Area, Volume, and Center of Mass
|
|
- Arron Reeves
- 6 years ago
- Views:
Transcription
1 Area, Volume, and Center of Mass MATH 311, Calculus III J. obert Buchanan Department of Mathematics Fall 211
2 Introduction Today we will apply the double integral to the problems of finding the area of regions in the plane, the volume of a bounded region in three dimensions, the center of mass of a region in the plane.
3 Area If is a region in the xy-plane, the area A of is A 1 da provided the value of the double integral is finite.
4 Example (1 of 6) Find the area of the triangular region with vertices at (, 2), (3, 2), and (1, 1) x
5 Example (2 of 6) The region is bounded by the graphs of the lines x 2 y and x 2y 1 for 1 y 2. A 1 da y y 1 dx dy (2y 1) (2 y) dy 3y 3 dy 3 2 y 2 2 3y 1 ( )
6 Example (3 of 6) Find the area of the bounded region in the xy-plane between y x and y x y
7 Example (4 of 6) A 1 da 1 x 1 x 2 1 dy dx x x 2 dx 2 3 x 3/2 1 3 x
8 Example (5 of 6) Find the area of the bounded region in the xy-plane between y x and x y 2 y. 2 y x
9 Example (6 of 6) A 1 da y y 2 y 1 dx dy y (y 2 y) dy 2y y 2 dy y y
10 Volume of a Solid egion If f (x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the region in the xy-plane is V f (x, y) da.
11 Volume of a Solid egion If f (x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the region in the xy-plane is V f (x, y) da. If f (x, y) g(x, y) for all (x, y) in a region, the volume of the solid below the surface z f (x, y) and above the surface z g(x, y) within the the region in the xy-plane is V (f (x, y) g(x, y)) da.
12 Example (1 of 4) Find the volume of the solid region below the graph of z e y 2 and above the triangle in the xy-plane with vertices at (, ), (1, 1), (, 1). z y 1. x.5 1..
13 Example (2 of 4) V 1 y 1 1 f (x, y) da e y 2 da xe y 2 y 1 2 ey 2 1 ye y 2 dy 1 2 e 1 2 e y 2 dx dy dy
14 Example (3 of 4) Find the volume of the solid region above the graph of z 2x 2 + y 2 and below the graph of z 8 x 2 2y 2 and within the rectangle {(x, y) x 1, y 1}. z y 1. x.5 1..
15 Example (4 of 4) V f (x, y) da (8 x 2 2y 2 ) (2x 2 + y 2 ) dx dy (8 3x 2 3y 2 ) dx dy 8x x 3 3xy 2 1 (8 1 3y 2 ) dy (7 3y 2 ) dy 7y y dy
16 Moments and Center of Mass lamina: a thin sheet of material in the shape of a region 2. ρ(x, y): the density of the lamina at (x, y). m: mass of the lamina m ρ(x, y) da. M x : moment with respect to the x-axis M x yρ(x, y) da. M y : moment with respect to the y-axis M y xρ(x, y) da. (x, y) center of mass (x, y) ( My m, M ) x m
17 Example (1 of 2) A lamina with density ρ(x, y) y + 3 is bounded by the curves x y 2 and x 4. Find its mass, moments, and center of mass. 1 y x
18 Example (2 of 2) m M x y 2 (y + 3) dx dy y 2 y(y + 3) dx dy M y y 2 x(y + 3) dx dy
19 Example (2 of 2) m M x y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12 + 4y 3y 2 y 3 ) dy 32 M y y 2 x(y + 3) dx dy
20 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy y 2 x(y + 3) dx dy (12 + 4y 3y 2 y 3 ) dy 32
21 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy x(y + 3) dx dy y ( y 32 y 4 12 ) y 5 (12 + 4y 3y 2 y 3 ) dy 32 dy 384 5
22 Example (2 of 2) m M x M y y 2 (y + 3) dx dy y 2 y(y + 3) dx dy (12y + 4y 2 3y 3 y 4 ) dy x(y + 3) dx dy y ( y 32 y 4 12 ) y 5 (x, y) (12 + 4y 3y 2 y 3 ) dy 32 dy ( My m, M ) ( x 12 m 5, 4 ) 15
23 Moment of Inertia Inertia: tendency of an object to resist a change in motion. I x : moment of inertia about the x-axis I x y 2 ρ(x, y) da. I y : moment of inertia about the y-axis I y x 2 ρ(x, y) da.
