Experiment 6: Friction


 Shonda Montgomery
 1 years ago
 Views:
Transcription
1 Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht friction must be tken into ccount for relistic description of prcticl situtions  it is something we cnnot ignore. Frictionl forces ct between two surfces nd oppose their reltive motion. They occur becuse of surfce irregulrities, such s defects, nd moleculr forces (or bonds) between the mterils. In this lb we will study frictionl forces between vrious objects on different types of surfces. There re two types of friction: kinetic nd sttic. Kinetic friction is the friction between surfces in reltive motion. When sliding n object cross nother surfce, microscopic bumps nd defects tend to impede nd resist the motion (even the smoothest surfces re rough on the microscopic scle). This is the type of force tht brings rolling bll to rest or costing cr to stop. Experimentlly, it is observed tht the force of kinetic friction is proportionl to the norml force cting between the surfces: if you increse the norml force, the surfces re crushed more together, incresing the contct re, nd thus incresing the frictionl force. Mthemticlly we cn write the force of kinetic friction s F k = µ k F N (1) where F N is the norml force between the two surfces in contct with one nother nd µ k is the coefficient of kinetic friction. The coefficient of kinetic friction is dimensionless quntity (no units) tht depends on the properties of the two surfces. µ k rnges from 0.01 for very smooth surfces to 1.5 for very rough surfces. So, for exmple, if we wnt to push n object with constnt speed on very smooth horizontl surfce (such s ice), we must pply round 1% of its weight, wheres if we wnted to push the object on rough surfce (dry concrete) we might need to push the object with greter force thn its own weight. Sttic friction describes the frictionl forces between the surfces of two objects tht re t rest with respect to ech other. The sttic friction between the two surfces is described by the coefficient of sttic friction µ s. Experimentlly, is it found tht the mximum vlue for the sttic frictionl force is proportionl to the norml force between the two surfces. Thus the sttic frictionl force F s is F s µ s F N (2) Since the objects re t rest with one nother, more moleculr bonds re ble to form mking the object hrder to move nd so greter force is needed to strt motion when compred to the kinetic friction cse. Therefore µ s is generlly greter thn µ k. Grphiclly, this is shown in Figure 1: As you increse the force, the sttic friction force increses linerly until the pplied force F equls µ s F N. After this point the object breks wy nd the friction force flls to the kinetic friction vlue. 1
2 fr fr = μsfn sttic kinetic 0 no motion sliding F Figure 1: Force of friction (fr) s function of n externl force F pplied to n object tht is initilly t rest. Experimentl Objectives The purpose of this lb is to construct reltionship between frictionl forces nd the norml force on n object, to clculte the kinetic nd sttic coefficients of friction for vrious objects nd surfces nd to ultimtely gin solid understnding of sttic vs kinetic friction. In this lb you re given pulley sensor tht cn mesure ccelertion, force sensor, string, friction crts with different surfces (cork, felt, nd plstic), nd different surfces (sheet of luminum, construction pper, nd the tble top) to drg the crts on. 1: Coefficient of Kinetic Friction To study nd clculte vrious coefficients of kinetic friction, we will use pulley system s shown in Figure 2(). The pulley ( smrt pulley ) is equipped with sensor tht llows you to mesure nd grph the velocity of the msses s function of time vi the Dt Studio softwre. With the velocity grph you cn obtin the ccelertion of the mss system by finding the slope of the pproprite liner fit, similr to the Atwood lb (lb 5). Looking t the free body digrms of our system (Figure 2(b)), we cn write Newton s second lw for ech mss s m 1 = T fr (3) m 2 = m 2 g T (4) Here we hve ssumed tht the ccelertions of the two msses re sme by neglecting ny frictionl effects on the pulley mking the tension in the string uniform. The kinetic frictionl force fr is given by fr = µ k F N = µ k m 1 g (5) 2
