeball he speed a firs hen when you hink he speed is nearly consan, use he phoogae (se on Pendulum mode) o ime a revoluion early on and o ime a revoluion oward he end of he moion (see Figure.3 for seup). Clearly hese wo periods should be he same if a consan angular speed is aained. Try o ge he periods o mach o hree significan figures. Record he fricion in able., under he appropriae column. Finding he Acceleraion of he Poin Mass and Apparaus 1) You will no use he above phoogae unil you repea he procedure wih he Apparaus Alone. Carefully clamp/ape he oher wo phoogae heads ono a verical sand abou 50+ cm apar and connec hem o a phoogae imer se on PULSE and 1 ms (which means he precision of he phoogae imer is o he neares millisecond, bu he unis of he reading are sill seconds). See Figure.4 for seup. The op phoogae beam mus be broken by he hanging immediaely when i begins o fall, since he iniial velociy of he hanging is assumed o be ZERO in equaion (6) of he heory. Measure he separaion beween he phoogae heads,! y, and record i in able.. Make sure he sring suspending he hanging does no pass hrough he beams of he phoogae heads, as his can break he beam premaurely! When he phoogae imers reach 0 seconds hey rese o 10 seconds and hen coninue o coun every 10 seconds hereafer. Unforunaely hey are inended o coun shorer inervals. Do a pracice run and jus wach he phoogae imer o learn how o correcly read and record he ime of fall. ) To find he acceleraion, place abou 50 g (record he hanging in able. under he appropriae column) over he pulley and wind he hread around he 8/06 5
6 plaform. Using he verical phoogaes, ime he hanging as i falls hrough he phoogaes and record he ime in he appropriae column in able.. Do his for a oal of 15 rials. Measuring he Radius, r Using calipers, measure he diameer of he cylinder abou which he hread is wrapped and calculae he radius. Record his in Table.. Finding he Acceleraion of he Apparaus Alone Since in he Par II procedure Finding he Acceleraion of he Poin Mass and Apparaus, he apparaus is roaing as well as he poin, i is necessary o deermine he acceleraion and roaional ineria of he apparaus by iself so ha his roaional ineria can be subraced from he oal, leaving only he roaional ineria of he poin. 1) Take he poin off he roaional apparaus and repea he Par II procedure Finding he Acceleraion of he Poin Mass and Apparaus for he apparaus ALONE! ) Record he daa in Table. under Apparaus Alone. 8/06 6
7 Table. Poin Mass & Apparaus Apparaus Alone Fricion Mass Hanging Mass Mass, m (for use in equaions) Radius, r (of cylinder) Acceleraion, a (of hanging ) Disance beween Phoogaes, Time, (o fall hrough phoogaes) ! y Average ime, Poin Mass & Apparaus Apparaus Alone 8/06 7
8 Calculaions 1) Subrac he fricion from he hanging used o accelerae he apparaus o deermine he, m, o be used in he equaions and record in Table.. ) Use he average ime of fall o deermine he acceleraion of he sysem from equaion (6). Record he acceleraion in he appropriae column of Table.. 3) Calculae he experimenal value of he roaional ineria of he poin and apparaus ogeher and record in Table.3. 4) Calculae he experimenal value of he roaional ineria of he apparaus alone. Record in Table.3. 5) Apply equaion (7) in he heory o ge he roaional ineria of he poin alone. Record in Table.3. 6) Calculae he heoreical value of he roaional ineria of he poin. Record in Table.3. 7) Compare he experimenal value o he heoreical value. Record in Table.3. I po in I apparaus & alone apparaus Resuls I in (Experimenal Value) po I in (Theoreical Value) po % Difference in I po in Table.3 8/06 8
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 information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationDiscussion Examples Chapter 10: Rotational Kinematics and Energy
Discussion Examples Chaper : Roaional Kinemaics and Energy 9. The Crab Nebula One o he mos sudied objecs in he nigh sky is he Crab nebula, he remains o a supernova explosion observed by he Chinese in 54.
