# 11. Tire pressure. Here we always work with relative pressure. That s what everybody always does.

Save this PDF as:

Size: px
Start display at page:

Download "11. Tire pressure. Here we always work with relative pressure. That s what everybody always does."

## Transcription

1 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 P agains ime should look like. 400 P I hrow his quesion ou o he class and hey draw a graph like ha a he righ. [They learned heir waer-ank lesson well.] As air flows ou, P decreases, bu as he pressure in he ire ges less, here s less force pushing he air ou and so he air will flow ou more slowly, so herefore he pressure will go down more slowly. Tha seems o make he graph concaveup. Evenually he pressure drops o zero and he graph his he -axis. The problem I give hem now is o consruc a model for he P- relaionship. Tha is, you are required o use wha undersanding of air pressure you already have, or can acquire, o deermine exacly how he pressure in he leaking ire should change over ime, and hence find he form of an equaion relaing P o. Le s sar wih a conjecure. Wha sor of equaion do you hink i migh be? Isn i a parabola? Why would you hink ha? Remember he waer ank experimen? We go a parabola ou of ha, and isn his really he same hing? Is i? Is a ire he same as a waer ank? Grea nosalgia for me when hey ask his quesion because I am reminded of a cerain Sunday afernoon, long ago, when I was young and green and hinking abou his very quesion, he waer ank and he ire boh wih a small hole, how hey were he same and how hey were differen. And a one poin I convinced myself ha hey should be prey much he same and ha he air pressure graph should herefore a parabola. 0 Relaive and absolue pressure When you pu your pressure gauge on he ire you read 400 kpa and you hink ha s he pressure inside bu you re wrong. I s acually 500 kpa. The gauge measures pressure relaive o he ouside air, and ambien air pressure is 100 kpa. If you ook he same ire ino ouer space, he gauge would read 500 kpa. Here we always work wih relaive pressure. Tha s wha everybody always does. And ha s all very well, bu he rouble was ha I had anoher compleely reasonable argumen ha i wasn a parabola, an argumen ha was prey imporan o me because i was based on my concepion of wha air really was, a whole collecion of iny randomly moving molecules. So was I abou o lose his long-held concepion? I was in quie a quandary and had o call up one of my physics profs o ge my hinking sraigh. Read on! 11. ire pressure 1

2 Is a ire he same as a waer ank? Le s sar wih he waer ank. The waer ges pushed ou of he hole because of he pressure, and ha s deermined by he amoun of he waer in he ank, and as he waer flows ou he pressure ges less so he flow rae decreases. Now ha s wha happens in he ire as well. The air ges pushed ou because of he pressure, and ha s deermined by he amoun of air in he ire, and as he air flows ou he pressure ges less so he flow rae decreases. There seems o be a big similariy here. Bu i urns ou ha waer and air are differen in an essenial way. Waer pours ou of a hole in he ank in quie a differen way from he rush of air ou of a hole in he ire. And he difference, in a word, is graviy! The main facor behind he flow of waer ou of he ank is he force of graviy. For example i makes a difference o he flow rae wheher he hole is a he boom of he ank or half-way up. Bu for he ire, he force of graviy is negligible i doesn maer wheher he hole is a he op or he boom of he ire he flow rae will be he same. Le s ge down o basics. The air in he ire consiss of a large number of molecules, which are consanly in moion. Now wha happens when one of hose molecules his he inside surface of he ire? Well, i bounces off. And in fac ha s wha causes he pressure in he ire. When you poke he ire and push i in, why does i pop back ou when you ake your finger off? because all hose molecules are colliding wih he inside surface of he ire and pushing i back. Now. Wha happens if here s a hole so ha a molecule heading for he inside surface of he ire his he hole insead? Well i shoos ou and escapes. And ha s wha causes he flow ou of he hole. Surprisingly enough, ha s all here is o i. Every molecule ha flows ou is one ha was heading for he surface of he ire, minding is own molecular business, and found a hole insead. I lead my sudens hrough his analysis wih as lile promping as possible. In fac I am impressed by how much hey seem o be able o do, a leas he ones who are prepared o conribue. The sudens have a endency o hink of he ire as a balloon, wih he air being pushed ou because of he elasic force of he ube as i conracs. Bu when he ube is imprisoned inside he ire, i doesn srech as i ges filled. If i did, i would be weaker. So i s no like a balloon a all. In fac i s beer o hink of he ire as a rigid srucure made of hard plasic wih air pressure inside. Pu your finger jus above he hole and feel ha lile je of air. Those molecules are jus escaping by chance? Wow. Jus o emphasize he difference, suppose we ook he waer ank ou ino space where here was no graviy. Then here d be nohing o sop he waer molecules from wandering around he inside of he ank and we d have o pu a lid on he ank or hey d wander ou he op. Bu wha abou he hole? Well hey d wander ou of ha oo, in jus he same way ha he air molecules wandered ou of he hole in he ire. And in his case, he z-curve of he amoun of waer remaining in he ank wouldn be a parabola anymore, i would be he same kind of curve as we ge for he ire. 11. ire pressure 2

