Problem 1: Multiple Choice Questions

Size: px
Start display at page:

Download "Problem 1: Multiple Choice Questions"

Transcription

1 Mathematics 102 Review Questions Poblem 1: Multiple Choice Questions 1: Conside the function y = f(x) = 3e 2x 5e 4x (a) The function has a local maximum at x = (1/2)ln(10/3) (b) The function has a local minimum at x = (1/2)ln(10/3) (c) The function has a local maximum at x = ( 1/2)ln(3/5) (d) The function has a local minimum at x = (1/2)ln(3/5) (e) The function has a local maximum at x = ( 1/2)ln(3/20) 2: Let m 1 be the slope of the function y = 3 x at the point x = 0 and let m 2 be the slope of the function y = log 3 x at x = 1 Then (a) m 1 = ln(3)m 2 (b) m 1 = m 2 (c) m 1 = m 2 (d) m 1 = 1/m 2 (e) m 1 = m 2 /ln(3) 3: Conside the cuve whose equation is x 4 +y 4 +3xy = 5. The slope of the tangent line, dy/dx, at the point (1,1) is (a) 1 (b) -1 (c) 0 (d) -4/7 (e) 1/7 4: Two kinds of bacteia ae found in a sample of tainted food. It is found that the population size of type 1, N 1 and of type 2, N 2 satisfy the equations dn 1 dt = 0.2N 1, N 1 (0) = 1000, dn 2 dt = 0.8N 2, N 2 (0) = 10. Then the population sizes ae equal N 1 = N 2 at the following time: (a) t = ln(40) (b) t = ln(60) (c) t = ln(80) (d) t = ln(90) (e) t = ln(100) 5: In a conical pile of sand the atio of the height to the base adius is always /h = 3. (Recall that the volume of a cone with height h and adius is V = (π/3) 2 h.) If the volume is inceasing at ate 3 m 3 /min, how fast (in m/min) is the height changing when h = 2m? (a) 1/(12π) (b) (1/π) 1/3 (c) 27/(4π) (d) 1/(4π) (e) 1/(36π) 6: Newton s Law of cooling leads to a diffeential equation that pedicts the tempeatue T(t) of an object whose initial tempeatue is T 0 in an envionment whose tempeatue is E. The pedicted tempeatue is given by T(t) = E + (T 0 E)e kt whee t is time and k is a constant. Shown in Fig 1 on the following page is some data points plotted as ln(t(t) E) vesus time in minutes. The ambient tempeatue was E = 22 C. Also shown on the gaph is the line that best fits those 11 points. Accoding to this gaph, the value of the constant k is appoximately (a) -1/27 (b) e 1/27 (c) 1/27 (d) 4/27 (e) ln(1/27) 1

2 ln ( T(t) - E ) Bestfitline time in minutes Figue 1: Figue fo Multiple Choice poblem 6 2

3 Long Answe Poblems Poblem 2a: Fig. 2 shows a 1 km ace tack with cicula ends. Find the values of x and y that will maximize the aea of the ectangle. y x Figue 2: This shape is investigated in both poblems 1 and 2. Poblem 2b: Now suppose that Fig 2 shows the shape of a leaf of some plant. If the plant gows so that x inceases at the ate 2 cm/yea and y inceases at the ate 1 cm/yea, at what ate will the leaf s entie aea be inceasing? Poblem 3: Find the dimensions of the lagest ectangle that can fit exactly into a cicle whose adius is 10 cm. Poblem 4: A cell of the bacteium E.coli has the shape of a cylinde with two hemispheical h Figue 3: Shape of the object descibed in Poblem 4. Note: Useful volumes and suface aeas: Fo a hemisphee, V = (2/3)π 3,S = 2π 2. Fo a cylinde, V = π 2 h and S = 2πh (not including end caps) caps, as shown in Fig 3. Conside this shape, with h the height of the cylinde, and the adius of the cylinde and hemisphees. (a) Find the values of and h that lead to the lagest volume fo a fixed constant suface aea, S= constant. (b) Descibe o sketch the shape you found in (a). (c) A typical E. coli cell has h = 1µm and = 0.5µm. Based on you esults in (a) and (b), would you agee that E. coli has a shape that maximizes its volume fo a fixed suface aea? (Explain you answe). 3