3 m1 FN T m2 fr m1 W1 T m2 W2 () (b) Figure 2: () Pulley system used to clculte u k. (b) Free body digrms for the pulley setup. W i is the weight of the object nd fr is the frictionl force. The system of equtions (Eqs. 34) cn be solved for the ccelertion in terms of the msses (m 1, m 2 ), g, nd µ k = (m 2 µ k m 1 )g m 1 + m 2 (6) Devise n experiment to clculte µ k for vrious surfces, mking use of the smrt pulley system, the friction crts, nd the different surfces. Tke mesurements with ll 3 crts (felt, plstic, nd cork) on one of the surfces. Use up to 3 different msses for ech crt. Remember to do severl trils for ech run to obtin consistent dt. 2: Coefficient of Sttic Friction To mesure the sttic coefficient of friction µ s we will use the force sensor. The force sensor records the pulling (or pushing) in Newtons vi the Dt Studio softwre. Connect the force sensor to one of the friction crts using string. With no force on the sensor press, the zero (tre) button before tking ny mesurements (this should only be performed once). Open the grph under the disply section in Dt Studio. With the force sensor setup nd ttched to the crt, strt to slowly nd crefully pull on the crt on of the surfces while monitoring the force vlue with the grph. You wnt to record the minimum force needed for the friction crt to brek wy nd strt moving. Your grph should look similr to Figure 1 (note: depending on the force sensor setup, your grph might be upside down!). Hve every member of the lb group try this. Repet this procedure for ll the friction crts (felt, plstic, cork) using t lest 5 different msses for ech crt. 3
4 3: Coefficient of Kinetic Friction by Force Sensor We cn check our µ k vlues obtined from the first experiment by mking use of the force sensor. Devise nd experiment to mesure the coefficient of kinetic friction using the force sensor. Record the force, norml force nd obtin µ k grphiclly. How do your vlues compre with those from experiment 1? Repet this experiment for the vrious crts with 3 different msses. Hint: This prt my be difficult t first, but drw free body digrm of the crt nd force sensor nd then convince yourself why you wnt the crt to move with constnt velocity (look t the velocity grph s guide). A full lb report is not necessry for this lb. Answer the questions on the following pge nd turn it in with your signed dtsheet. 4
5 PHYS 123, Lb 6 Questions Nme: CWID: Write your nswers on seprte sheet nd ttch your signed dtsheet when turning it in. You must show ll of your work for full credit. Mke it cler to me you understnd wht you re doing. Any grphs or tbles should be mde vi computer softwre nd ttched to this hndout. 1. Answer the following questions using the dt you cquired in this experiment: () For the first experiment, crete dt tble for the different msses (M 1, M 2 ), the ccelertion, nd the clculted coefficient of friction µ k. Remember to lbel the crt types (felt, cork, plstic) in your tble nd describe the surfce. (b) Do your mesured vlues of µ k mke sense? Compre them with smple coefficients of friction (for vrious mterils) found in your textbook. (c) For the second experiment, wht is the force tht you re mesuring? Crete plot of this mesured force vs the norml force of the friction crt. Find its slope nd explin wht it represents. (d) For the third experiment, mke dt tble consisting of the crt msses, ny pplied force, nd the norml force. Using your dt, crete grph tht represents the coefficient of friction. (e) How does your coefficient of friction from the third experiment compre with the one you obtined from the first experiment? Wht re the sources of error? 5
Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationTheory of Forces. Forces and Motion
his eek extbook  Red Chpter 4, 5 Competent roblem Solver  Chpter 4 relb Computer Quiz ht s on the next Quiz? Check out smple quiz on web by hurs. ht you missed on first quiz Kinemtics  Everything
More informationEinstein. Mechanics. In Grade 10 we investigated kinematics, or movement described in terms of velocity, acceleration, displacement, and so on.
Cmbridge University Press 9780521683593  Study nd Mster Physicl Sciences Grde 11 Lerner s Book Krin Kelder More informtion MODULE 1 Einstein Mechnics motion force Glileo Newton decelerte moment of
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More information1. 1 m/s m/s m/s. 5. None of these m/s m/s m/s m/s correct m/s
Crete ssignment, 99552, Homework 5, Sep 15 t 10:11 m 1 This printout should he 30 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. The due time
More informationNewton s Three Laws. d dt F = If the mass is constant, this relationship becomes the familiar form of Newton s Second Law: dv dt
Newton s Three Lws For couple centuries before Einstein, Newton s Lws were the bsic principles of Physics. These lws re still vlid nd they re the bsis for much engineering nlysis tody. Forml sttements
More informationNet Change and Displacement
mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the
More informationPHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
More informationAnswer, Key Homework 4 David McIntyre Mar 25,
Answer, Key Homework 4 Dvid McIntyre 45123 Mr 25, 2004 1 his printout should hve 18 questions. Multiplechoice questions my continue on the next column or pe find ll choices before mkin your selection.