3 So wha is he air pressure curve anyway? I s ime o ask jus wha he ire curve migh be. Consider he following quesion. Suppose your ire has a small leak. A one poin you measure he pressure o be 400 kpa. Suppose over he nex minue i drops o 384 kpa a loss of 16 kpa. Now you leave i for a while unil i s dropped o 200 kpa. Half of wha i sared wih. So here s he quesion How much will i lose in one minue now? I hrow his problem ou o he class and am impressed by how many of hem ge i righ and even seem o have a good feeling for he reason. Tha s he nice hing abou molecules hey re quie inuiive. Well, here s a simple argumen. We need o hink in erms of our model he reason ha molecules escape from he hole is ha hey happen o encouner i in heir random moion. When he pressure is 200, here s half as many molecules in he ire as here were when i was 400, so here will be half as many collisions wih he hole, so he flow rae ou should be cu in half. So insead of losing 16 kpa in he nex minue, i loses 8. And so forh. When i has dropped o 100 kpa, i will lose 4 kpa in he nex minue. Wha his argumen is really saying is ha he amoun los in a minue ough o be always in proporion o he amoun in he ire a he sar of he minue. A formula for P. Well, ha s a beauifully simple argumen, and we can even ge an equaion ou of i. Le s firs formulae his air loss principle precisely. Wha have we said? Tha he pressure loss over a one minue inerval is proporional o he pressure a he sar of he minue. Alernaively saed: every minue he ire pressure drops by a fixed percenage. Wha is his percenage for our hypoheical example? Well, when P = 400, he loss is 16, and when P = 200, he loss is 8, and when P = 100, he loss is 4, so he 1-minue loss is always 4% of he saring pressure. Thus: every minue he ire pressure drops by 4%. Does his give us a formula? Yes i does. Think in erms of he muliplier. A 4% loss is he same as muliplicaion by 96%, so ha every minue he pressure muliplied by 0.96 If he pressure sars a 400, hen afer minues i will be: P() = 400(0.96). We have our P-funcion! Our pressure loss curve is no a parabola i s an exponenial decay funcion. Parabola and exponenial An ineresing ake on he difference beween hese wo modes of decrease is o ask how he flow rae ou of he hole depends on he amoun in he vessel. For he ire (exponenial decay), as we now see, he flow rae ou is proporional o he pressure iself. Wha abou he waer ank (parabola)? In his case he flow rae ou is proporional o he square roo of he waer deph. In he firs case, he flow rae will be cu in half when he pressure is halved. In he second case, o cu he flow rae in half we have o cu he amoun of waer o ¼. 11. ire pressure 3

4 Experimen. The quesion we are sudying is how he pressure P in your ire goes down if here s a small leak, and in he las secion we developed an exponenial decay model. I s ime o perform an experimen o check his ou. In fac i s no so easy o work wih a ire. The problem is ha he very ac of measuring he pressure alers is value (hiss!) and for somehing as small as a ire his can be significan. So we used an old car ire insead which we go from a garage. We pumped i up o 400 kpa (ha s a very hard car ire hey are normally jus over 200) and one of he sudens drilled a hole ino he side wih a small drill, and i worked perfecly. Wih a large ire, he amoun of air los in he ac of measuring he pressure is negligible relaive o he amoun in he ire. In fac he pump we used had an in-line gauge and all we had o do was monior ha. We used he ire in anoher class a couple of days laer and he hole had o be poked, bu i worked fine again. The ire now sis in a safe place in he school and is a resource for all ime. Ever school should have an old car ire. We ook readings every 5 minues for an hour. The daa are recorded below and ploed a he righ. The gauge was such ha one could ake a reading accurae o 5 kpa. ime (min) pressure P (kpa) Who wans he drill? I held he elecric drill up in he air and asked who waned o drill he hole. We had expeced a grea rush o he fron bu here was sillness. Hasn everybody always waned o drill a hole in a car ire? Perhaps hey were jus awed by he prospec. Of course afer a few momens here was a graifying response P The graph cerainly looks reasonable. I has he concave-up shape ha our heoreical model prediced. Now wha do we wan o do wih i? We wan o es wheher i has he exponenial form. 11. ire pressure 4