24 Example (1 of 2) A lamina with density ρ(x, y) y + 3 is bounded by the curves x y 2 and x 4. Find its moments of inertia. 1 y x
25 Example (2 of 2) I x y 2 y 2 (y + 3) dx dy I y y 2 x 2 (y + 3) dx dy
26 Example (2 of 2) I x I y y 2 y 2 (y + 3) dx dy 12y 2 + 4y 3 3y 4 y 5 dy 4y 3 + y y y 6 2 y 2 x 2 (y + 3) dx dy 128 5
27 Example (2 of 2) I x I y y 2 y 2 (y + 3) dx dy 12y 2 + 4y 3 3y 4 y 5 dy 4y 3 + y y y 6 2 y 2 x 2 (y + 3) dx dy y y y 7 dy 64y y y y
28 Homework ead Section Exercises: 1 17 odd, odd, 37, 43, 45
Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope
There Is Only One Line Passing Through a Given Point with a Given Slope Classwork Opening Exercise Examine each of the graphs and their equations. Identify the coordinates of the point where the line intersects
More informationTotal 70
Math 120 A Spring 2017 Final Exam June 3, 2017 Name: Student ID no. : Signature: Section: 1 10 2 10 3 10 4 10 5 10 6 10 7 10 Total 70 Complete all seven questions. Show all work for full credit. The only
More informationHomework Helpers Sampler
Homework Helpers Sampler This sampler includes s for Algebra I, Lessons 1-3. To order a full-year set of s visit >>> http://eurmath.link/homework-helpers Published by the non-profit Great Minds. Copyright
More informationAP Physics 1 - Test 04 - Projectile Motion
P Physics 1 - Test 04 - Projectile Motion Score: 1. stone thrown from the top of a tall building follows a path that is circular made of two straight line segments hyperbolic parabolic a straight line
More information- a set of known masses, - four weight hangers, - tape - a fulcrum upon which the meter stick can be mounted and pivoted - string - stopwatch
1. In the laboratory, you are asked to determine the mass of a meter stick without using a scale of any kind. In addition to the meter stick, you may use any or all of the following equipment: - a set
More informationOrganizing Quantitative Data
Organizing Quantitative Data MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2018 Objectives At the end of this lesson we will be able to: organize discrete data in
More informationFor example, the velocity at t = 10 is given by the gradient of the curve at t = 10, 10 t
R15 INTERPRET THE GRADIENT AT A POINT ON A CURVE AS THE INSTANTANEOUS RATE OF CHANGE; APPLY THE CONCEPTS OF AVERAGE AND INSTANTANEOUS RATE OF CHANGE (GRADIENTS OF CHORDS AND TANGENTS) IN NUMERICAL, ALGEBRAIC
More informationFC-CIV HIDRCANA: Channel Hydraulics Flow Mechanics Review Fluid Statics
FC-CIV HIDRCANA: Channel Hydraulics Flow Mechanics Review Fluid Statics Civil Engineering Program, San Ignacio de Loyola University Objective Calculate the forces exerted by a fluid at rest on plane or
More informationJeddah Knowledge International School. Science Revision Pack Answer Key Quarter 3 Grade 10
Jeddah Knowledge International School Science Revision Pack Answer Key 2016-2017 Quarter 3 Grade 10 Name: Section: ANSWER KEY- SCIENCE GRADE 10, QUARTER 3 1 1. What are the units for mass? A Kilograms
More informationProblem 5: Platform Diving
Problem 5: Platform Diving In the 22 World Cup Trials, Kyle Prandi set up a diving record with a back 3 ½ somersault pike from the 1 m board. He pushed off from the board at an angle of θ = 46 with an
More informationPhysics 11 Unit III Practice Test Projectile Motion. Instructions: Pick the best answer available in Part A and Show all your work for Part B
Physics 11 Unit III Practice Test Projectile Motion Instructions: Pick the best answer available in Part A and Show all your work for Part B 1. Which of the following is constant for all projectiles? A.
