Chapter 3 HW Solution


 Merilyn Kennedy
 1 years ago
 Views:
Transcription
1 Chapter 3 HW Solution Problem 3.6: I placed an xy coordinate system at a convenient point (origin doesn t really matter). y 173 x The positions of both planes are given by r B = v B ti + 173j mi (1) r A = 1i + v A t(.77i +.77j) mi (2) The distance between A and B is the length (magnitude) of vector r AB, which is the vector from B to A (or the reverse; distance is the same). So we have r BA = r B r A (3) However, we re really interested in the magnitude of the vector r BA, denote that magnitude by length l: l(t) = r BA = (74t 1)i + ( 276t + 173)j mi (4) The magnitude (Euclidean norm) of a vector is simply the square root of the sum of the components squared, so l(t) = (74t 1) 2 + ( 276t + 173) 2 mi (5) Parts (a) and (b) of the problem should have been reversed: first you have to find the time, then you can find the distance. This is just a standard function minimization problem; e.g. take the first derivative and set it equal to zero. (b) The differentiation is simpler if you realize that when l(t) is at a minimum, so is [l(t)] 2. This removes the square root, and d [l(t)] 2 = d [ 81, 652t 2 11, 296t + 39, 929 ] = 163, 34t 11, 296 = (6) dt dt Solving (13) yields the time when the minimum separation occurs, which is If the planes leave at 6: p.m., then the time of minimum separation is t = 11, 296 =.675 hr = 4.52 min (7) 163, 34 t min = 6 : 4 : 31 p.m. (8) (a) To find the actual separation distance, substitute t min (expressed in fractional hours of (14)) into equation (12). The result I found was l min = mi (9) Although not required, I couldn t resist plotting separation distance vs time (MATLAB plot on next page); it seems to agree with my result. 1
2 Separation distance (mi) Time (min) (c) Although not required, you can also do this problem with ADAMS, you have to set up two bodies with the velocities of the planes, and a PointtoPoint measure for the distance between the two planes. You get the following plot: The minimum of the ADAMS plot occurs at t = 4.6 minutes, which is pretty close to the previous result. ADAMS separation distance is miles; again pretty close. The Problem 3.8: Do this analytically. Velocity of A along this line Velocity of B is known Points A and B are both on link 3, so they re related by the 2 pts on a body equation: v A = v B + ω 3 r BA (1) Velocity v B is known (along lower plane), the direction of velocity v A is known (along upper plane), and the angular velocity ω 3 is in the k direction (perpendicular to the plane). From the angles given, the upper plane is at angle of 2
3 15 from the horizontal. From inspection, block A is moving to the left, so we have v A ( cos 15 i sin 15 j) = 4i + ω 3 k.4( cos 3 i + sin 3 j) (11) Separating the i and j equations, there are i :.9659v a = 4.2ω 3 (12) j :.2588v a =.3463ω 3 (13) In matrix form, equations (12) (13) are Solving, we get [ ] [ ] va = ω 3 [ ] [ ] va m/s = ω rad/s [ ] 4 (14) (15) In terms of vectors, we have v A = v A ( cos 15 i sin 15 j), so v A = i j m/s ω 3 = k rad/s (16) (17) So link 3 is rotating CCW, which I think agrees with the sketch. And block A is sliding a little faster than block B (49 m/s compared with 4 m/s). Problem 3.9 In the 4bar mechanism shown below, link 2 is driven at a constant angular velocity of ω 2 = 45 rad/s CCW. We want to find the angular velocities ω 3 and ω You will need the angles I found above in the analysis. You are to do this problem both analytically and using ADAMS. (a) Analytical Solution. This can be done using only 2 point on a body throughout. Start by finding the velocity of A: Next relate the velocities of A and B: v A = v O2 +ω 2 r O2A = 45k ( 2i j) = 155.9i 9j in/s (18) }{{} = v B = v A + ω 3 r AB (19) 3
4 where ω 3 = ω 3 k rad/s and r AB = 6.78i j in. Substituting for v A and evaluating, we get Now relate the velocities of B and O 4 : v B = ( ω 3 )i + ( ω 3 )j in/s (2) where ω 4 = ω 4 k rad/s and r BA = 5.22i j in. Evaluting this, we get Equate (2) and (22) to obtain v B = v O4 +ω 4 r BO4 (21) }{{} = v B = 1.81ω 4 i 5.22ω 4 j in/s (22) i : ω 3 = 1.81ω 4 (23) j : ω 3 = 5.22ω 4 (24) I like to express these in matrix form: I solved these with MATLAB to yield [ ] [ ] ω3 = ω 4 ω 3 = 1.42k rad/s ω 4 = 15.39k rad/s [ ] (25) (26) (27) So both angular velocities are CCW, and link 4 is much faster than link 3. I guess that looks okay. (b) ADAMSSolution. An ADAMS screenshot of the mechanism (at θ 2 = 12 ) is shown below. The velocity plot is shown on the next page. 4