5 Tesing he exponenial model Now we es wheher our daa has he exponenial form: P = ac. So how do we do ha, and how do we find he righ values of a and c? A sandard calculaion shows ha P will be an exponenial funcion of ime if he logp graph is a sraigh line. So we abulae and plo logp. This is done below. And holy cow he poins do lie in a quie a wonderful sraigh line. We can conclude ha he P- daa is exponenial. Of course, his log ransform resul is no somehing you need o remember. Beer o reconsruc i each ime by hiing he arge equaion wih he log funcion. P = ac logp = log(ac ) logp = (loga) + log(c ) logp = (loga) + log(c). This ells us ha logp is linear in. P logp logp We fi a rend-line and ge he equaion: logp = Now o ge he P-equaion we exponeniae wih base 10: P = = = ( ) = 385 (0.974) Noe ha he saring value (=0) ha he formula gives us is p=385. Tha suggess ha a he beginning we didn ge he ire quie up o 400. And wha is he percenage loss ha our formula predics? In each minue he pressure is muliplied by a 2.6% loss per minue logp Acually ha saring value of 385 is no acually such a big error from our measured 400. The relaive error, 15/385, is only 4%, and if you abulae he exponenial formula you ll find ha some of he oher poins have almos he same inaccuracy.] 11. ire pressure 5

6 Problems 1. A he momen (=0) my ire has pressure P = 300 kpa, bu i has a slow leak and loses 5% of is pressure every hour. (a) Wha will is pressure be afer en hours (=10)? [Answer: ] (b) Find a formula for P afer hours. [Answer: P = 300(0.95).] (c) A wha ime will P = 150 kpa? [Answer: =13.5 h.] 2. A he momen (=0) my ire has pressure 360 kpa, bu i has a slow leak and loses 2% of is pressure every hour. (a) Wha will is pressure be afer one hour (=1)? (b) Wha will is pressure be afer five hours (=5)? (c) Find a formula for is pressure P afer hours. (d) Wha is he percenage loss in pressure over a en hour period? (e) How long unil he pressure drops o 10 kpa? 3. My bike ire has pressure 360 kpa, bu afer a week his has fallen o 320 kpa. (a) Wha will i be afer anoher week? (b) Find a formula for he pressure P afer weeks. (c) I can be bohered o fix he ire and I refuse o ride i when i s below 100 kpa and I don have a pump and I can only go o he gas saion once a week on Saurday mornings, and you have o pay 25 cens for air, so I only fill i up when I have o. Assuming ha each pump-up brings i up o 360 kpa, how long is he inerval beween pump-ups? 4. The daa on he lef was colleced from a leaking ire. Plo logp agains and deermine ha his daa lies in a reasonable sraigh line. Draw he bes-fi line, eiher wih eyeball and ruler or wih a regression rouine, and find is equaion and ransform his equaion ino an exponenial equaion for P. 5. I measure my car ire pressure a noon, a 1 PM and a 2 PM. During he firs hour i los 30 kpa and during he second hour i los 26 kpa. Wha was he pressure a noon? 6. I measure my bike ire pressure a noon, a 1 PM a 2 PM and a 3 PM. During he second hour i los 80% of wha i los during he firs hour and during he hird hour i los 50 kpa. Wha was he pressure a noon? P ire pressure 6

### 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

### 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

### Entropy: 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

### 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:

### 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

### Rotational Inertia of a Point Mass

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

### 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

### Chabot 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

### 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

### 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

### Section 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

### Chapter 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

### A 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

### Duration 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 UVA-F-38 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

### 1 HALF-LIFE EQUATIONS

R.L. Hanna Page HALF-LIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of half-lives, and / log / o calculae he age (# ears): age (half-life)

### 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

### 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

### RC, 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.

### 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

### PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE

Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees

### 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

### 11/6/2013. Chapter 14: Dynamic AD-AS. Introduction. Introduction. Keeping track of time. The model s elements

Inroducion Chaper 14: Dynamic D-S 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 cuing-edge