More informationChapter 2 Problems. s = d t up. = 40km / hr d t down. 60km / hr. d t total. + t down. = t up. = 40km / hr + d. 60km / hr + 40km / hr
Chaper 2 Problems 2.2 A car ravels up a hill a a consan speed of 40km/h and reurns down he hill a a consan speed of 60 km/h. Calculae he average speed for he rip. This problem is a bi more suble han i
More informationAP1 Kinematics (A) (C) (B) (D) Answer: C
1. A ball is hrown verically upward from he ground. Which pair of graphs bes describes he moion of he ball as a funcion of ime while i is in he air? Neglec air resisance. y a v a (A) (C) y a v a (B) (D)
More informationChapter 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 informationChapter 11A Angular Motion. A PowerPoint Presentation by Paul E. Tippens, Professor of Physics Southern Polytechnic State University
Chaper 11A Angular Moion A PowerPoin Presenaion by Paul E. Tippens, Proessor o Physics Souhern Polyechnic Sae Universiy 007 WIND TUBINES such as hese can generae signiican energy in a way ha is environmenally
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationChapter 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 informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationWeek #9  The Integral Section 5.1
Week #9  The Inegral Secion 5.1 From Calculus, Single Variable by HughesHalle, Gleason, McCallum e. al. Copyrigh 005 by John Wiley & Sons, Inc. This maerial is used by permission of John Wiley & Sons,
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationSection 7.1 Angles and Their Measure
Secion 7.1 Angles and Their Measure Greek Leers Commonly Used in Trigonomery Quadran II Quadran III Quadran I Quadran IV α = alpha β = bea θ = hea δ = dela ω = omega γ = gamma DEGREES The angle formed
More informationRC, RL and RLC circuits
Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.
More informationCHARGE 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 informationSection A: Forces and Motion
I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and
More informationNewton's second law in action
Newon's second law in acion In many cases, he naure of he force acing on a body is known I migh depend on ime, posiion, velociy, or some combinaion of hese, bu is dependence is known from experimen In
More informationLAB 6: SIMPLE HARMONIC MOTION
1 Name Dae Day/Time of Lab Parner(s) Lab TA Objecives LAB 6: SIMPLE HARMONIC MOTION To undersand oscillaion in relaion o equilibrium of conservaive forces To manipulae he independen variables of oscillaion:
More informationSection 2.3 Linear and Angular Velocities
Secion 2.3 Linear and Angular Velociies The mos inuiive measure of he rae a which he rider is raveling around he wheel is wha we call linear velociy. Anoher way o specify how fas he rider is raveling around
More informationMOTION ALONG A STRAIGHT LINE
Chaper 2: MOTION ALONG A STRAIGHT LINE 1 A paricle moes along he ais from i o f Of he following alues of he iniial and final coordinaes, which resuls in he displacemen wih he larges magniude? A i =4m,
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationName: Teacher: DO NOT OPEN THE EXAMINATION PAPER UNTIL YOU ARE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINAL EXAMINATION. June 2009.
Name: Teacher: DO NOT OPEN THE EXMINTION PPER UNTIL YOU RE TOLD BY THE SUPERVISOR TO BEGIN PHYSICS 2204 FINL EXMINTION June 2009 Value: 100% General Insrucions This examinaion consiss of wo pars. Boh pars
More informationcooking 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 informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder 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 informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More information( ) in the following way. ( ) < 2
Sraigh Line Moion  Classwork Consider an obbec moving along a sraigh line eiher horizonally or verically. There are many such obbecs naural and manmade. Wrie down several of hem. Horizonal cars waer
More informationM3. Experiment 3 NEWTON S LAWS OF MOTION M3. Purpose: Investigation of Newton s Laws of Motion using air track rail.