5 Here s the velocity plot, with lines drawn at 12. The results at the angle agree with the analytical. 2. ADAMS Analysis of Problem 3.9 Angular Velocity of Links 3 & Angular Velocity (rad/sec) Link 3 Angular Velocity (rad/s) Link 4 Angular Velocity (rad/s) Angle (deg) Problem 3.11 (ADAMS Only). A screenshot of my linkage in the initial position is shown below: 5
6 The velocity plot for this problem is shown below. Note that the velocity of point C is quite large near the limits of motion (typical). 8 Velocity of Point C and Angular Velocity of Link Velocity (ft/sec) Velocity of C (Xcomponent) Velocity of C (Y component) Omega 3 (rad/sec) Angular Velocity (rad/sec) Angle (deg) Problem 3.15: A position analysis using the loop closure equation shows that r AO4 = mm (28) Angle of AB with horizontal = , (29) and both these values will be needed. The figure is shown below, with those numerical values mm n n (a) Analytical Velocity: For the analytical velocity analysis, you ll need to use both the 2 points on a body and the one point moving on a body equations. Find the velocity of A using points O 2 (stationary) and A and the two points on a body relationship: So the velocity of A is known. v A = ω 2 r O2A = 225i 3897j mm/s (3) 6
7 Next find the velocity of A again, but now you relate links 3 and 4. You know the path of A relative to body 4. For this situation use the one point moving on a body equation, with A as the point, and 4 as the body, therefore written as follows: v A = v A4 + 4 v A (31) Consider equation (31) very carefully!! Point A 4 is point A in the figure. However...point A 4 is a point that is coincident with A, but FIXED TO BODY 4. You may think of it as a hypothetical extension of body 4 up to point A. The path of A 4 is a circular arc centered at O 4. Therefore, the velocity v A4 is tangential to that circle, and hence perpendicular to AB. So we know the direction of v A4, but not its magnitude. Finally, term 4 v A is the velocity of A relative to body 4. I visualize this by mentally fixing body 4, then examining the motion of A. All right, let s solve the problem. Referring to equation (31), we know velocity v A, it s given in equation (3). Next express v A4 as an unknown magnitude in a known direction. This can either be done using v A4 multiplied by the direction of the velocity, or using ω 4 and the cross product. Since the problem statement asks for the angular velocities of 3 and 4 (they re equal), I ll do that: v A4 = ω 4 r O4A = ω 4 k ( i j) = 37.51ω 4 i ω 4 j (32) Now express 4 v A as an unknown magnitude in a known direction: 4 v A = 4 v A (cos(11.17 i sin(11.17 j) = 4 v A (.9811i.1937j) (33) }{{} along AB Substituting into (31) and separating the i and j components, we get Angular velocities: Solving (34) and (35), we get results So the angular velocity vector of links 3 and 4 is i : 37.5ω v A3/4 = 225 (34) j : ω v A3/4 = 3897 (35) ω 4 = ω 3 = 22 rad/s (36) v A3/4 = 1453 mm/s (37) ω 3 = ω 4 = 22 k rad/s (CCW) (38) Velocity of point B: Knowing ω 3 we can relate the velocity of B to the velocity of A: v B = v A + ω 3 r AB = v A + 22k (392.42i 77.49j) = 545.3i j mm/s =.5453i j m/s = m/s (39) (4) (41) I expressed the last result for v B in polar form; this may be easier to visualize. (b) ADAMS Analysis. The path of Point B is shown at right. The yellow bar across the center is simply the initial position of link 3. What is NOT shown in this plot is the speed (magnitude of velocity) of point B as it moves along the path. In particular, the y velocity is quite large as θ 2 is near zero. Hopefully this will be shown in the velocity plots on the next page. 7
8 The plot of the velocity of point B appears below; the y velocity is large near θ 2 =. 15 Velocity of Point B 1 5 Velocity (m/sec) X Velocity of B (m/s) Y Velocity of B (m/s) Link 2 Angle (deg) At θ 2 = 15 the values for the velocity components are (v B ) x =.546 m/s (v B ) y = m/s (42) (43) which agree quite well with the analytical solution. The plot of ω 3 (same as ω 4 ) is: 25. Angular Velocity of Links 3 & 4 Same for both links. Angular Velocity (rad/sec) Link 2 Angle (deg) At θ 2 = 15 the value of the angular velocity is which also agrees well. ω 3 = ω 4 = rad/s (44) 8
Chapter 5 HW Solution
ME 314 Chapter 5 HW March 6, 1 Chapter 5 HW Solution Problem 5.: The reciprocating flatface follower motion is a rise of in with SHM in 18 of cam rotation, followed by a return with SHM in the remaining
More informationPHYSICS. Chapter 8 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.
PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 8 Lecture RANDALL D. KNIGHT Chapter 8. Dynamics II: Motion in a Plane IN THIS CHAPTER, you will learn to solve problems about motion
More informationChapter 5 Introduction to Trigonometric Functions
Chapter 5 Introduction to Trigonometric Functions 5.1 Angles Section Exercises Verbal 1. Draw an angle in standard position. Label the vertex, initial side, and terminal side. 2. Explain why there are
More informationMOTION IN TWO OR THREE DIMENSIONS
MOTION IN TWO OR THREE DIMENSIONS 3 Sections Covered 3.1 : Position & velocity vectors 3.2 : The acceleration vector 3.3 : Projectile motion 3.4 : Motion in a circle 3.5 : Relative velocity 3.1 Position
More informationPlanar Rigid Body Kinematics Homework
Chapter 2: Planar Rigid ody Kinematics Homework Chapter 2 Planar Rigid ody Kinematics Homework Freeform c 2018 21 Chapter 2: Planar Rigid ody Kinematics Homework 22 Freeform c 2018 Chapter 2: Planar
More informationFORCE TABLE INTRODUCTION
FORCE TABLE INTRODUCTION All measurable quantities can be classified as either a scalar 1 or a vector 2. A scalar has only magnitude while a vector has both magnitude and direction. Examples of scalar
More informationChapter 3 Velocity Analysis
Chapter 3 Velocity nalysis The position of point with respect to 0, Fig 1 may be defined mathematically in either polar or Cartesian form. Two scalar quantities, the length R and the angle θ with respect
More informationChapter 4. Motion in Two Dimensions
Chapter 4 Motion in Two Dimensions Kinematics in Two Dimensions Will study the vector nature of position, velocity and acceleration in greater detail Will treat projectile motion and uniform circular motion
More informationQ 100 OA = 30 AB = 80 BQ = 100 BC = rpm. All dimensions are in mm
rolem 6: A toggle mechanism is shown in figure along with the diagrams of the links in mm. find the velocities of the points B and C and the angular velocities of links AB, BQ and BC. The crank rotates
More informationPhysics 207 Lecture 10. Lecture 10. Employ Newton s Laws in 2D problems with circular motion
Lecture 10 Goals: Employ Newton s Laws in 2D problems with circular motion Assignment: HW5, (Chapters 8 & 9, due 3/4, Wednesday) For Tuesday: Finish reading Chapter 8, start Chapter 9. Physics 207: Lecture
More informationChapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians = 180º Therefore, o 180 π 1 rad =, or π 1º = 180 rad Angle Measure Conversions π 1. To convert degrees to radians,
More informationChapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics
Chapter 1: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational / / 1/ m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv / / 1/ I
More informationDATE: MATH ANALYSIS 2 CHAPTER 12: VECTORS & DETERMINANTS
NAME: PERIOD: DATE: MATH ANALYSIS 2 MR. MELLINA CHAPTER 12: VECTORS & DETERMINANTS Sections: v 12.1 Geometric Representation of Vectors v 12.2 Algebraic Representation of Vectors v 12.3 Vector and Parametric
More informationNormal Force. W = mg cos(θ) Normal force F N = mg cos(θ) F N
Normal Force W = mg cos(θ) Normal force F N = mg cos(θ) Note there is no weight force parallel/down the include. The car is not pressing on anything causing a force in that direction. If there were a person
More informationTIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 1112, Exam 3 Section 1 Version 2 April 23, 2013 Total Weight: 100 points
TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS, Exam 3 Section Version April 3, 03 Total Weight: 00 points. Check your examination for completeness prior to starting. There are a
More informationForces on a banked airplane that travels in uniform circular motion.