More information3. Answer the following questions with your group. How high do you think he was at the top of the stairs? How did you estimate that elevation?
Classwork Exploratory Challenge 1. Watch the first 1:08 minutes of the video below and describe in words the motion of the man. Elevation vs. Time #2 [http://www.mrmeyer.com/graphingstories1/graphingstories2.mov.
More informationSet the Sails! 3. You will study patterns on the graphs and tables and make generalizations about the transformations presented.
The sails on a sailboat are set according to the direction of the wind and the bearing. When one jibes or tacks, the main sail is moved from one side of the boat to the opposite about the mast. As the
More informationMath 116 Practice for Exam 1
Math 116 Practice for Exam 1 Generated September 3, 215 Name: SOLUTIONS Instructor: Section Number: 1. This exam has 4 questions. Note that the problems are not of equal difficulty, so you may want to
More informationCONCEPTUAL PHYSICS LAB
PURPOSE The purpose of this lab is to determine the density of an unknown solid by direct calculation and by graphing mass vs. volume for several samples of the solid. INTRODUCTION Which is heavier, a
More informationHomework #10 Due Monday, May Compute the moment of inertia of the objects below.
Homework #10 Due Monday, May 1. Compute the moment of inertia of the objects below. θ (a) Compute the moment of inertia of a thin rod of mass M and length R for rotations about its end. Explain why your
More informationExploration Series. MODEL ROCKET Interactive Physics Simulation Page 01
MODEL ROCKET ------- Interactive Physics Simulation ------- Page 01 How high will your model rocket fly? At liftoff, the rocket engine is ignited and a thrust force is generated. The rocket accelerates
More information2 Available: 1390/08/02 Date of returning: 1390/08/17 1. A suction cup is used to support a plate of weight as shown in below Figure. For the conditio
1. A suction cup is used to support a plate of weight as shown in below Figure. For the conditions shown, determine. 2. A tanker truck carries water, and the cross section of the truck s tank is shown
More informationx 2 = (60 m) 2 + (60 m) 2 x 2 = 3600 m m 2 x = m
3.1 Track Question a) Distance Traveled is 1600 m. This is length of the path that the person took. The displacement is 0 m. The person begins and ends their journey at the same position. They did not
More informationPreliminary design of a high-altitude kite. A flexible membrane kite section at various wind speeds
Preliminary design of a high-altitude kite A flexible membrane kite section at various wind speeds This is the third paper in a series that began with one titled A flexible membrane kite section at high
More informationDescribing a journey made by an object is very boring if you just use words. As with much of science, graphs are more revealing.
Distance vs. Time Describing a journey made by an object is very boring if you just use words. As with much of science, graphs are more revealing. Plotting distance against time can tell you a lot about
More information3. Answer the following questions with your group. How high do you think he was at the top of the stairs? How did you estimate that elevation?
J Hart Interactive Algebra 1 Classwork Exploratory Challenge 1. Watch the first 1:08 minutes of the video below and describe in words the motion of the man. Elevation vs. Time #2 [http://www.mrmeyer.com/graphingstories1/graphingstories2.mov.