### RC Circuit and Time Constant

ircui and Time onsan 8M Objec: Apparaus: To invesigae he volages across he resisor and capacior in a resisor-capacior circui ( circui) as he capacior charges and discharges. We also wish o deermine he

### The 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

### Usefulness of the Forward Curve in Forecasting Oil Prices

Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,

### 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

### 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

### 4. International Parity Conditions

4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency

### 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

### 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

### 17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides

7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion

### 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

### 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

### Period 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

### C Fast-Dealing Property Trading Game C

If you are already an experienced MONOPOLY dealer and wan a faser game, ry he rules on he back page! AGES 8+ C Fas-Dealing Propery Trading Game C Y Original MONOPOLY Game Rules plus Special Rules for his

### Part 1: White Noise and Moving Average Models

Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical

### INTRODUCTION TO EMAIL MARKETING PERSONALIZATION. How to increase your sales with personalized triggered emails

INTRODUCTION TO EMAIL MARKETING PERSONALIZATION How o increase your sales wih personalized riggered emails ECOMMERCE TRIGGERED EMAILS BEST PRACTICES Triggered emails are generaed in real ime based on each

### Two Compartment Body Model and V d Terms by Jeff Stark

Two Comparmen Body Model and V d Terms by Jeff Sark In a one-comparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics - By his, we mean ha eliminaion is firs order and ha pharmacokineic

### 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

### MTH6121 Introduction to Mathematical Finance Lesson 5

26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random

### MA261-A Calculus III 2006 Fall Homework 4 Solutions Due 9/29/2006 8:00AM

MA6-A Calculus III 006 Fall Homework 4 Soluions Due 9/9/006 00AM 97 #4 Describe in words he surface 3 A half-lane in he osiive x and y erriory (See Figure in Page 67) 97 # Idenify he surface cos We see

### A 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

### 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

### C Fast-Dealing Property Trading Game C

AGES 8+ C Fas-Dealing Propery Trading Game C Y Collecor s Ediion Original MONOPOLY Game Rules plus Special Rules for his Ediion. CONTENTS Game board, 6 Collecible okens, 28 Tile Deed cards, 16 Wha he Deuce?

### Chapter 4: Exponential and Logarithmic Functions

Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion

### EDEXCEL 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 resisor-capacior (R) and resisor-inducor (RL) D circuis

### A Probability Density Function for Google s stocks

A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural

### 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)

### MOTION 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,

### AP Calculus AB 2007 Scoring Guidelines

AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a no-for-profi membership associaion whose mission is o connec sudens o college success and

### A Further Examination of Insurance Pricing and Underwriting Cycles

A Furher Examinaion of Insurance ricing and Underwriing Cycles AFIR Conference, Sepember 2005, Zurich, Swizerland Chris K. Madsen, GE Insurance Soluions, Copenhagen, Denmark Svend Haasrup, GE Insurance

### Lectures # 5 and 6: The Prime Number Theorem.

Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζ-funcion o skech an argumen which would give an acual formula for π( and sugges

### Vector Autoregressions (VARs): Operational Perspectives

Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101-115. Macroeconomericians

### Individual Health Insurance April 30, 2008 Pages 167-170

Individual Healh Insurance April 30, 2008 Pages 167-170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve

### THE 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.

### AP Calculus AB 2013 Scoring Guidelines

AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a mission-driven no-for-profi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was

### 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

### Why Did the Demand for Cash Decrease Recently in Korea?

Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in

### Principal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.

Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one

### Full-wave rectification, bulk capacitor calculations Chris Basso January 2009

ull-wave 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

### 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

### Circuit 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

### Revisions 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

### 6.5. Modelling Exercises. Introduction. Prerequisites. Learning Outcomes

Modelling Exercises 6.5 Inroducion This Secion provides examples and asks employing exponenial funcions and logarihmic funcions, such as growh and decay models which are imporan hroughou science and engineering.

### GoRA. For more information on genetics and on Rheumatoid Arthritis: Genetics of Rheumatoid Arthritis. Published work referred to in the results:

For more informaion on geneics and on Rheumaoid Arhriis: Published work referred o in he resuls: The geneics revoluion and he assaul on rheumaoid arhriis. A review by Michael Seldin, Crisopher Amos, Ryk

### The Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas

The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he

### 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

### OPERATION MANUAL. Indoor unit for air to water heat pump system and options EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1

OPERAION MANUAL Indoor uni for air o waer hea pump sysem and opions EKHBRD011ABV1 EKHBRD014ABV1 EKHBRD016ABV1 EKHBRD011ABY1 EKHBRD014ABY1 EKHBRD016ABY1 EKHBRD011ACV1 EKHBRD014ACV1 EKHBRD016ACV1 EKHBRD011ACY1

### AP Calculus AB 2010 Scoring Guidelines

AP Calculus AB 1 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 1, he College

### Density Dependence. births are a decreasing function of density b(n) and deaths are an increasing function of density d(n).