Question (60) Forces on a banked airplane that travels in uniform circular motion. A propellerdriven airplane of mass 680 kg is turning in a horizontal circle with a constant speed of 280 km/h. Its bank
More informationRevision Guide for Chapter 15
Revision Guide for Chapter 15 Contents tudent s Checklist Revision otes Transformer... 4 Electromagnetic induction... 4 Generator... 5 Electric motor... 6 Magnetic field... 8 Magnetic flux... 9 Force on
More informationMath 3c Solutions: Exam 2 Fall 2017
Math 3c Solutions: Exam Fall 07. 0 points) The graph of a smooth vectorvalued function is shown below except that your irresponsible teacher forgot to include the orientation!) Several points are indicated
More informationDEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS BADDEST CLASS ON CAMPUS IB PHYSICS OPTION B1A: ROTATIONAL DYNAMICS Essential Idea: The basic laws of mechanics have an extension when equivalent principles are applied to rotation. Actual
More informationExam 1 January 31, 2012
Exam 1 Instructions: You have 60 minutes to complete this exam. This is a closedbook, closednotes exam. You are allowed to use a calculator during the exam. Usage of mobile phones and other electronic
More informationContents. Objectives Circular Motion Velocity and Acceleration Examples Accelerating Frames Polar Coordinates Recap. Contents
Physics 121 for Majors Today s Class You will see how motion in a circle is mathematically similar to motion in a straight line. You will learn that there is a centripetal acceleration (and force) and
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More information1 (20 pts) Nyquist Exercise
EE C128 / ME134 Problem Set 6 Solution Fall 2011 1 (20 pts) Nyquist Exercise Consider a close loop system with unity feedback. For each G(s), hand sketch the Nyquist diagram, determine Z = P N, algebraically
More informationIt is convenient to think that solutions of differential equations consist of a family of functions (just like indefinite integrals ).
Section 1.1 Direction Fields Key Terms/Ideas: Mathematical model Geometric behavior of solutions without solving the model using calculus Graphical description using direction fields Equilibrium solution
More informationChapter 4: Newton s Second Law F = m a. F = m a (4.2)
Lecture 7: Newton s Laws and Their Applications 1 Chapter 4: Newton s Second Law F = m a First Law: The Law of Inertia An object at rest will remain at rest unless, until acted upon by an external force.
More informationChapter 8. Dynamics II: Motion in a Plane
Chapter 8. Dynamics II: Motion in a Plane Chapter Goal: To learn how to solve problems about motion in a plane. Slide 82 Chapter 8 Preview Slide 83 Chapter 8 Preview Slide 84 Chapter 8 Preview Slide
More informationPhysics A  PHY 2048C
Physics A  PHY 2048C Newton s Laws & Equations of 09/27/2017 My Office Hours: Thursday 2:003:00 PM 212 Keen Building Warmup Questions 1 In uniform circular motion (constant speed), what is the direction
More informationThere are two types of multiplication that can be done with vectors: = +.
Section 7.5: The Dot Product Multiplying Two Vectors using the Dot Product There are two types of multiplication that can be done with vectors: Scalar Multiplication Dot Product The Dot Product of two
More informationMATH H53 : MidTerm1
MATH H53 : MidTerm1 22nd September, 215 Name: You have 8 minutes to answer the questions. Use of calculators or study materials including textbooks, notes etc. is not permitted. Answer the questions
More informationProjectile Motion. directions simultaneously. deal with is called projectile motion. ! An object may move in both the x and y
Projectile Motion! An object may move in both the x and y directions simultaneously! The form of twodimensional motion we will deal with is called projectile motion Assumptions of Projectile Motion! The
More informationCourse Updates. Reminders: 1) Quiz today. 2) Written problems: 21.96, 22.6, 22.58, ) Complete Chapter 22 (all this information on web page)
Course Updates http://www.phys.hawaii.edu/~varner/phy272pr10/physics272.html Reminders: 1) Quiz today 2) Written problems: 21.96, 22.6, 22.58, 22.30 3) Complete Chapter 22 (all this information on web
More informationPolynomials; Add/Subtract
Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x
More information3. ANALYTICAL KINEMATICS
In planar mechanisms, kinematic analysis can be performed either analytically or graphically In this course we first discuss analytical kinematic analysis nalytical kinematics is based on projecting the
More information= y(x, t) =A cos (!t + kx)
A harmonic wave propagates horizontally along a taut string of length L = 8.0 m and mass M = 0.23 kg. The vertical displacement of the string along its length is given by y(x, t) = 0. m cos(.5 t + 0.8
More informationChapter 6. Trigonometric Functions of Angles. 6.1 Angle Measure. 1 radians = 180º. π 1. To convert degrees to radians, multiply by.