More information2.6 Related Rates Worksheet Calculus AB. dy /dt!when!x=8
Two Rates That Are Related(1-7) In exercises 1-2, assume that x and y are both differentiable functions of t and find the required dy /dt and dx /dt. Equation Find Given 1. dx /dt = 10 y = x (a) dy /dt
More informationREAL LIFE GRAPHS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Real Life Graphs Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier REAL LIFE GRAPHS Version: 2.1 Date: 20-10-2015 Mathematics Revision Guides
More informationA NEW GOLF-SWING ROBOT MODEL UTILIZING SHAFT ELASTICITY
Journal of Sound and Vibration (1998) 17(1), 17 31 Article No. sv981733 A NEW GOLF-SWING ROBOT MODEL UTILIZING SHAFT ELASTICITY S. SUZUKI Department of Mechanical System Engineering, Kitami Institute of
More informationCalculus 12: Evaluation 3 Outline and Review
Calculus 12: Evaluation 3 Outline and Review You should be able to: 1. Differentiate various types of functions including trigonometric, exponential and logarithmic functions, 2. Solve various related
More informationJefferson Township Public Schools Mathematics Department
Jefferson Township Public Schools Mathematics Department Dear Student of Math Investigations, Your first assignment as a Math Investigations student will be the summer assignment. This packet is a review
More informationSTEM SPORTS.
STEM SPORTS Mr. B. Pezzuto Email: bpezzuto@revere.mec.edu 781-388-7520 ext: 54217 Staying Connected to STEM Sports Twitter: @SBA_Pezzuto Instagram: SBA_Pezzuto Classroom Policies Grading Policy: Tests:
More informationG.C.E (A/L) Examination March In collaboration with
; G.C.E (A/L) Examination March - 2018 Conducted by Field Work Centre, Thondaimanaru In collaboration with FWC Provincial Department of Education Northern Province Grade:- 12 (2019) Physics Part - II Structured
More informationA NOVEL FLOATING OFFSHORE WIND TURBINE CONCEPT: NEW DEVELOPMENTS
A NOVEL FLOATING OFFSHORE WIND TURBINE CONCEPT: NEW DEVELOPMENTS L. Vita, U.S.Paulsen, T.F.Pedersen Risø-DTU Technical University of Denmark, Roskilde, Denmark luca.vita@risoe.dk Abstract: A novel concept
More informationMotion in 1 Dimension
A.P. Physics 1 LCHS A. Rice Unit 1 Displacement, Velocity, & Acceleration: Motion in 1 Dimension In-Class Example Problems and Lecture Notes 1. Freddy the cat started at the 3 meter position. He then walked
More information1 8 Practice Perimeter Circumference And Area Answers Form G
1 8 Practice Perimeter Circumference And Area Answers Form G We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer,
More informationUnit 7. Math Problem 1. This segment will go through the endpoint of the original line segment, perpendicular to the line segment.
Math 1007 Unit 7 1 Construct a square with sides equal to r. 1: Extend the segment and draw a circle centered at one of the endpoints of the segment 2: Draw two larger congruent circles centered where
More informationShedding Light on Motion Episode 4: Graphing Motion
Shedding Light on Motion Episode 4: Graphing Motion In a 100-metre sprint, when do athletes reach their highest speed? When do they accelerate at the highest rate and at what point, if any, do they stop
More informationPhysics Wave Problems. Science and Mathematics Education Research Group
F FA ACULTY C U L T Y OF O F EDUCATION E D U C A T I O N Department of Curriculum and Pedagogy Physics Wave Problems Science and Mathematics Education Research Group Supported by UBC Teaching and Learning
More information1 8 Practice Perimeter Circumference And Area Form K Answers
1 8 Practice Perimeter Circumference And Area Form K Answers We have made it easy for you to find a PDF Ebooks without any digging. And by having access to our ebooks online or by storing it on your computer,
More informationRelated Rates - Classwork
Related Rates - Classwork Earlier in the year, we used the basic definition of calculus as the mathematics of change. We defined words that meant change: increasing, decreasing, growing, shrinking, etc.