FW 662 Densiy-dependen populaion models In he previous lecure we considered densiy independen populaion models ha assumed ha birh and deah raes were consan and no a funcion of populaion size. Long-erm

The Roos of Lisp paul graham Draf, January 18, 2002. In 1960, John McCarhy published a remarkable paper in which he did for programming somehing like wha Euclid did for geomery. 1 He showed how, given

### Using RCtime to Measure Resistance

Basic Express Applicaion Noe Using RCime o Measure Resisance Inroducion One common use for I/O pins is o measure he analog value of a variable resisance. Alhough a buil-in ADC (Analog o Digial Converer)

### µ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ

Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high

### PHYS245 Lab: RC circuits

PHYS245 Lab: C circuis Purpose: Undersand he charging and discharging ransien processes of a capacior Display he charging and discharging process using an oscilloscope Undersand he physical meaning of

### Term Structure of Prices of Asian Options

Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 1-1-1 Nojihigashi, Kusasu, Shiga 525-8577, Japan E-mail:

### INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES

INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchange-raded ineres rae fuures and heir opions are described. The fuure opions include hose paying

### Making Use of Gate Charge Information in MOSFET and IGBT Data Sheets

Making Use of ae Charge Informaion in MOSFET and IBT Daa Shees Ralph McArhur Senior Applicaions Engineer Advanced Power Technology 405 S.W. Columbia Sree Bend, Oregon 97702 Power MOSFETs and IBTs have

### 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

### CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE

CLASSIFICATION OF REINSURANCE IN LIFE INSURANCE Kaarína Sakálová 1. Classificaions of reinsurance There are many differen ways in which reinsurance may be classified or disinguished. We will discuss briefly

### Table of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities

Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17

### 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.

### 4.2 Trigonometric Functions; The Unit Circle

4. Trigonomeric Funcions; The Uni Circle Secion 4. Noes Page A uni circle is a circle cenered a he origin wih a radius of. Is equaion is as shown in he drawing below. Here he leer represens an angle measure.

### Stochastic Calculus, Week 10. Definitions and Notation. Term-Structure Models & Interest Rate Derivatives

Sochasic Calculus, Week 10 Term-Srucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Term-srucure models 3. Ineres rae derivaives Definiions and Noaion Zero-coupon

### 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,

### Physics 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

### 3 Runge-Kutta Methods

3 Runge-Kua Mehods In conras o he mulisep mehods of he previous secion, Runge-Kua mehods are single-sep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he

### 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.

### CLASSICAL TIME SERIES DECOMPOSITION

Time Series Lecure Noes, MSc in Operaional Research Lecure CLASSICAL TIME SERIES DECOMPOSITION Inroducion We menioned in lecure ha afer we calculaed he rend, everyhing else ha remained (according o ha

### Present Value Methodology

Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer

### Fourier Series & The Fourier Transform

Fourier Series & The Fourier Transform Wha is he Fourier Transform? Fourier Cosine Series for even funcions and Sine Series for odd funcions The coninuous limi: he Fourier ransform (and is inverse) The

### Second Order Linear Differential Equations

Second Order Linear Differenial Equaions Second order linear equaions wih consan coefficiens; Fundamenal soluions; Wronskian; Exisence and Uniqueness of soluions; he characerisic equaion; soluions of homogeneous

### Mortality Variance of the Present Value (PV) of Future Annuity Payments

Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role

### Morningstar Investor Return

Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion

### WHAT ARE OPTION CONTRACTS?

WHAT ARE OTION CONTRACTS? By rof. Ashok anekar An oion conrac is a derivaive which gives he righ o he holder of he conrac o do 'Somehing' bu wihou he obligaion o do ha 'Somehing'. The 'Somehing' can be

### USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES

USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were

### Hedging with Forwards and Futures

Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buy-side of a forward/fuures

### 4 Convolution. Recommended Problems. x2[n] 1 2[n]

4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discree-ime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.1-1.

### 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,