Chapter 6. Trigonometric Functions of Angles 6.1 Angle Measure Radian Measure 1 radians 180º Therefore, o 180 π 1 rad, or π 1º 180 rad Angle Measure Conversions π 1. To convert degrees to radians, multiply
More information9/4/2017. Motion: Acceleration
Velocity Velocity (m/s) Position Velocity Position 9/4/217 Motion: Acceleration Summary Last : Find your clicker! Scalars: Distance, Speed Vectors: Position velocity Speed = Distance covered/time taken
More informationTIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES. PHYS 1112, Exam 3 Section 1 Version 1 April 23, 2013 Total Weight: 100 points
TIME OF COMPLETION NAME SOLUTION DEPARTMENT OF NATURAL SCIENCES PHYS, Exam 3 Section Version April 3, 03 Total Weight: 00 points. Check your examination for completeness prior to starting. There are a
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES Before starting this section, you might need to review the trigonometric functions. DIFFERENTIATION RULES In particular, it is important to remember that,
More informationFigure 10: Tangent vectors approximating a path.
3 Curvature 3.1 Curvature Now that we re parametrizing curves, it makes sense to wonder how we might measure the extent to which a curve actually curves. That is, how much does our path deviate from being
More information6. Vectors. Given two points, P 0 = (x 0, y 0 ) and P 1 = (x 1, y 1 ), a vector can be drawn with its foot at P 0 and
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationIntroduction  Motivation. Many phenomena (physical, chemical, biological, etc.) are model by differential equations. f f(x + h) f(x) (x) = lim
Introduction  Motivation Many phenomena (physical, chemical, biological, etc.) are model by differential equations. Recall the definition of the derivative of f(x) f f(x + h) f(x) (x) = lim. h 0 h Its
More informationPhysics 8 Wednesday, October 14, 2015
Physics 8 Wednesday, October 14, 2015 HW5 due Friday (problems from Ch9 and Ch10.) Bill/Camilla switch HW sessions this week only (same rooms, same times what changes is which one of us is there): Weds
More informationUNIT V: MultiDimensional Kinematics and Dynamics Page 1
UNIT V: MultiDimensional Kinematics and Dynamics Page 1 UNIT V: MultiDimensional Kinematics and Dynamics As we have already discussed, the study of the rules of nature (a.k.a. Physics) involves both
More information5.3 GRAPHICAL VELOCITY ANALYSIS Instant Center Method
ME GRHL VELOTY NLYSS GRHL VELOTY NLYSS nstant enter Method nstant center of velocities is a simple graphical method for performing velocity analysis on mechanisms The method provides visual understanding
More informationKinematics. Vector solutions. Vectors
Kinematics Study of motion Accelerated vs unaccelerated motion Translational vs Rotational motion Vector solutions required for problems of 2 directional motion Vector solutions Possible solution sets
More informationTorque. Physics 6A. Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB
Physics 6A Torque is what causes angular acceleration (just like a force causes linear acceleration) Torque is what causes angular acceleration (just like a force causes linear acceleration) For a torque
More informationDescription: Using conservation of energy, find the final velocity of a "yo yo" as it unwinds under the influence of gravity.
Chapter 10 [ Edit ] Overview Summary View Diagnostics View Print View with Answers Chapter 10 Due: 11:59pm on Sunday, November 6, 2016 To understand how points are awarded, read the Grading Policy for
More informationConservation of Angular Momentum
Physics 101 Section 3 March 3 rd : Ch. 10 Announcements: Monday s Review Posted (in Plummer s section (4) Today start Ch. 10. Next Quiz will be next week Test# (Ch. 79) will be at 6 PM, March 3, Lockett6
More informationPhysics 101 Lecture 12 Equilibrium and Angular Momentum
Physics 101 Lecture 1 Equilibrium and Angular Momentum Ali ÖVGÜN EMU Physics Department www.aovgun.com Static Equilibrium q Equilibrium and static equilibrium q Static equilibrium conditions n Net external
More informationSlide 1 / 37. Rotational Motion
Slide 1 / 37 Rotational Motion Slide 2 / 37 Angular Quantities An angle θ can be given by: where r is the radius and l is the arc length. This gives θ in radians. There are 360 in a circle or 2π radians.