More informationFUNCTIONAL SKILLS MATHEMATICS (level 1)
FUNCTIONAL SKILLS MATHEMATICS (level 1) Detailed Marking Instructions Version: May 2011 Question Marking Scheme Illustrations of evidence No Give for each for awarding a mark 1 (a) Ans: 675 represent:
More information1ACE Exercise 4. Name Date Class
1ACE Exercise 4 Investigation 1 4. A farm wants to add a small rectangular petting zoo for the public. They have a fixed amount of fencing to use for the zoo. This graph shows the lengths and areas of
More informationForce and Motion Test Review
Name: Period: Force and Motion Test Review 1. I can tell you that force is.. 2. Force is measured in units called. 3. Unbalanced forces acting on an object will MOST LIKELY cause the object to A. remain
More informationGary Delia AMS151 Fall 2009 Homework Set 3 due 10/07/2010 at 11:59pm EDT
Gary Delia AMS151 Fall 2009 Homework Set 3 due 10/07/2010 at 11:59pm EDT 1. (2 pts) The angle of elevation to the top of a building is found to be 9 from the ground at a distance of 6000 feet from the
More informationMEASURING VOLUME & MASS
MEASURING VOLUME & MASS In this laboratory you will have the opportunity to apply your measuring skills in gathering data, processing it, and interpreting the results. For this experiment you will: 1)
More informationIGCSE - Cumulative Frequency Questions
Dulwich College Shanghai IGCSE - Cumulative Frequency Questions 85 min 72 marks 1. Answer the whole of this question on one sheet of graph paper. The heights (h cm) of 270 students in a school are measured
More informationFor a tennis ball brand to be approved for tournament play
L A B 12 ANALYZING A BOUNCING TENNIS BALL Infinite Series For a tennis ball brand to be approved for tournament play by the United States Tennis Association (USTA), it must satisfy several specifications.
More informationChapter 3 Hydrostatics and Floatation
Chapter 3 Hydrostatics and Floatation Naval Architecture Notes 3.1 Archimedes Law of Floatation Archimedes (born 287 B.C) Law states that An object immersed in a liquid experience a lift equivalent to
More informationFurman University Wylie Mathematics Tournament Ciphering Competition. March 10, 2007
Furman University Wylie Mathematics Tournament Ciphering Competition March 10, 2007 House Rules 1. All answers are integers(!) 2. All answers must be written in standard form. For example, 8 not 2 3, and
More informationA position graph will give the location of an object at a certain time.
Calculus 3.4 Notes A position graph will give the location of an object at a certain time. At t = 4, the car is 20 miles away from where it started. A position function is usually written as or. If the
More information1. Five more than three times a number x. 3. Seventeen percent of a number x.
Pre-Calculus: 1.7.1 Mastery Practice Name: Date Period Write a mathematical expression for the quantity described verbally: 1. Five more than three times a number x. 3. Seventeen percent of a number x.
More informationHomework: Turn in Tortoise & the Hare
Your Learning Goal: After students experienced speed in the Runner s Speed Lab, they will be able to describe how different speeds look like on a graph with 100% accuracy. Table of Contents: Notes: Graphs
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and
More informationBIOMECHANICAL MOVEMENT
SECTION PART 5 5 CHAPTER 12 13 CHAPTER 12: Biomechanical movement Practice questions - text book pages 169-172 1) For which of the following is the athlete s centre of mass most likely to lie outside of
More informationCalculate the horizontal component of the baseball's velocity at an earlier time calculated in part (a).
Ch3 Supplemental [ Edit ] Overview Summary View Diagnostics View Print View with Answers Ch3 Supplemental Due: 6:59pm on Monday, February 13, 2017 To understand how points are awarded, read the Grading
More informationChapter 11 Waves. Waves transport energy without transporting matter. The intensity is the average power per unit area. It is measured in W/m 2.
Chapter 11 Waves Energy can be transported by particles or waves A wave is characterized as some sort of disturbance that travels away from a source. The key difference between particles and waves is a
More informationAP Physics B Summer Homework (Show work)
#1 NAME: AP Physics B Summer Homework (Show work) #2 Fill in the radian conversion of each angle and the trigonometric value at each angle on the chart. Degree 0 o 30 o 45 o 60 o 90 o 180 o 270 o 360 o
More informationGX-S. 2-jaw self centering radial pneumatic gripper (series GX-S) GX-S
GX-S 2-jaw self centering radial pneumatic gripper (series GX-S) Double acting. Very high gripping force at the end of the stroke. Long life and reliability, maintenance free. Various options for fastening.