4 Poblem 5: If the cell shown above in Fig 3 is gowing so that the height inceases twice as fast as the adius. If the adius is gowing at 1 µm pe day at what ate will the volume of the cell incease? (Leave you answe in tems of the height and adius of the cell.) Poblem 6:(a) It takes you 1 hs (total) to tavel to and fom UBC evey day to study Philosophy 101. The amount of new leaning (in abitay units) that you can get by spending t hous at the univesity is given appoximately by L P (t) = 10t 9+t. How long should you stay at UBC on a given day if you want to maximize you leaning pe time spent? (Time spent includes tavel time.) (b) If you take Math instead of Philosophy, you leaning at time t is L M (t) = t 2. How long should you stay at UBC to maximize you leaning in that case? Poblem 7: The atoms of some adioactive mateial ae known to have a pobability k of decaying pe unit time. We will use y(t) to denote the amount of adioactivity emaining at time t. Suppose that thee is 100 gm of this adioactive substance initially, at time t = 0. Conside what happens duing a small time inteval t. How much adioactive mateial is left at time t 1 = t? At time t s = 2 t? Wite down an equation that links y(t n+1 ) to y(t n ) whee t n = n t. Convet this equation to a diffeential equation. If the half life of this substance is 1 day, find out how much is left afte 5 days. What is the value of k in this case, and how much adioactivity is left at time t? Poblem 8: Given a population of 6 billion people on Planet Eath, and using the appoximate gowth ate of = pe yea, how long ago was this population only 1 million? Assume that the gowth has been the same thoughout histoy (which is not actually tue). Poblem 9: Find citical points fo the function y = e x (1 ln(x)) fo 0.1 x 2 and classify thei types. Poblem 10: The function y = ln(x) e x has a citical point in the inteval 0.1 x 2. It is not possible to solve fo the value of x at that point, but it is possible to find out what kind of citical point that is. Detemine whethe that point is a local maximum, minimum, o inflection point. Poblem 11: (a) Conside the polynomial y = 4x 5 15x 4. Find all local minima maxima, and inflection points fo this function. (b) Find the global minimum and maximum fo the function in poblem (1) on the inteval [-1,1]. Poblem 12: Conside the polynomial y = x 5 x 4 +3x 3. Use calculus to find all local minima maxima, and inflection points fo this function. (b) Find the global minimum and maximum fo the function in Poblem (3) on the inteval [-1,1]. Poblem 13: Find a polynomial of thid degee that has a local maximum at x = 1, a zeo and an inflection point at x = 0, and goes though the point (1,2). Hint: assume p(x) = ax 3 +bx 2 +cx+d and find the values of a, b, c, d. 4

5 Poblem 14: Find a linea appoximation to the function y = x 2 at the point whose x coodinate is x = 2. Use you esult to appoximate the value of (2.0001) 2. Poblem 15: The Lennad-Jones potential, V(x) is the potential enegy associated with two unchaged molecules a distance x apat, and is given by the fomula V(x) = a x 12 b x 6 Molecules would tend to adjust thei sepaation distance so as to minimize this potential. Find any local maxima o minima of this potential. Find the distance between the molecules, x, at which V(x) is minimized and use the second deivative test to veify that this is a local minimum. Poblem 16: Conside an object thown upwads with initial velocity v 0 > 0 and initial height h 0 > 0. Then the height of the object at time t is given by y = f(t) = 1 2 gt2 +v 0 t+h 0. Find citical points of f(t) and use both the second and fist deivative tests to establish that this is a local maximum. Poblem 17: The figue (not dawn to scale) shows a tumo mass containing a necotic (dead) coe (adius 2 ), suounded by a laye of actively dividing tumo cells. The entie tumo can be assumed to be spheical, and the coe is also spheical 1. (a) If the necotic coe inceases at the ate 3 cm 3 /yea and the volume of the active cells inceases by 4 cm 3 /yea, at what ate is the oute adius of the tumo ( 1 ) changing when 1 = 1 cm. (NOTE: Show all you wok, leave you answe as a faction in tems of π; indicate units with you answe.) necotic coe 2 1 active cells (b) At what ate (in cm 2 /y) does the oute suface aea of the tumo incease when 1 = 1cm? Poblem 18: Shown below is a majo atey, (adius R) and one of its banches (adius ). A labeledschematic diagamisalso shown (ight). Thelength0AisL, andthedistance between 0and P is d, whee 0P is pependicula to 0A. The location of the banch point (B) is to be detemined so that the total esistance to blood flow in the path ABP is as small as possible. (R,,d,L ae positive constants, and R >.) 1 Recall that the volume and suface aea of a sphee ae V = (4/3)π 3, S = 4π 2 5

6 R L d 0 B A P (a) Let the distance between 0 and B be x. What is the length of the segment BA and what is the length of the segment BP? (b) The esistance of any blood vessel is popotional to its length and invesely popotional to its adius to the fouth powe 2. Based on this fact, what is the esistance, T 1, of segment BA and what is the esistance, T 2, of the segment BP? (c) Find the value of the vaiable x fo which the total esistance, T(x) = T 1 +T 2 is a minimum. 2 z is invesely popotional to y means that z = k/y fo some constant k 6

Review Exercise Set 16

Review Exercise Set 16 Review Execise Set 16 Execise 1: A ectangula plot of famland will be bounded on one side by a ive and on the othe thee sides by a fence. If the fame only has 600 feet of fence, what is the lagest aea that

More information

Motithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100

Motithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100 Motithang Highe Seconday School Thimphu Thomde Mid Tem Examination 016 Subject: Mathematics Full Maks: 100 Class: IX Witing Time: 3 Hous Read the following instuctions caefully In this pape, thee ae thee

More information

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam

Math 259 Winter Handout 6: In-class Review for the Cumulative Final Exam Math 259 Winte 2009 Handout 6: In-class Review fo the Cumulative Final Exam The topics coveed by the cumulative final exam include the following: Paametic cuves. Finding fomulas fo paametic cuves. Dawing