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationAnswer, Key Homework 8 David McIntyre 1
Answer, Key Homework 8 Dvid McIntyre 1 This printout should hve 17 questions, check tht it is complete. Multiplechoice questions my continue on the net column or pge: find ll choices before mking your
More informationHelicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
More informationPhys 207. Announcements. Hwk3 is posted on course website Quizzes & answers will be posted on course website Formula sheets.
Phys 07 Announcements Hwk3 is posted on course website Quizzes & nswers will be posted on course website ormul sheets Newton s 3 lws Tody s Agend How nd why do objects move? Dynmics 1 Dynmics Isc Newton
More informationMechanics Cycle 1 Chapter 5. Chapter 5
Chpter 5 Contct orces: ree Body Digrms nd Idel Ropes Pushes nd Pulls in 1D, nd Newton s Second Lw Neglecting riction ree Body Digrms Tension Along Idel Ropes (i.e., Mssless Ropes) Newton s Third Lw Bodies
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics. W02D3_0 Group Problem: Pulleys and Ropes Constraint Conditions
MSSCHUSES INSIUE OF ECHNOLOGY Deprtment of hysics 8.0 W02D3_0 Group roblem: ulleys nd Ropes Constrint Conditions Consider the rrngement of pulleys nd blocks shown in the figure. he pulleys re ssumed mssless
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationYour Thoughts. Does the moment of inertia play a part in determining the amount of time it takes an object to get to the bottom of an incline.
Your Thoughts Suppose bll rolls down rmp with coefficient of friction just big enough to keep the bll from slipping. An identicl bll rolls down n identicl rmp with coefficient of friction of. Do both blls
More informationr 2 F ds W = r 1 qe ds = q
Chpter 4 The Electric Potentil 4.1 The Importnt Stuff 4.1.1 Electricl Potentil Energy A chrge q moving in constnt electric field E experiences force F = qe from tht field. Also, s we know from our study
More informationPlotting and Graphing
Plotting nd Grphing Much of the dt nd informtion used by engineers is presented in the form of grphs. The vlues to be plotted cn come from theoreticl or empiricl (observed) reltionships, or from mesured
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationSolutions to Section 1
Solutions to Section Exercise. Show tht nd. This follows from the fct tht mx{, } nd mx{, } Exercise. Show tht = { if 0 if < 0 Tht is, the bsolute vlue function is piecewise defined function. Grph this
More informationAP QUIZ #2 GRAPHING MOTION 1) POSITION TIME GRAPHS DISPLACEMENT Each graph below shows the position of an object as a function of time.
AP QUIZ # GRAPHING MOTION ) POSITION TIME GRAPHS DISPLAEMENT Ech grph below shows the position of n object s function of time. A, B,, D, Rnk these grphs on the gretest mgnitude displcement during the time
More informationMath 22B Solutions Homework 1 Spring 2008
Mth 22B Solutions Homework 1 Spring 2008 Section 1.1 22. A sphericl rindrop evportes t rte proportionl to its surfce re. Write differentil eqution for the volume of the rindrop s function of time. Solution
More informationUnderstanding 22. 23. The frictional force acting to the left is missing. It is equal in magnitude to the applied force acting to the right.
Chpter 3 Review, pges 154 159 Knowledge 1. (c) 2. () 3. (d) 4. (d) 5. (d) 6. (c) 7. (b) 8. (c) 9. Flse. One newton is equl to 1 kg /s 2. 10. Flse. A norl force is perpendiculr force cting on n object tht
More informationVersion 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This printout should he 35 questions. Multiplechoice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationTests for One Poisson Mean
Chpter 412 Tests for One Poisson Men Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson distribution
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationTreatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the ttest is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the ttest cn be used to compre the mens of only
More informationModule 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationSection 2.3. Motion Along a Curve. The Calculus of Functions of Several Variables
The Clculus of Functions of Severl Vribles Section 2.3 Motion Along Curve Velocity ccelertion Consider prticle moving in spce so tht its position t time t is given by x(t. We think of x(t s moving long
More informationSOLUTIONS TO CONCEPTS CHAPTER 5
1. m k S 10m Let, ccelertion, Initil velocity u 0. S ut + 1/ t 10 ½ ( ) 10 5 m/s orce: m 5 10N (ns) 40000. u 40 km/hr 11.11 m/s. 3600 m 000 k ; v 0 ; s 4m v u ccelertion s SOLUIONS O CONCEPS CHPE 5 0 11.11
More informationAAPT UNITED STATES PHYSICS TEAM AIP 2010
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS  75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
More informationVersion 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More information4.0 5Minute Review: Rational Functions
mth 130 dy 4: working with limits 1 40 5Minute Review: Rtionl Functions DEFINITION A rtionl function 1 is function of the form y = r(x) = p(x) q(x), 1 Here the term rtionl mens rtio s in the rtio of two
More informationHomework #4: Answers. 1. Draw the array of world outputs that free trade allows by making use of each country s transformation schedule.