More informationGCSE Mathematics Practice Tests: Set 5
GCSE Mathematics Practice Tests: Set 5 Paper 2H (Calculator) Time: 1 hour 30 minutes You should have: Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil, eraser,
More informationAdvanced Subsidiary / Advanced Level
GCE Examinations Mechanics Module M1 Advanced Subsidiary / Advanced Level Paper K Time: 1 hour 30 minutes Instructions and Information Candidates may use any calculator except those with a facility for
More informationAtmosphere, Ocean and Climate Dynamics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 12.003 Atmosphere, Ocean and Climate Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Problem
More informationModule 3: Hydrostatic forces on submerged bodies Lecture 7: Calculation of horizontal component, buoyancy. Forces on submerged bodies (continued)
Forces on submerged bodies (continued) Buoyancy file:///d /Web%20Course/Dr.%20Nishith%20Verma/local%20server/fluid_mechanics/lecture7/7_1.htm[5/9/2012 3:05:26 PM] Forces on submerged bodies (continued)
More informationAdaptor Core Technology:
Adaptor Core Technology: The Inception and Adapting of Calculus Based Truths within Geometric Entities Written By: Nick Siefers (Nicks@900global.com) Director of Operations 900 Global would like to introduce
More information+ t1 t2 moment-time curves
Part 6 - Angular Kinematics / Angular Impulse 1. While jumping over a hurdle, an athlete s hip angle was measured to be 2.41 radians. Within 0.15 seconds, the hurdler s hip angle changed to be 3.29 radians.
More informationWhich student do you agree with (if any)? Please explain your reasoning.
Practice Test 1 1. Shown below is a speed-time graph for a cart moving in front of the motion sensor. For convenience it has been divided into five sections (A,B,C,D,E). B C D A E During each of the five
More informationLesson 22: Average Rate of Change
Student Outcomes Students know how to compute the average rate of change in the height of water level when water is poured into a conical container at a constant rate. MP.1 Lesson Notes This lesson focuses
More informationSimplifying Radical Expressions and the Distance Formula
1 RD. Simplifying Radical Expressions and the Distance Formula In the previous section, we simplified some radical expressions by replacing radical signs with rational exponents, applying the rules of
More informationCH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES
CH 34 MORE PYTHAGOREAN THEOREM AND RECTANGLES 317 Recalling The Pythagorean Theorem a 2 + b 2 = c 2 a c 90 b The 90 angle is called the right angle of the right triangle. The other two angles of the right
More informationParallel grippers HGPM, micro
Key features G6: G8: G9: stroke compensation clamping spigot flange mounting At a glance Compact, handy design With open or closed gripper jaws Versatility thanks to externally adaptable gripper fingers
More information5.1. Data Displays Batter Up. My Notes ACTIVITY
SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Marking the Text, Group Presentation, Interactive Word Wall Henry Hank Aaron and Harmon Killebrew are among the alltime leaders in home runs in
More informationUsing GPOPS-II to optimize sum of squared torques of a double pendulum as a prosthesis leg. Abstract
Using GPOPS-II to optimize sum of squared torques of a double pendulum as a prosthesis leg Abstract Milad Zarei MCE 593 Prosthesis Design & Control A two-dimensional, two links pendulum is developed to
More informationThe purpose of this experiment is to find this acceleration for a puck moving on an inclined air table.