More information

Universal Gravitation

Universal Gravitation Chapte 1 Univesal Gavitation Pactice Poblem Solutions Student Textbook page 580 1. Conceptualize the Poblem - The law of univesal gavitation applies to this poblem. The gavitational foce, F g, between

More information

Math 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3

Math 1105: Calculus I (Math/Sci majors) MWF 11am / 12pm, Campion 235 Written homework 3 Math : alculus I Math/Sci majos MWF am / pm, ampion Witten homewok Review: p 94, p 977,8,9,6, 6: p 46, 6: p 4964b,c,69, 6: p 47,6 p 94, Evaluate the following it by identifying the integal that it epesents:

More information

Ch 13 Universal Gravitation

Ch 13 Universal Gravitation Ch 13 Univesal Gavitation Ch 13 Univesal Gavitation Why do celestial objects move the way they do? Keple (1561-1630) Tycho Bahe s assistant, analyzed celestial motion mathematically Galileo (1564-1642)

More information

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4!" or. r ˆ = points from source q to observer

$ i. !((( dv vol. Physics 8.02 Quiz One Equations Fall q 1 q 2 r 2 C = 2 C! V 2 = Q 2 2C F = 4! or. r ˆ = points from source q to observer Physics 8.0 Quiz One Equations Fall 006 F = 1 4" o q 1 q = q q ˆ 3 4" o = E 4" o ˆ = points fom souce q to obseve 1 dq E = # ˆ 4" 0 V "## E "d A = Q inside closed suface o d A points fom inside to V =

More information

Physics 107 TUTORIAL ASSIGNMENT #8

Physics 107 TUTORIAL ASSIGNMENT #8 Physics 07 TUTORIAL ASSIGNMENT #8 Cutnell & Johnson, 7 th edition Chapte 8: Poblems 5,, 3, 39, 76 Chapte 9: Poblems 9, 0, 4, 5, 6 Chapte 8 5 Inteactive Solution 8.5 povides a model fo solving this type

More information

B. Spherical Wave Propagation

B. Spherical Wave Propagation 11/8/007 Spheical Wave Popagation notes 1/1 B. Spheical Wave Popagation Evey antenna launches a spheical wave, thus its powe density educes as a function of 1, whee is the distance fom the antenna. We

More information

Between any two masses, there exists a mutual attractive force.

Between any two masses, there exists a mutual attractive force. YEAR 12 PHYSICS: GRAVITATION PAST EXAM QUESTIONS Name: QUESTION 1 (1995 EXAM) (a) State Newton s Univesal Law of Gavitation in wods Between any two masses, thee exists a mutual attactive foce. This foce

More information

Math Notes on Kepler s first law 1. r(t) kp(t)

Math Notes on Kepler s first law 1. r(t) kp(t) Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is

More information

Related Rates - the Basics

Related Rates - the Basics Related Rates - the Basics In this section we exploe the way we can use deivatives to find the velocity at which things ae changing ove time. Up to now we have been finding the deivative to compae the

More information

Central Force Motion

Central Force Motion Cental Foce Motion Cental Foce Poblem Find the motion of two bodies inteacting via a cental foce. Examples: Gavitational foce (Keple poblem): m1m F 1, ( ) =! G ˆ Linea estoing foce: F 1, ( ) =! k ˆ Two

More information

4.3 Area of a Sector. Area of a Sector Section

4.3 Area of a Sector. Area of a Sector Section ea of a Secto Section 4. 9 4. ea of a Secto In geomety you leaned that the aea of a cicle of adius is π 2. We will now lean how to find the aea of a secto of a cicle. secto is the egion bounded by a cental

More information

Physics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G =

Physics 111. Ch 12: Gravity. Newton s Universal Gravity. R - hat. the equation. = Gm 1 m 2. F g 2 1. ˆr 2 1. Gravity G = ics Announcements day, embe 9, 004 Ch 1: Gavity Univesal Law Potential Enegy Keple s Laws Ch 15: Fluids density hydostatic equilibium Pascal s Pinciple This week s lab will be anothe physics wokshop -

More information

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0

2 E. on each of these two surfaces. r r r r. Q E E ε. 2 2 Qencl encl right left 0 Ch : 4, 9,, 9,,, 4, 9,, 4, 8 4 (a) Fom the diagam in the textbook, we see that the flux outwad though the hemispheical suface is the same as the flux inwad though the cicula suface base of the hemisphee

More information

Chapter 13 Gravitation

Chapter 13 Gravitation Chapte 13 Gavitation In this chapte we will exploe the following topics: -Newton s law of gavitation, which descibes the attactive foce between two point masses and its application to extended objects

More information

Magnetic Field. Conference 6. Physics 102 General Physics II

Magnetic Field. Conference 6. Physics 102 General Physics II Physics 102 Confeence 6 Magnetic Field Confeence 6 Physics 102 Geneal Physics II Monday, Mach 3d, 2014 6.1 Quiz Poblem 6.1 Think about the magnetic field associated with an infinite, cuent caying wie.