Text questions, Chpter 5, problems 15: Homework #4: Answers 1. Drw the rry of world outputs tht free trde llows by mking use of ech country s trnsformtion schedule.. Drw it. This digrm is constructed
More information1. 0 m/s m/s m/s m/s
Version PREVIEW Kine Grphs PRACTICE burke (1111) 1 This printout should he 30 questions. Multiplechoice questions m continue on the next column or pge find ll choices before nswering. Distnce Time Grph
More informationDiffraction and Interference of Light
rev 12/2016 Diffrction nd Interference of Light Equipment Qty Items Prt Number 1 Light Sensor CI6504 1 Rotry Motion Sensor CI6538 1 Single Slit Set OS8523 1 Multiple Slit Set OS8523 1 Liner Trnsltor
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationChapter 6 Solving equations
Chpter 6 Solving equtions Defining n eqution 6.1 Up to now we hve looked minly t epressions. An epression is n incomplete sttement nd hs no equl sign. Now we wnt to look t equtions. An eqution hs n = sign
More informationN Mean SD Mean SD Shelf # Shelf # Shelf #
NOV xercises smple of 0 different types of cerels ws tken from ech of three grocery store shelves (1,, nd, counting from the floor). summry of the sugr content (grms per serving) nd dietry fiber (grms
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationElectric Circuits. Simple Electric Cell. Electric Current
Electric Circuits Count Alessndro olt (74587) Georg Simon Ohm (787854) Chrles Augustin de Coulomb (736 806) André Mrie AMPÈRE (775836) Crbon Electrode () Simple Electric Cell wire Zn Zn Zn Zn Sulfuric
More informationEcon 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
More information14.2. The Mean Value and the RootMeanSquare Value. Introduction. Prerequisites. Learning Outcomes
he Men Vlue nd the RootMenSqure Vlue 4. Introduction Currents nd voltges often vry with time nd engineers my wish to know the men vlue of such current or voltge over some prticulr time intervl. he men
More informationnot to be republished NCERT POLYNOMIALS CHAPTER 2 (A) Main Concepts and Results (B) Multiple Choice Questions
POLYNOMIALS (A) Min Concepts nd Results Geometricl mening of zeroes of polynomil: The zeroes of polynomil p(x) re precisely the xcoordintes of the points where the grph of y = p(x) intersects the xxis.
More informationNumerical Solutions of Linear Systems of Equations
EE 6 Clss Notes Numericl Solutions of Liner Systems of Equtions Liner Dependence nd Independence An eqution in set of equtions is linerly independent if it cnnot e generted y ny liner comintion of the
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationProject 6 Aircraft static stability and control
Project 6 Aircrft sttic stbility nd control The min objective of the project No. 6 is to compute the chrcteristics of the ircrft sttic stbility nd control chrcteristics in the pitch nd roll chnnel. The
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationWell say we were dealing with a weak acid K a = 1x10, and had a formal concentration of.1m. What is the % dissociation of the acid?
Chpter 9 Buffers Problems 2, 5, 7, 8, 9, 12, 15, 17,19 A Buffer is solution tht resists chnges in ph when cids or bses re dded or when the solution is diluted. Buffers re importnt in Biochemistry becuse
More informationThe Laws of Motion. chapter
chpter The Lws of Motion 5 5.1 The Concept of Force 5.2 Newton s First Lw nd Inertil Frmes 5.3 Mss 5.4 Newton s econd Lw 5.5 The Grvittionl Force nd Weight 5.6 Newton s Third Lw 5.7 Anlysis Models Using
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationHomework #6: Answers. a. If both goods are produced, what must be their prices?