Experien 3 NEWTON S LAWS O OTION Purpose: Invesigaion of Newon s Laws of oion using air rack rail. Equipens: Air rack, blower (air source), ier, phoogaes, s wih differen asses, asses (0g), rope, pencil,
More information2.6 Limits at Infinity, Horizontal Asymptotes Math 1271, TA: Amy DeCelles. 1. Overview. 2. Examples. Outline: 1. Definition of limits at infinity
.6 Limis a Infiniy, Horizonal Asympoes Mah 7, TA: Amy DeCelles. Overview Ouline:. Definiion of is a infiniy. Definiion of horizonal asympoe 3. Theorem abou raional powers of. Infinie is a infiniy This
More informationA 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 noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy.
More informationNewton 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 informationTEACHER NOTES HIGH SCHOOL SCIENCE NSPIRED
Radioacive Daing Science Objecives Sudens will discover ha radioacive isoopes decay exponenially. Sudens will discover ha each radioacive isoope has a specific halflife. Sudens will develop mahemaical
More informationVelocity & Acceleration Analysis
Velociy & Acceleraion Analysis Secion 4 Velociy analysis deermines how fas pars of a machine are moving. Linear Velociy (v) Sraigh line, insananeous speed of a poin. ds s v = d Linear velociy is a vecor.
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationMotion 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 informationDigital Data Acquisition
ME231 Measuremens Laboraory Spring 1999 Digial Daa Acquisiion Edmundo Corona c The laer par of he 20h cenury winessed he birh of he compuer revoluion. The developmen of digial compuer echnology has had
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More information2. Waves in Elastic Media, Mechanical Waves
2. Waves in Elasic Media, Mechanical Waves Wave moion appears in almos ever branch of phsics. We confine our aenion o waves in deformable or elasic media. These waves, for eample ordinar sound waves in
More informationImagine 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 informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO
Profi Tes Modelling in Life Assurance Using Spreadshees, par wo PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART TWO Erik Alm Peer Millingon Profi Tes Modelling in Life Assurance Using Spreadshees,
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More information11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.
11. Tire pressure. The graph You have a hole in your ire. You pump i up o P=400 kilopascals (kpa) and over he nex few hours i goes down ill he ire is quie fla. Draw wha you hink he graph of ire pressure
More informationPhysics 111 Fall 2007 Electric Currents and DC Circuits
Physics 111 Fall 007 Elecric Currens and DC Circuis 1 Wha is he average curren when all he sodium channels on a 100 µm pach of muscle membrane open ogeher for 1 ms? Assume a densiy of 0 sodium channels
More informationRC (ResistorCapacitor) Circuits. AP Physics C
(ResisorCapacior 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 informationSOLID 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 informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationLenz's Law. Definition from the book:
Lenz's Law Definiion from he book: The induced emf resuling from a changing magneic flux has a polariy ha leads o an induced curren whose direcion is such ha he induced magneic field opposes he original
More informationA Mathematical Description of MOSFET Behavior
10/19/004 A Mahemaical Descripion of MOSFET Behavior.doc 1/8 A Mahemaical Descripion of MOSFET Behavior Q: We ve learned an awful lo abou enhancemen MOSFETs, bu we sill have ye o esablished a mahemaical
More informationCircuit Types. () i( t) ( )
Circui Types DC Circuis Idenifying feaures: o Consan inpus: he volages of independen volage sources and currens of independen curren sources are all consan. o The circui does no conain any swiches. All
More informationName: 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 information4kq 2. D) south A) F B) 2F C) 4F D) 8F E) 16F
efore you begin: Use black pencil. Wrie and bubble your SU ID Number a boom lef. Fill bubbles fully and erase cleanly if you wish o change! 20 Quesions, each quesion is 10 poins. Each quesion has a mos
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More information9. 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 informationChapter 15: Superposition and Interference of Waves
Chaper 5: Superposiion and Inerference of Waves Real waves are rarely purely sinusoidal (harmonic, bu hey can be represened by superposiions of harmonic waves In his chaper we explore wha happens when
More informationMachine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Flywheel. A flywheel is an inertial energystorage device. It absorbs mechanical
Flywheel A lywheel is an inerial energysorage device. I absorbs mechanical energy and serves as a reservoir, soring energy during he period when he supply o energy is more han he requiremen and releases
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationRandom Walk in 1D. 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 informationApplication of kinematic equation:
HELP: See me (office hours). There will be a HW help session on Monda nigh from 78 in Nicholson 109. Tuoring a #10 of Nicholson Hall. Applicaion of kinemaic equaion: a = cons. v= v0 + a = + v + 0 0 a
More informationHANDOUT 14. A.) Introduction: Many actions in life are reversible. * Examples: Simple One: a closed door can be opened and an open door can be closed.