More information

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8

2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8 5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

Circular Orbits. and g =

Circular Orbits. and g = using analyse planetay and satellite motion modelled as unifom cicula motion in a univesal gavitation field, a = v = 4π and g = T GM1 GM and F = 1M SATELLITES IN OBIT A satellite is any object that is

More information

MAP4C1 Exam Review. 4. Juno makes and sells CDs for her band. The cost, C dollars, to produce n CDs is given by. Determine the cost of making 150 CDs.

MAP4C1 Exam Review. 4. Juno makes and sells CDs for her band. The cost, C dollars, to produce n CDs is given by. Determine the cost of making 150 CDs. MAP4C1 Exam Review Exam Date: Time: Room: Mak Beakdown: Answe these questions on a sepaate page: 1. Which equations model quadatic elations? i) ii) iii) 2. Expess as a adical and then evaluate: a) b) 3.

More information

Newton s Laws, Kepler s Laws, and Planetary Orbits

Newton s Laws, Kepler s Laws, and Planetary Orbits Newton s Laws, Keple s Laws, and Planetay Obits PROBLEM SET 4 DUE TUESDAY AT START OF LECTURE 28 Septembe 2017 ASTRONOMY 111 FALL 2017 1 Newton s & Keple s laws and planetay obits Unifom cicula motion

More information

Objectives: After finishing this unit you should be able to:

Objectives: After finishing this unit you should be able to: lectic Field 7 Objectives: Afte finishing this unit you should be able to: Define the electic field and explain what detemines its magnitude and diection. Wite and apply fomulas fo the electic field intensity

More information

Objective Notes Summary

Objective Notes Summary Objective Notes Summay An object moving in unifom cicula motion has constant speed but not constant velocity because the diection is changing. The velocity vecto in tangent to the cicle, the acceleation

More information

Chapter 10 Sample Exam

Chapter 10 Sample Exam Chapte Sample Exam Poblems maked with an asteisk (*) ae paticulaly challenging and should be given caeful consideation.. Conside the paametic cuve x (t) =e t, y (t) =e t, t (a) Compute the length of the

More information

Lecture 1a: Satellite Orbits

Lecture 1a: Satellite Orbits Lectue 1a: Satellite Obits Outline 1. Newton s Laws of Motion 2. Newton s Law of Univesal Gavitation 3. Calculating satellite obital paametes (assuming cicula motion) Scala & Vectos Scala: a physical quantity

More information

Math 209 Assignment 9 Solutions

Math 209 Assignment 9 Solutions Math 9 Assignment 9 olutions 1. Evaluate 4y + 1 d whee is the fist octant pat of y x cut out by x + y + z 1. olution We need a paametic epesentation of the suface. (x, z). Now detemine the nomal vecto:

More information

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1)

EM-2. 1 Coulomb s law, electric field, potential field, superposition q. Electric field of a point charge (1) EM- Coulomb s law, electic field, potential field, supeposition q ' Electic field of a point chage ( ') E( ) kq, whee k / 4 () ' Foce of q on a test chage e at position is ee( ) Electic potential O kq

More information

10.2 Parametric Calculus

10.2 Parametric Calculus 10. Paametic Calculus Let s now tun ou attention to figuing out how to do all that good calculus stuff with a paametically defined function. As a woking eample, let s conside the cuve taced out by a point

More information

Unit 6 Test Review Gravitation & Oscillation Chapters 13 & 15

Unit 6 Test Review Gravitation & Oscillation Chapters 13 & 15 A.P. Physics C Unit 6 Test Review Gavitation & Oscillation Chaptes 13 & 15 * In studying fo you test, make sue to study this eview sheet along with you quizzes and homewok assignments. Multiple Choice

More information

F g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N

F g. = G mm. m 1. = 7.0 kg m 2. = 5.5 kg r = 0.60 m G = N m 2 kg 2 = = N Chapte answes Heinemann Physics 4e Section. Woked example: Ty youself.. GRAVITATIONAL ATTRACTION BETWEEN SMALL OBJECTS Two bowling balls ae sitting next to each othe on a shelf so that the centes of the

More information

AP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws.

AP-C WEP. h. Students should be able to recognize and solve problems that call for application both of conservation of energy and Newton s Laws. AP-C WEP 1. Wok a. Calculate the wok done by a specified constant foce on an object that undegoes a specified displacement. b. Relate the wok done by a foce to the aea unde a gaph of foce as a function

More information

Physics 2212 GH Quiz #2 Solutions Spring 2016

Physics 2212 GH Quiz #2 Solutions Spring 2016 Physics 2212 GH Quiz #2 Solutions Sping 216 I. 17 points) Thee point chages, each caying a chage Q = +6. nc, ae placed on an equilateal tiangle of side length = 3. mm. An additional point chage, caying

More information

Physics 181. Assignment 4

Physics 181. Assignment 4 Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This

More information

K.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper-1 (2015) Mathematics

K.S.E.E.B., Malleshwaram, Bangalore SSLC Model Question Paper-1 (2015) Mathematics K.S.E.E.B., Malleshwaam, Bangaloe SSLC Model Question Pape-1 (015) Mathematics Max Maks: 80 No. of Questions: 40 Time: Hous 45 minutes Code No. : 81E Fou altenatives ae given fo the each question. Choose

More information

Chapter 7. Rotational Motion Angles, Angular Velocity and Angular Acceleration Universal Law of Gravitation Kepler s Laws

Chapter 7. Rotational Motion Angles, Angular Velocity and Angular Acceleration Universal Law of Gravitation Kepler s Laws Chapte 7 Rotational Motion Angles, Angula Velocity and Angula Acceleation Univesal Law of Gavitation Keple s Laws Angula Displacement Cicula motion about AXIS Thee diffeent measues of angles: 1. Degees.