Experiment : Motion in an Inclined Plane PURPOSE The purpose of this experiment is to find this acceleration for a puck moving on an inclined air table. GENERAL In Experiment-1 you were concerned with
More informationHomework of chapter (3)
The Islamic University of Gaza, Civil Engineering Department, Fluid mechanics-discussion, Instructor: Dr. Khalil M. Al Astal T.A: Eng. Hasan Almassri T.A: Eng. Mahmoud AlQazzaz First semester, 2013. Homework
More informationINCLINOMETER DEVICE FOR SHIP STABILITY EVALUATION
Proceedings of COBEM 2009 Copyright 2009 by ABCM 20th International Congress of Mechanical Engineering November 15-20, 2009, Gramado, RS, Brazil INCLINOMETER DEVICE FOR SHIP STABILITY EVALUATION Helena
More informationAbstract. 1 Introduction
A computational method for calculatingthe instantaneous restoring coefficients for a ship moving in waves N. El-Simillawy College of Engineering and Technology, Arab Academyfor Science and Technology,
More informationVariable Face Milling to Normalize Putter Ball Speed and Maximize Forgiveness
Proceedings Variable Face Milling to Normalize Putter Ball Speed and Maximize Forgiveness Jacob Lambeth *, Dustin Brekke and Jeff Brunski Cleveland Golf, 5601 Skylab Rd. Huntington Beach, CA 92647, USA;
More informationWHAT DO THESE GRAPHS MEAN?
SOLUBILITY CURVES WHAT DO THESE GRAPHS MEAN? Each line shows how much stuff (CALLED SOLUTE) can be dissolved in 100 ml OF WATER (CALLED SOLVENT) at a bunch of different temperatures. Why at a bunch of
More informationFinding the Area of a Major League Baseball Field
Undergraduate Journal of Mathematical Modeling: One + Two Volume 5 2012 Fall Issue 1 Article 7 Finding the Area of a Major League Baseball Field Jacob Courchaine University of South Florida Advisors: Arcadii
More informationcomputed using Equation 3-18 by setting the 2nd term equal to 0 and K A equal to K o and using the pressure distribution as shown in Figure 3-23.
computed using Equation 3-18 by setting the 2nd term equal to 0 and K A equal to K o and using the pressure distribution as shown in Figure 3-23. (2) For the resisting side, passive pressure theory indicates
More informationHonors Physics Rotation HW, statics (Homework)
Honors Physics Rotation HW, statics (Homework) For answers, send email to: admin@tutor-homework.com. Include file name: Physics_Worksheet_0058 Price: $3 (c) 2012 www.tutor-homework.com: Tutoring, homework
More informationMATH GRADE 6 UNIT 6 RATE ANSWERS FOR EXERCISES
MATH GRADE 6 UNIT 6 RATE FOR EXERCISES LESSON 2: PRICE AS A RATE 1. $6.25 2. $.625, or $.63 3. $5.25 4. $.3125, or $.31 5. a. $2.5 b. $13.75 6. a. Amount (pt) 1 2 3 4 5 6 Cost non-organic ($) $.75 $1.5
More informationLab 5: Forces on Submerged Objects
A. Background Information Lab 5: Forces on Submerged Objects When an object is submerged in a fluid (liquid or gas), it will experience a force due to the pressure exerted by the fluid on the object. The
More informationFinding the Area of a Major League Baseball Field
Undergraduate Journal of Mathematical Modeling: One + Two Volume 5 2012 Fall Issue 1 Article 7 Finding the Area of a Major League Baseball Field Jacob Courchaine University of South Florida Advisors: Arcadii
More informationForce, Motion and Energy Review
NAME Force, Motion and Energy Review 1 In the picture to the right, two teams of students are playing tug-of-war. Each team is pulling in the opposite direction, but both teams are moving in the same direction.