Inverse Funcions Reference Angles Inverse Trig Problems Trig Indeniies HANDOUT 4 INVERSE FUNCTIONS KEY POINTS A.) Inroducion: Many acions in life are reversible. * Examples: Simple One: a closed door can
More informationMathematics 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 informationPermutations 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 kvalue for he middle erm, divide
More informationForm measurement systems from HommelEtamic Geometrical tolerancing in practice DKDK02401. Precision is our business.
Form measuremen sysems from HommelEamic Geomerical olerancing in pracice DKDK02401 Precision is our business. Drawing enries Tolerance frame 0.01 0.01 Daum leer Tolerance value in mm Symbol for he oleranced
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationDamped Harmonic Motion Closing Doors and Bumpy Rides
Prerequisies and Goal Damped Harmonic Moion Closing Doors and Bumpy Rides Andrew Forreser May 4, 21 Assuming you are familiar wih simple harmonic moion, is equaion of moion, and is soluions, we will now
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationPhysics 107 HOMEWORK ASSIGNMENT #2
Phsics 7 HOMEWORK ASSIGNMENT # Cunell & Johnson, 7 h ediion Chaper : Problem 5 Chaper : Problems 44, 54, 56 Chaper 3: Problem 38 *5 MulipleConcep Example 9 deals wih he conceps ha are imporan in his problem.
More informationTHE PRESSURE DERIVATIVE
Tom Aage Jelmer NTNU Dearmen of Peroleum Engineering and Alied Geohysics THE PRESSURE DERIVATIVE The ressure derivaive has imoran diagnosic roeries. I is also imoran for making ye curve analysis more reliable.
More informationRC Circuit and Time Constant
ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisorcapacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he
More informationChapter 7: Estimating the Variance of an Estimate s Probability Distribution
Chaper 7: Esimaing he Variance of an Esimae s Probabiliy Disribuion Chaper 7 Ouline Review o Clin s Assignmen o General Properies of he Ordinary Leas Squares (OLS) Esimaion Procedure o Imporance of he
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA UNIT 67  FURTHER ELECTRICAL PRINCIPLES NQF LEVEL 3 OUTCOME 2 TUTORIAL 1  TRANSIENTS
EDEXEL NAIONAL ERIFIAE/DIPLOMA UNI 67  FURHER ELERIAL PRINIPLE NQF LEEL 3 OUOME 2 UORIAL 1  RANIEN Uni conen 2 Undersand he ransien behaviour of resisorcapacior (R) and resisorinducor (RL) D circuis
More informationm m m m m correct
Version 055 Miderm 1 OConnor (05141) 1 This prinou should have 36 quesions. Muliplechoice quesions ma coninue on he ne column or pae find all choices before answerin. V1:1, V:1, V3:3, V4:, V5:1. 001
More informationDifferential 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 informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
More informationEntropy: From the Boltzmann equation to the Maxwell Boltzmann distribution
Enropy: From he Bolzmann equaion o he Maxwell Bolzmann disribuion A formula o relae enropy o probabiliy Ofen i is a lo more useful o hink abou enropy in erms of he probabiliy wih which differen saes are
More informationFullwave rectification, bulk capacitor calculations Chris Basso January 2009
ullwave recificaion, bulk capacior calculaions Chris Basso January 9 This shor paper shows how o calculae he bulk capacior value based on ripple specificaions and evaluae he rms curren ha crosses i. oal
More informationBrown University PHYS 0060 Physics Department LAB B Measuring the Earth s Magnetic Field
Measuring he Earh s Magneic Field As is well known he Earh has a small magneic field on he order of 0.5 Gauss. I is an ineresing and challenging lab exercise o ry and measure i. elow we lis 5 mehods wih
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationWhen one talks about a 'projectile', the implicabion is itrai we give an object
LAB, PROJECTLE MOruO.^\ 45 Lab Projecile Moion 1 nroducion n his lab we will look a he moion of a projecile in wo dimensions. When one alks abou a 'projecile', he implicabion is irai we give an objec an
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 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 1415, Gibbons
More informationKinematics in 1D From Problems and Solutions in Introductory Mechanics (Draft version, August 2014) David Morin, morin@physics.harvard.