More information

Midterm Exam #2, Part A

Midterm Exam #2, Part A Physics 151 Mach 17, 2006 Midtem Exam #2, Pat A Roste No.: Scoe: Exam time limit: 50 minutes. You may use calculatos and both sides of ONE sheet of notes, handwitten only. Closed book; no collaboation.

More information

Practice. Understanding Concepts. Answers J 2. (a) J (b) 2% m/s. Gravitation and Celestial Mechanics 287

Practice. Understanding Concepts. Answers J 2. (a) J (b) 2% m/s. Gravitation and Celestial Mechanics 287 Pactice Undestanding Concepts 1. Detemine the gavitational potential enegy of the Eath Moon system, given that the aveage distance between thei centes is 3.84 10 5 km, and the mass of the Moon is 0.0123

More information

Gravitation. AP/Honors Physics 1 Mr. Velazquez

Gravitation. AP/Honors Physics 1 Mr. Velazquez Gavitation AP/Honos Physics 1 M. Velazquez Newton s Law of Gavitation Newton was the fist to make the connection between objects falling on Eath and the motion of the planets To illustate this connection

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

Chap 5. Circular Motion: Gravitation

Chap 5. Circular Motion: Gravitation Chap 5. Cicula Motion: Gavitation Sec. 5.1 - Unifom Cicula Motion A body moves in unifom cicula motion, if the magnitude of the velocity vecto is constant and the diection changes at evey point and is

More information

m1 m2 M 2 = M -1 L 3 T -2

m1 m2 M 2 = M -1 L 3 T -2 GAVITATION Newton s Univesal law of gavitation. Evey paticle of matte in this univese attacts evey othe paticle with a foce which vaies diectly as the poduct of thei masses and invesely as the squae of

More information

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1

AST 121S: The origin and evolution of the Universe. Introduction to Mathematical Handout 1 Please ead this fist... AST S: The oigin and evolution of the Univese Intoduction to Mathematical Handout This is an unusually long hand-out and one which uses in places mathematics that you may not be

More information

Physics C Rotational Motion Name: ANSWER KEY_ AP Review Packet

Physics C Rotational Motion Name: ANSWER KEY_ AP Review Packet Linea and angula analogs Linea Rotation x position x displacement v velocity a T tangential acceleation Vectos in otational motion Use the ight hand ule to detemine diection of the vecto! Don t foget centipetal

More information

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1

Phys102 Second Major-182 Zero Version Monday, March 25, 2019 Page: 1 Monday, Mach 5, 019 Page: 1 Q1. Figue 1 shows thee pais of identical conducting sphees that ae to be touched togethe and then sepaated. The initial chages on them befoe the touch ae indicated. Rank the

More information

MCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate.

MCV4U Final Exam Review. 1. Consider the function f (x) Find: f) lim. a) lim. c) lim. d) lim. 3. Consider the function: 4. Evaluate. lim. 5. Evaluate. MCVU Final Eam Review Answe (o Solution) Pactice Questions Conside the function f () defined b the following gaph Find a) f ( ) c) f ( ) f ( ) d) f ( ) Evaluate the following its a) ( ) c) sin d) π / π

More information

Central Force Problem. Central Force Motion. Two Body Problem: Center of Mass Coordinates. Reduction of Two Body Problem 8.01 W14D1. + m 2. m 2.

Central Force Problem. Central Force Motion. Two Body Problem: Center of Mass Coordinates. Reduction of Two Body Problem 8.01 W14D1. + m 2. m 2. Cental oce Poblem ind the motion of two bodies inteacting via a cental foce. Cental oce Motion 8.01 W14D1 Examples: Gavitational foce (Keple poblem): 1 1, ( ) G mm Linea estoing foce: ( ) k 1, Two Body

More information

AP * PHYSICS B. Circular Motion, Gravity, & Orbits. Teacher Packet

AP * PHYSICS B. Circular Motion, Gravity, & Orbits. Teacher Packet AP * PHYSICS B Cicula Motion, Gavity, & Obits Teache Packet AP* is a tademak of the College Entance Examination Boad. The College Entance Examination Boad was not involved in the poduction of this mateial.

More information

What Form of Gravitation Ensures Weakened Kepler s Third Law?