More informationRigid Object. Chapter 10. Angular Position. Angular Position. A rigid object is one that is nondeformable
Rigid Object Chapter 10 Rotation of a Rigid Object about a Fixed Axis A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant All real objects
More informationForce, Mass, and Acceleration
Introduction Force, Mass, and Acceleration At this point you append you knowledge of the geometry of motion (kinematics) to cover the forces and moments associated with any motion (kinetics). The relations
More information4.1 Implicit Differentiation
4.1 Implicit Differentiation Learning Objectives A student will be able to: Find the derivative of variety of functions by using the technique of implicit differentiation. Consider the equation We want
More informationKinematics in Two Dimensions; 2D Vectors
Kinematics in Two Dimensions; 2D Vectors Addition of Vectors Graphical Methods Below are two example vector additions of 1D displacement vectors. For vectors in one dimension, simple addition and subtraction
More informationEQUATIONS OF MOTION: CYLINDRICAL COORDINATES
Today s Objectives: Students will be able to: 1. Analyze the kinetics of a particle using cylindrical coordinates. EQUATIONS OF MOTION: CYLINDRICAL COORDINATES InClass Activities: Check Homework Reading
More informationFig. 6.1 Plate or disk cam.
CAMS INTRODUCTION A cam is a mechanical device used to transmit motion to a follower by direct contact. The driver is called the cam and the driven member is called the follower. In a cam follower pair,
More informationPrecalculus is the stepping stone for Calculus. It s the final hurdle after all those years of
Chapter 1 Beginning at the Very Beginning: PrePreCalculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope
More informationSection 1.6 Inverse Functions
0 Chapter 1 Section 1.6 Inverse Functions A fashion designer is travelling to Milan for a fashion show. He asks his assistant, Betty, what 7 degrees Fahrenheit is in Celsius, and after a quick search on
More information= constant of gravitation is G = N m 2 kg 2. Your goal is to find the radius of the orbit of a geostationary satellite.
Problem 1 Earth and a Geostationary Satellite (10 points) The earth is spinning about its axis with a period of 3 hours 56 minutes and 4 seconds. The equatorial radius of the earth is 6.38 10 6 m. The
More informationGeneral Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,
More informationUniform Circular Motion
Uniform Circular Motion Introduction Earlier we defined acceleration as being the change in velocity with time: = Until now we have only talked about changes in the magnitude of the acceleration: the speeding
More informationGiven an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r :
Given an arc of length s on a circle of radius r, the radian measure of the central angle subtended by the arc is given by θ = s r : To convert from radians (rad) to degrees ( ) and vice versa, use the
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
alculus III Preface Here are my online notes for my alculus III course that I teach here at Lamar University. espite the fact that these are my class notes, they should be accessible to anyone wanting
More informationCircular motion. Aug. 22, 2017
Circular motion Aug. 22, 2017 Until now, we have been observers to Newtonian physics through inertial reference frames. From our discussion of Newton s laws, these are frames which obey Newton s first
More informationMoving Reference Frame Kinematics Homework
Chapter 3 Moving Reference Frame Kinematics Homework Freeform c 2018 31 32 Freeform c 2018 Homework H.3. Given: The disk shown is rotating about its center with a constant rotation rate of Ω. Four slots
More informationSpacetime Diagrams Lab Exercise
Spacetime Diagrams Lab Exercise The spacetime diagram (also known as a Minkowski diagram) is a tool that can used to graphically describe complex problems in special relativity. In many cases, with a properly
More informationWhat path do the longest sparks take after they leave the wand? Today we ll be doing one more new concept before the test on Wednesday.