Chaper 2 Kinemaics in 1D 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 informationAmerican Spirit Sample
American Spiri Sample Inroducion 9 Philosophy 9 The Five Developmenal Sages of Spelling 10 Curriculum Sequence and Placemen Guidelines 13 Abou American Spiri 15 Geing Sared 15 Overview 15 Needed Iems 15
More informationPeriod 4 Activity Solutions: Transfer of Thermal Energy
Period 4 Aciviy Soluions: Transfer of Thermal nergy 4.1 How Does Temperaure Differ from Thermal nergy? a) Temperaure Your insrucor will demonsrae molecular moion a differen emperaures. 1) Wha happens o
More informationChapter 12 PURE TORSION
Chaper 1 PURE TORSION 1.1 GENERALS A member is subjeced o pure orsion if in any cross secion of his member he single sress differen from zero is he momen of orsion or wising (shorer TORQUE). Pure orsion
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationChapter 8 Copyright Henning Umland All Rights Reserved
Chaper 8 Copyrigh 19972004 Henning Umland All Righs Reserved Rise, Se, Twiligh General Visibiliy For he planning of observaions, i is useful o know he imes during which a cerain body is above he horizon
More informationRevisions to Nonfarm Payroll Employment: 1964 to 2011
Revisions o Nonfarm Payroll Employmen: 1964 o 2011 Tom Sark December 2011 Summary Over recen monhs, he Bureau of Labor Saisics (BLS) has revised upward is iniial esimaes of he monhly change in nonfarm
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationInductance 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 informationChapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )
Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y
More informationAP Physics Velocity and Linear Acceleration Unit 1 Problems:
Uni 1 Problems: Linear Velociy and Acceleraion This enire se of problems is due he day of he es. I will no accep hese for a lae grade. * = Problems we do ogeher; all oher problems are homework (bu we will
More information1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z 1 A B C D E F G H I J K L M N O P Q R S { U V W X Y Z
o ffix uden abel ere uden ame chool ame isric ame/ ender emale ale onh ay ear ae of irh an eb ar pr ay un ul ug ep c ov ec as ame irs ame lace he uden abel ere ae uden denifier chool se nly rined in he
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More information11. Properties of alternating currents of LCRelectric circuits
WS. Properies of alernaing currens of Lelecric circuis. Inroducion Socalled passive elecric componens, such as ohmic resisors (), capaciors () and inducors (L), are widely used in various areas of science
More informationChapter 6. First Order PDEs. 6.1 Characteristics The Simplest Case. u(x,t) t=1 t=2. t=0. Suppose u(x, t) satisfies the PDE.
Chaper 6 Firs Order PDEs 6.1 Characerisics 6.1.1 The Simples Case Suppose u(, ) saisfies he PDE where b, c are consan. au + bu = 0 If a = 0, he PDE is rivial (i says ha u = 0 and so u = f(). If a = 0,
More information