More informationtime v (vertical) time
NT4E-QRT20: PROJECTILE MOTION FOR TWO ROCKS VELOCITY AND ACCELERATION GRAPHS II Two identical rocks are thrown horizontally from a cliff with Rock A having a greater velocity at the instant it is released
More informationFLUID MECHANICS. Fluid Statics BUOYANCY. Fig. Buoyancy CENTER OF BUOYANCY
FLUID MECHANICS Fluid Statics BUOYANCY When a body is either wholly or partially immersed in a fluid, the hydrostatic lift due to the net vertical component of the hydrostatic pressure forces experienced
More informationNAVAL ARCHITECTURE 1. Class Notes
NAVAL ARCHITECTURE 1 Class Notes d G G 1 W tonnes d G 1 G w tonnes d G 1 G w tonnes Omar bin Yaakob Chapter 1 Introduction Naval Architecture Notes Introduction To carry out various activities at sea,
More informationProfessor Rachel Levy Mathematics Department Harvey Mudd College
Professor Rachel Levy Mathematics Department Harvey Mudd College Goals: bring slope to life and provide a window into current applied mathematics research Participants: Faculty wanting to create outreach
More informationCOURSE NUMBER: ME 321 Fluid Mechanics I Fluid statics. Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET
COURSE NUMBER: ME 321 Fluid Mechanics I Fluid statics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Fluid statics Fluid statics is the study of fluids in
More informationPITCHING EQUILIBRIUM, WING SPAN AND TAIL SPAN IN A GLIDING HARRIS' HAWK, PARABUTEO
/. exp. Biol. 165, 21-41 (1992) 21 Printed in Great Britain The Company of Biologists Limited 1992 PITCHING EQUILIBRIUM, WING SPAN AND TAIL SPAN IN A GLIDING HARRIS' HAWK, PARABUTEO UNICINCTUS BY VANCE
More informationVapour pressure of liquids SURFACE TENSION
Vapour pressure of liquids A liquid in a closed container is subjected to partial vapour pressure due to the escaping molecules from the surface; it reaches a stage of equilibrium when this pressure reaches
More informationYear 10 Term 2 Homework
Yimin Math Centre Year 10 Term 2 Homework Student Name: Grade: Date: Score: Table of contents 6 Year 10 Term 2 Week 6 Homework 1 6.1 Data analysis and evaluation............................... 1 6.1.1
More informationAge of Fans
Measures of Central Tendency SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Interactive Word Wall, Marking the Text, Summarize/Paraphrase/Retell, Think/Pair/Share Matthew is a student reporter
More informationHydrodynamics for Ocean Engineers Prof. A.H. Techet
13.012 Hydrodynamics for Ocean Engineers Prof. A.H. Techet Archimedes s Principle and tatic tability I. Archimedes s Principle: The force on a body due to pressure alone (in the absence of iscous forces)
More informationThe Pythagorean Theorem Diamond in the Rough
The Pythagorean Theorem SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Visualization, Interactive Word Wall Cameron is a catcher trying out for the school baseball team. He
More information2013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members:
013 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas and
More information2011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above)
011 Excellence in Mathematics Contest Team Project Level I (Precalculus and above) School Name: Solutions Group Members: Reference Sheet Formulas and Facts You may need to use some of the following formulas
More informationV mca (and the conditions that affect it)
V mca (and the conditions that affect it) V mca, the minimum airspeed at which an airborne multiengine airplane is controllable with an inoperative engine under a standard set of conditions, is arguably
More information11.4 Apply the Pythagorean
11.4 Apply the Pythagorean Theorem and its Converse Goal p and its converse. Your Notes VOCABULARY Hypotenuse Legs of a right triangle Pythagorean theorem THE PYTHAGOREAN THEOREM Words If a triangle is
More informationDo not turn this page until you are asked to.
YEAR 7 MATHEMATICS EXAMINATION SEMESTER 2 2016 QUESTION AND ANSWER BOOKLET STUDENT NAME: TEACHER: DATE: TIME ALLOWED FOR THIS PAPER Reading time before commencing work: 10 minutes Working time for this
More informationCHAPTER 6 PROJECTILE MOTION
CHAPTER 6 PROJECTILE MOTION 1 Basic principle of analyzing projecting motion Independency of vertical and horizontal motion 2 A simple case: Horizontally projected motion An angry bird is fired horizontally
More informationForces that govern a baseball s flight path
Forces that govern a baseball s flight path Andrew W. Nowicki Physics Department, The College of Wooster, Wooster, Ohio 44691 April 21, 1999 The three major forces that affect the baseball while in flight
More informationA Study of Olympic Winning Times
Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Updated: 05/15/ A Study of Olympic Winning Times Objective: Students will graph data, determine a line that models
More information