What path do the longest sparks take after they leave the wand? Today we ll be doing one more new concept before the test on Wednesday. Centripetal Acceleration and Newtonian Gravitation Reminders: 15
More informationExample 2.1. Draw the points with polar coordinates: (i) (3, π) (ii) (2, π/4) (iii) (6, 2π/4) We illustrate all on the following graph:
Section 10.3: Polar Coordinates The polar coordinate system is another way to coordinatize the Cartesian plane. It is particularly useful when examining regions which are circular. 1. Cartesian Coordinates
More information30. TRANSFORMING TOOL #1 (the Addition Property of Equality)
30 TRANSFORMING TOOL #1 (the Addition Property of Equality) sentences that look different, but always have the same truth values What can you DO to a sentence that will make it LOOK different, but not
More informationq = tan 1 (R y /R x )
Vector Addition Using Vector Components = + R x = A x + B x B y R y = A y + B y R = (R x 2 + R y 2 ) 1/2 B x q = tan 1 (R y /R x ) Example 1.7: Vector has a magnitude of 50 cm and direction of 30º, and
More information2.5 The Fundamental Theorem of Algebra.
2.5. THE FUNDAMENTAL THEOREM OF ALGEBRA. 79 2.5 The Fundamental Theorem of Algebra. We ve seen formulas for the (complex) roots of quadratic, cubic and quartic polynomials. It is then reasonable to ask:
More informationNotes on multivariable calculus
Notes on multivariable calculus Jonathan Wise February 2, 2010 1 Review of trigonometry Trigonometry is essentially the study of the relationship between polar coordinates and Cartesian coordinates in
More information8. TRANSFORMING TOOL #1 (the Addition Property of Equality)
8 TRANSFORMING TOOL #1 (the Addition Property of Equality) sentences that look different, but always have the same truth values What can you DO to a sentence that will make it LOOK different, but not change
More informationSequences & Functions
Ch. 5 Sec. 1 Sequences & Functions Skip Counting to Arithmetic Sequences When you skipped counted as a child, you were introduced to arithmetic sequences. Example 1: 2, 4, 6, 8, adding 2 Example 2: 10,
More informationMTH 277 Test 4 review sheet Chapter , 14.7, 14.8 Chalmeta
MTH 77 Test 4 review sheet Chapter 13.113.4, 14.7, 14.8 Chalmeta Multiple Choice 1. Let r(t) = 3 sin t i + 3 cos t j + αt k. What value of α gives an arc length of 5 from t = 0 to t = 1? (a) 6 (b) 5 (c)
More informationChapter 9: Rotational Dynamics Tuesday, September 17, 2013
Chapter 9: Rotational Dynamics Tuesday, September 17, 2013 10:00 PM The fundamental idea of Newtonian dynamics is that "things happen for a reason;" to be more specific, there is no need to explain rest
More informationa Write down the coordinates of the point on the curve where t = 2. b Find the value of t at the point on the curve with coordinates ( 5 4, 8).
Worksheet A 1 A curve is given by the parametric equations x = t + 1, y = 4 t. a Write down the coordinates of the point on the curve where t =. b Find the value of t at the point on the curve with coordinates
More information100 (s + 10) (s + 100) e 0.5s. s 100 (s + 10) (s + 100). G(s) =
1 AME 3315; Spring 215; Midterm 2 Review (not graded) Problems: 9.3 9.8 9.9 9.12 except parts 5 and 6. 9.13 except parts 4 and 5 9.28 9.34 You are given the transfer function: G(s) = 1) Plot the bode plot
More informationAlex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1
Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 What is a linear equation? It sounds fancy, but linear equation means the same thing as a line. In other words, it s an equation
More informationCALCULATING MAGNETIC FIELDS & THE BIOTSAVART LAW. Purdue University Physics 241 Lecture 15 Brendan Sullivan
CALCULATING MAGNETIC FIELDS & THE BIOTSAVAT LAW Purdue University Physics 41 Lecture 15 Brendan Sullivan Introduction Brendan Sullivan, PHYS89, sullivb@purdue.edu Office Hours: By Appointment Just stop
More information8.012 Physics I: Classical Mechanics Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE
More informationMultiple Choice  TEST III
Multiple Choice Test IIIClassical Mechanics Multiple Choice  TEST III 1) n atomic particle whose mass is 210 atomic mass units collides with a stationary atomic particle B whose mass is 12 atomic mass
More informationChapter 3 Motion in two or three dimensions
Chapter 3 Motion in two or three dimensions Lecture by Dr. Hebin Li Announcements As requested by the Disability Resource Center: In this class there is a student who is a client of Disability Resource
More informationMathematics for Graphics and Vision
Mathematics for Graphics and Vision Steven Mills March 3, 06 Contents Introduction 5 Scalars 6. Visualising Scalars........................ 6. Operations on Scalars...................... 6.3 A Note on
More informationShenandoah University. (PowerPoint) LESSON PLAN *
Shenandoah University (PowerPoint) LESSON PLAN * NAME DATE 10/28/04 TIME REQUIRED 90 minutes SUBJECT Algebra I GRADE 69 OBJECTIVES AND PURPOSE (for each objective, show connection to SOL for your subject
More informationCalculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions.