What Form of Gravitation Ensures Weakened Kepler s Third Law? Bulletin of Aichi Univ. of Education, 6(Natual Sciences, pp. - 6, Mach, 03 What Fom of Gavitation Ensues Weakened Keple s Thid Law? Kenzi ODANI Depatment of Mathematics Education, Aichi Univesity of Education,

More information

Δt The textbook chooses to say that the average velocity is

Δt The textbook chooses to say that the average velocity is 1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the

More information

Chapter 2: Basic Physics and Math Supplements

Chapter 2: Basic Physics and Math Supplements Chapte 2: Basic Physics and Math Supplements Decembe 1, 215 1 Supplement 2.1: Centipetal Acceleation This supplement expands on a topic addessed on page 19 of the textbook. Ou task hee is to calculate

More information

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,

More information

6.1: Angles and Their Measure

6.1: Angles and Their Measure 6.1: Angles and Thei Measue Radian Measue Def: An angle that has its vetex at the cente of a cicle and intecepts an ac on the cicle equal in length to the adius of the cicle has a measue of one adian.

More information

EN40: Dynamics and Vibrations. Midterm Examination Thursday March

EN40: Dynamics and Vibrations. Midterm Examination Thursday March EN40: Dynamics and Vibations Midtem Examination Thusday Mach 9 2017 School of Engineeing Bown Univesity NAME: Geneal Instuctions No collaboation of any kind is pemitted on this examination. You may bing

More information

Physics 4A Chapter 8: Dynamics II Motion in a Plane

Physics 4A Chapter 8: Dynamics II Motion in a Plane Physics 4A Chapte 8: Dynamics II Motion in a Plane Conceptual Questions and Example Poblems fom Chapte 8 Conceptual Question 8.5 The figue below shows two balls of equal mass moving in vetical cicles.

More information

Potential Energy and Conservation of Energy

Potential Energy and Conservation of Energy Potential Enegy and Consevation of Enegy Consevative Foces Definition: Consevative Foce If the wok done by a foce in moving an object fom an initial point to a final point is independent of the path (A

More information

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2 THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace

More information

4. Some Applications of first order linear differential

4. Some Applications of first order linear differential August 30, 2011 4-1 4. Some Applications of fist ode linea diffeential Equations The modeling poblem Thee ae seveal steps equied fo modeling scientific phenomena 1. Data collection (expeimentation) Given

More information

Circular Motion & Torque Test Review. The period is the amount of time it takes for an object to travel around a circular path once.

Circular Motion & Torque Test Review. The period is the amount of time it takes for an object to travel around a circular path once. Honos Physics Fall, 2016 Cicula Motion & Toque Test Review Name: M. Leonad Instuctions: Complete the following woksheet. SHOW ALL OF YOUR WORK ON A SEPARATE SHEET OF PAPER. 1. Detemine whethe each statement

More information

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if

Question Bank. Section A. is skew-hermitian matrix. is diagonalizable. (, ) , Evaluate (, ) 12 about = 1 and = Find, if Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7

More information

Supplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in

Supplementary Figure 1. Circular parallel lamellae grain size as a function of annealing time at 250 C. Error bars represent the 2σ uncertainty in Supplementay Figue 1. Cicula paallel lamellae gain size as a function of annealing time at 50 C. Eo bas epesent the σ uncetainty in the measued adii based on image pixilation and analysis uncetainty contibutions

More information

Chapter. s r. check whether your calculator is in all other parts of the body. When a rigid body rotates through a given angle, all

Chapter. s r. check whether your calculator is in all other parts of the body. When a rigid body rotates through a given angle, all conveted to adians. Also, be sue to vanced to a new position (Fig. 7.2b). In this inteval, the line OP has moved check whethe you calculato is in all othe pats of the body. When a igid body otates though

More information

Tutorial Exercises: Central Forces

Tutorial Exercises: Central Forces Tutoial Execises: Cental Foces. Tuning Points fo the Keple potential (a) Wite down the two fist integals fo cental motion in the Keple potential V () = µm/ using J fo the angula momentum and E fo the total

More information

Problem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by

Problem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,

More information

Liquid gas interface under hydrostatic pressure

Liquid gas interface under hydrostatic pressure Advances in Fluid Mechanics IX 5 Liquid gas inteface unde hydostatic pessue A. Gajewski Bialystok Univesity of Technology, Faculty of Civil Engineeing and Envionmental Engineeing, Depatment of Heat Engineeing,

More information

MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: MOTION OF PLANETS AND SATELLITES VISUAL PHYSICS ONLINE

MODULE 5 ADVANCED MECHANICS GRAVITATIONAL FIELD: MOTION OF PLANETS AND SATELLITES VISUAL PHYSICS ONLINE VISUAL PHYSICS ONLIN MODUL 5 ADVANCD MCHANICS GRAVITATIONAL FILD: MOTION OF PLANTS AND SATLLITS SATLLITS: Obital motion of object of mass m about a massive object of mass M (m

More information

11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below.

11) A thin, uniform rod of mass M is supported by two vertical strings, as shown below. Fall 2007 Qualifie Pat II 12 minute questions 11) A thin, unifom od of mass M is suppoted by two vetical stings, as shown below. Find the tension in the emaining sting immediately afte one of the stings

More information

When two numbers are written as the product of their prime factors, they are in factored form.