Text questions, hpter 7, problems 12. Homework #6: Answers 1. Suppose there is only one technique tht cn be used in clothing production. To produce one unit of clothing requires four lborhours nd one
More informationLecture 1: Introduction to Economics
E111 Introduction to Economics ontct detils: E111 Introduction to Economics Ginluigi Vernsc Room: 5B.217 Office hours: Wednesdy 2pm to 4pm Emil: gvern@essex.c.uk Lecture 1: Introduction to Economics I
More informationv T R x m Version PREVIEW Practice 7 carroll (11108) 1
Version PEVIEW Prctice 7 crroll (08) his printout should he 5 questions. Multiplechoice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationPHYS1231 Higher Physics 1B Solutions Tutorial 2
PHYS3 Higher Physics Solutions Tutoril sic info: lthough the term voltge is use every y, in physics it is mesure of firly bstrct quntity clle Electric Potentil. It s importnt to istinguish electric potentil
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More information9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is soclled becuse when the sclr product of two vectors
More informationPicture Match Words. Strobe pictures. Stopping distance. Following. Safety
Tuesdy: Picture Mtch + Spelling Pyrmid Homework [the hndout for it is two pges down] Mterils: 1 bord + 1 set of words per 2 students (totl: 12 of ech) Routine: () once the Pictionry is completed; pirs
More informationRotating DC Motors Part II
Rotting Motors rt II II.1 Motor Equivlent Circuit The next step in our consiertion of motors is to evelop n equivlent circuit which cn be use to better unerstn motor opertion. The rmtures in rel motors
More informationRotational Equilibrium: A Question of Balance
Prt of the IEEE Techer InService Progrm  Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of
More informationMATLAB Workshop 13  Linear Systems of Equations
MATLAB: Workshop  Liner Systems of Equtions pge MATLAB Workshop  Liner Systems of Equtions Objectives: Crete script to solve commonly occurring problem in engineering: liner systems of equtions. MATLAB
More informationLesson 10. Parametric Curves
Return to List of Lessons Lesson 10. Prmetric Curves (A) Prmetric Curves If curve fils the Verticl Line Test, it cn t be expressed by function. In this cse you will encounter problem if you try to find
More informationWeek 11  Inductance
Week  Inductnce November 6, 202 Exercise.: Discussion Questions ) A trnsformer consists bsiclly of two coils in close proximity but not in electricl contct. A current in one coil mgneticlly induces n
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More informationModule 9. DC Machines. Version 2 EE IIT, Kharagpur
Module 9 DC Mchines Lesson 39 D.C Motors Contents 39 D.C Shunt Motor (Lesson39) 4 39.1 Gols of the lesson.. 4 39.2 Introduction. 4 39.3 Importnt Ides. 5 39.4 Strting of D.C shunt motor. 7 39.4.1 Problems
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationWritten Homework 6 Solutions
Written Homework 6 Solutions Section.10 0. Explin in terms of liner pproximtions or differentils why the pproximtion is resonble: 1.01) 6 1.06 Solution: First strt by finding the liner pproximtion of f
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationASTR 170! 2010 S1 " Daniel Zucker! E7A 317 Mathematics as a tool for understanding physics
A New Scientific Er Grity nd Tides ASTR 170! 2010 S1 " Dniel Zucker! E7A 317 Mthemtics s tool for understnding physics zucker@science.mq.edu.u! 1 2 Velocity nd Accelertion Isc Newton (16431727)! Building
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationQuick Reference Guide: Onetime Account Update
Quick Reference Guide: Onetime Account Updte How to complete The Quick Reference Guide shows wht existing SingPss users need to do when logging in to the enhnced SingPss service for the first time. 1)
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationOn the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding
Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl
More informationWorksheet 24: Optimization
Worksheet 4: Optimiztion Russell Buehler b.r@berkeley.edu 1. Let P 100I I +I+4. For wht vlues of I is P mximum? P 100I I + I + 4 Tking the derivtive, www.xkcd.com P (I + I + 4)(100) 100I(I + 1) (I + I
More informationProjectile Motion CHAPTER 1
CHAPTER 1 PHYSICS ESSENTIALS STAGE 2 Projectile Motion Subject Outline In the bsence of ir resistnce nd moing under the ction of constnt grittionl force, projectile hs constnt ccelertion in the direction
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationPerfect competition model (PCM)
18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,
More informationChapter G  Problems
Chpter G  Problems Blinn College  Physics 2426  Terry Honn Problem G.1 A plne flies horizonlly t speed of 280 mês in position where the erth's mgnetic field hs mgnitude 6.0µ105 T nd is directed t n
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationQuadratic Equations. Math 99 N1 Chapter 8
Qudrtic Equtions Mth 99 N1 Chpter 8 1 Introduction A qudrtic eqution is n eqution where the unknown ppers rised to the second power t most. In other words, it looks for the vlues of x such tht second degree
More information4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information