All exercises are from Section 4.7 of the textbook. 1. Calculus with Analytic Geometry I Exam 10, Take Home Friday, November 8, 2013 Solutions. 2. Solution. The picture suggests using the angle θ as variable;
More informationSolutions for the Practice Final  Math 23B, 2016
olutions for the Practice Final  Math B, 6 a. True. The area of a surface is given by the expression d, and since we have a parametrization φ x, y x, y, f x, y with φ, this expands as d T x T y da xy
More informationMEI STRUCTURED MATHEMATICS 4763
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MEI STRUCTURED MATHEMATICS 76 Mechanics Tuesday JANUARY 6 Afternoon
More informationCalculus: Preparation Problem Solutions
Calculus: Preparation Problem Solutions 1. If f(t) = 4e 0.75t, for what t does f(t) = 0.5? Leave your answer in terms of a logarithm and fractions. Solution The equation implied by this question is and
More informationAssignment 7 Solutions
Assignment 7 Solutions PY 106 1. A singleturn rectangular wire loop measures 6.00 cm wide by 10.0 cm long. The loop carries a current of 5.00 A. The loop is in a uniform magnetic field with B = 5.00 10
More informationThe Plane of Complex Numbers
The Plane of Complex Numbers In this chapter we ll introduce the complex numbers as a plane of numbers. Each complex number will be identified by a number on a real axis and a number on an imaginary axis.
More informationTest, Lesson 7 Waves  Answer Key Page 1
Test, Lesson 7 Waves  Answer Key Page 1 1. Match the proper units with the following: W. wavelength 1. nm F. frequency 2. /sec V. velocity 3. m 4. ms 1 5. Hz 6. m/sec (A) W: 1, 3 F: 2, 4, 5 V: 6 (B)
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University Abstract This handout defines the trigonometric function of angles and discusses the relationship between trigonometric
More informationVectors and Coordinate Systems
Vectors and Coordinate Systems In Newtonian mechanics, we want to understand how material bodies interact with each other and how this affects their motion through space. In order to be able to make quantitative
More informationTranslational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work
Translational vs Rotational / / 1/ Δ m x v dx dt a dv dt F ma p mv KE mv Work Fd / / 1/ θ ω θ α ω τ α ω ω τθ Δ I d dt d dt I L I KE I Work / θ ω α τ Δ Δ c t s r v r a v r a r Fr L pr Connection Translational
More informationAnnouncements Oct 16, 2014
Announcements Oct 16, 2014 1. Prayer 2. While waiting, see how many of these blanks you can fill out: Centripetal Accel.: Causes change in It points but not Magnitude: a c = How to use with N2: Always
More informationCircular Motion Dynamics
Circular Motion Dynamics 8.01 W04D2 Today s Reading Assignment: MIT 8.01 Course Notes Chapter 9 Circular Motion Dynamics Sections 9.19.2 Announcements Problem Set 3 due Week 5 Tuesday at 9 pm in box outside
More informationMAT1193 1f. Linear functions (most closely related to section 1.4) But for now, we introduce the most important equation in this class:
MAT1193 1f. Linear functions (most closely related to section 1.4) Linear functions are some of the simplest functions we ll consider. They have special properties and play an important role in many areas
More informationMTH 133: Plane Trigonometry
MTH 133: Plane Trigonometry Radian Measure, Arc Length, and Area Angular and Linear Velocity Thomas W. Judson Department of Mathematics & Statistics Stephen F. Austin State University Fall 2017 Plane Trigonometry
More informationECE382/ME482 Spring 2005 Homework 7 Solution April 17, K(s + 0.2) s 2 (s + 2)(s + 5) G(s) =
ECE382/ME482 Spring 25 Homework 7 Solution April 17, 25 1 Solution to HW7 AP9.5 We are given a system with open loop transfer function G(s) = K(s +.2) s 2 (s + 2)(s + 5) (1) and unity negative feedback.
More information