When two numbers are written as the product of their prime factors, they are in factored form. 10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

More information

Right-handed screw dislocation in an isotropic solid

Right-handed screw dislocation in an isotropic solid Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We

More information

Modeling Ballistics and Planetary Motion

Modeling Ballistics and Planetary Motion Discipline Couses-I Semeste-I Pape: Calculus-I Lesson: Lesson Develope: Chaitanya Kuma College/Depatment: Depatment of Mathematics, Delhi College of Ats and Commece, Univesity of Delhi Institute of Lifelong

More information

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50

working pages for Paul Richards class notes; do not copy or circulate without permission from PGR 2004/11/3 10:50 woking pages fo Paul Richads class notes; do not copy o ciculate without pemission fom PGR 2004/11/3 10:50 CHAPTER7 Solid angle, 3D integals, Gauss s Theoem, and a Delta Function We define the solid angle,

More information

Our Universe: GRAVITATION

Our Universe: GRAVITATION Ou Univese: GRAVITATION Fom Ancient times many scientists had shown geat inteest towads the sky. Most of the scientist studied the motion of celestial bodies. One of the most influential geek astonomes

More information

PHYSICS NOTES GRAVITATION

PHYSICS NOTES GRAVITATION GRAVITATION Newton s law of gavitation The law states that evey paticle of matte in the univese attacts evey othe paticle with a foce which is diectly popotional to the poduct of thei masses and invesely

More information

On the Sun s Electric-Field

On the Sun s Electric-Field On the Sun s Electic-Field D. E. Scott, Ph.D. (EE) Intoduction Most investigatos who ae sympathetic to the Electic Sun Model have come to agee that the Sun is a body that acts much like a esisto with a

More information

- 5 - TEST 1R. This is the repeat version of TEST 1, which was held during Session.

- 5 - TEST 1R. This is the repeat version of TEST 1, which was held during Session. - 5 - TEST 1R This is the epeat vesion of TEST 1, which was held duing Session. This epeat test should be attempted by those students who missed Test 1, o who wish to impove thei mak in Test 1. IF YOU

More information

Gauss s Law Simulation Activities

Gauss s Law Simulation Activities Gauss s Law Simulation Activities Name: Backgound: The electic field aound a point chage is found by: = kq/ 2 If thee ae multiple chages, the net field at any point is the vecto sum of the fields. Fo a

More information

1. Show that the volume of the solid shown can be represented by the polynomial 6x x.

1. Show that the volume of the solid shown can be represented by the polynomial 6x x. 7.3 Dividing Polynomials by Monomials Focus on Afte this lesson, you will be able to divide a polynomial by a monomial Mateials algeba tiles When you ae buying a fish tank, the size of the tank depends

More information

Circular motion. Objectives. Physics terms. Assessment. Equations 5/22/14. Describe the accelerated motion of objects moving in circles.

Circular motion. Objectives. Physics terms. Assessment. Equations 5/22/14. Describe the accelerated motion of objects moving in circles. Cicula motion Objectives Descibe the acceleated motion of objects moving in cicles. Use equations to analyze the acceleated motion of objects moving in cicles.. Descibe in you own wods what this equation

More information

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006

Qualifying Examination Electricity and Magnetism Solutions January 12, 2006 1 Qualifying Examination Electicity and Magnetism Solutions Januay 12, 2006 PROBLEM EA. a. Fist, we conside a unit length of cylinde to find the elationship between the total chage pe unit length λ and

More information

Recap. Centripetal acceleration: v r. a = m/s 2 (towards center of curvature)

Recap. Centripetal acceleration: v r. a = m/s 2 (towards center of curvature) a = c v 2 Recap Centipetal acceleation: m/s 2 (towads cente of cuvatue) A centipetal foce F c is equied to keep a body in cicula motion: This foce poduces centipetal acceleation that continuously changes

More information

Flux. Area Vector. Flux of Electric Field. Gauss s Law

Flux. Area Vector. Flux of Electric Field. Gauss s Law Gauss s Law Flux Flux in Physics is used to two distinct ways. The fist meaning is the ate of flow, such as the amount of wate flowing in a ive, i.e. volume pe unit aea pe unit time. O, fo light, it is

More information

1) Consider a particle moving with constant speed that experiences no net force. What path must this particle be taking?

1) Consider a particle moving with constant speed that experiences no net force. What path must this particle be taking? Chapte 5 Test Cicula Motion and Gavitation 1) Conside a paticle moving with constant speed that expeiences no net foce. What path must this paticle be taking? A) It is moving in a paabola. B) It is moving

More information

6.4 Period and Frequency for Uniform Circular Motion

6.4 Period and Frequency for Uniform Circular Motion 6.4 Peiod and Fequency fo Unifom Cicula Motion If the object is constained to move in a cicle and the total tangential foce acting on the total object is zeo, F θ = 0, then (Newton s Second Law), the tangential

More information

4/18/2005. Statistical Learning Theory

4/18/2005. Statistical Learning Theory Statistical Leaning Theoy Statistical Leaning Theoy A model of supevised leaning consists of: a Envionment - Supplying a vecto x with a fixed but unknown pdf F x (x b Teache. It povides a desied esponse

More information

GRAVITATION. Contents. Theory Exercise Exercise Exercise Exercise Answer Key

GRAVITATION. Contents. Theory Exercise Exercise Exercise Exercise Answer Key GAVITATION Contents Topic Page No. Theoy 0-0 Execise - 0 - Execise - - 8 Execise - 8 - Execise - 4 - Answe Key - 4 Syllabus Law of gavitation; Gavitational potential and field; Acceleation due to gavity;

More information

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an

! E da = 4πkQ enc, has E under the integral sign, so it is not ordinarily an Physics 142 Electostatics 2 Page 1 Electostatics 2 Electicity is just oganized lightning. Geoge Calin A tick that sometimes woks: calculating E fom Gauss s law Gauss s law,! E da = 4πkQ enc, has E unde

More information

Mark answers in spaces on the answer sheet

Mark answers in spaces on the answer sheet Mak answes in spaces 31-43 on the answe sheet PHYSICS 1 Summe 005 EXAM 3: July 5 005 9:50pm 10:50pm Name (pinted): ID Numbe: Section Numbe: INSTRUCTIONS: Some questions ae one point, othes ae two points,

More information

CHAPTER 3. Section 1. Modeling Population Growth

CHAPTER 3. Section 1. Modeling Population Growth CHAPTER 3 Section 1. Modeling Population Gowth 1.1. The equation of the Malthusian model is Pt) = Ce t. Apply the initial condition P) = 1. Then 1 = Ce,oC = 1. Next apply the condition P1) = 3. Then 3

More information

Calculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008

Calculus I Section 4.7. Optimization Equation. Math 151 November 29, 2008 Calculus I Section 4.7 Optimization Solutions Math 151 Novembe 9, 008 The following poblems ae maimum/minimum optimization poblems. They illustate one of the most impotant applications of the fist deivative.

More information

TAMPINES JUNIOR COLLEGE 2009 JC1 H2 PHYSICS GRAVITATIONAL FIELD

TAMPINES JUNIOR COLLEGE 2009 JC1 H2 PHYSICS GRAVITATIONAL FIELD TAMPINES JUNIOR COLLEGE 009 JC1 H PHYSICS GRAVITATIONAL FIELD OBJECTIVES Candidates should be able to: (a) show an undestanding of the concept of a gavitational field as an example of field of foce and

More information

Rewriting Equations and Formulas. Write original equation.

Rewriting Equations and Formulas. Write original equation. Page 1 of 7 1.4 Rewiting Equations and Fomulas What you should lean GOAL 1 Rewite equations with moe than one vaiale. GOAL Rewite common fomulas, as applied in Example 5. Why you should lean it To solve

More information

Recall from last week:

Recall from last week: Recall fom last week: Length of a cuve '( t) dt b Ac length s( t) a a Ac length paametization ( s) with '( s) 1 '( t) Unit tangent vecto T '(s) '( t) dt Cuvatue: s ds T t t t t t 3 t ds u du '( t) dt Pincipal

More information

ω = θ θ o = θ θ = s r v = rω

ω = θ θ o = θ θ = s r v = rω Unifom Cicula Motion Unifom cicula motion is the motion of an object taveling at a constant(unifom) speed in a cicula path. Fist we must define the angula displacement and angula velocity The angula displacement

More information

SMT 2013 Team Test Solutions February 2, 2013

SMT 2013 Team Test Solutions February 2, 2013 1 Let f 1 (n) be the numbe of divisos that n has, and define f k (n) = f 1 (f k 1 (n)) Compute the smallest intege k such that f k (013 013 ) = Answe: 4 Solution: We know that 013 013 = 3 013 11 013 61

More information

Inverse Square Law and Polarization

Inverse Square Law and Polarization Invese Squae Law and Polaization Objectives: To show that light intensity is invesely popotional to the squae of the distance fom a point light souce and to show that the intensity of the light tansmitted

More information

Math Section 4.2 Radians, Arc Length, and Area of a Sector

Math Section 4.2 Radians, Arc Length, and Area of a Sector Math 1330 - Section 4. Radians, Ac Length, and Aea of a Secto The wod tigonomety comes fom two Geek oots, tigonon, meaning having thee sides, and mete, meaning measue. We have aleady defined the six basic

More information

Exam 3, vers Physics Spring, 2003

Exam 3, vers Physics Spring, 2003 1 of 9 Exam 3, ves. 0001 - Physics 1120 - Sping, 2003 NAME Signatue Student ID # TA s Name(Cicle one): Michael Scheffestein, Chis Kelle, Paisa Seelungsawat Stating time of you Tues ecitation (wite time

More information

Gravitational Potential Energy in General

Gravitational Potential Energy in General Gavitational Potential Enegy in Geneal 6.3 To exploe such concepts as how much enegy a space pobe needs to escape fom Eath s gavity, we must expand on the topic of gavitational potential enegy, which we

More information

Rotational Motion. Lecture 6. Chapter 4. Physics I. Course website:

Rotational Motion. Lecture 6. Chapter 4. Physics I. Course website: Lectue 6 Chapte 4 Physics I Rotational Motion Couse website: http://faculty.uml.edu/andiy_danylov/teaching/physicsi Today we ae going to discuss: Chapte 4: Unifom Cicula Motion: Section 4.4 Nonunifom Cicula

More information