# Chapter 5: Distributed Forces; Centroids and Centers of Gravity

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 CE297-FA09-Ch5 Page 1 Wednesday, October 07, :39 PM Chapter 5: Distributed Forces; Centroids and Centers of Gravity What are distributed forces? Forces that act on a body per unit length, area or volume. They are not discrete forces that act at specific points. Rather they act over a continuous region. Examples: 5.2 Center of Gravity Gravity pulls each and every particle of a body vertically downwards. What is the location of the equivalent single force that replaces all the distributed forces. z y z y x x Let the location be z y x

2 CE297-FA09-Ch5 Page 2 Tuesday, October 13, :07 PM Example Find the Center of Gravity of the area shown below.

3 CE297-FA09-Ch5 Page 3 Wednesday, October 14, :21 PM Example: Centroid of a Quarter or Semi Ellipse.

4 CE297-FA09-Ch5 Page 4 Friday, October 16, :38 AM Centroids of Lines z y z y x x Exercise 5.45 Find the Centroid of the wire shown.

5 CE297-FA09-Ch5 Page 5

6 CE297-FA09-Ch5 Page 6 Wednesday, October 14, :13 PM Centroids and First Moments of Areas & Lines. Definition: First Moment of an Area Definition: First Moment of a Line Properties of Symmetry An area is symmetric with respect to an axis BB if for every point P there exists a point P such that PP is perpendicular to BB and is divided into two equal parts by BB. The first moment of an area with respect to a line of symmetry is zero. If an area possesses a line of symmetry, its centroid lies on that axis If an area possesses two lines of symmetry, its centroid lies at their intersection. An area is symmetric with respect to a center O if for every element da at (x,y) there exists an area da of equal area at (-x,-y). The centroid of the area coincides with the center of symmetry. NOTE: Centroid of any area always exists. But, a center of symmetry may or may not exist.

7 CE297-FA09-Ch5 Page 7 Friday, October 16, :53 AM 5.5 Composite Areas and Lines The Centroid of an area (or line) that is made up of several simple shapes can be found easily using the centroids of the individual shapes. Note: If an area is composed by adding some shapes and subtracting other shapes, then the moments of the subtracted shapes need to be subtracted as well.

8 CE297-FA09-Ch5 Page 8 Friday, October 16, :04 AM Exercise 5.7 Find the centroid of the figure shown. Find the reactions at A & B. (specific weight γ = 0.28 lb/in 3 ; thickness t = 1 in) Exercise 5.28 A uniform circular rod of weight 8 lb and radius r=10 in is shown. Determine the tension in the cable AB & the reaction at C.

9 CE297-FA09-Ch5 Page 9 Sunday, October 18, :55 PM 5.7 Surfaces & Volumes of Revolution: Theorems of Pappus-Guldinus Surfaces of revolution are obtained when one "sweeps" a 2-D curve about a fixed axis. Theorem 1 Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation. Surface areas of revolution Rotating about y-axis: Rotating about x-axis: General 3D surfaces (aside) The concepts of area, centers of areas, and Moments of areas can also be extended to general 3D surfaces. The same integral formulas still hold:

10 CE297-FA09-Ch5 Page 10 Sunday, October 18, :00 PM Theorem 2 Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation. Volumes of Revolution Rotating about y-axis: Rotating about x-axis: Exercise 5.59 Find the internal surface area and the volume of the punch bowl. Given R= 250 mm.

11 CE297-FA09-Ch5 Page 11 Wednesday, October 21, :33 AM 5.8 Distributed Loads on Beams In several applications, engineers have to design beams that carry distributed loads along their length. w(x) is weight per unit length. Simply supported beam. Cantilever Beam Total weight: Point of action: Exercise 5.70 Find the reactions at the supports.

12 CE297-FA09-Ch5 Page 12 Friday, October 23, :51 AM 5.9 Distributed forces on submerged surfaces. Objects that are submerged in water (or in any liquid) are subjected to distributed force per unit area which is called pressure. In water, this pressure always acts perpendicular (normal) to the submerged surface and its magnitude is given by: Note: The buoyancy force is the resultant of all these distributed forces acting on the body. Recall the buoyancy force is equal to the weight of the water displaced. Aside: If the liquid is viscous, then in addition the normal pressure the viscous fluid may also apply a tangential traction to the body. This traction is also a force per unit area and is a more general form of pressure. Resultant force To obtain the resultant force acting on a submerged surface: For inclined surfaces: Exercise m x 4m wall of the tank is hinged at A and held by rod BC. Find the tension in the rod as a function of the water depth d.

13 CE297-FA09-Ch5 Page 13 Friday, October 23, :42 AM For Curved Surfaces: Forces on curved submerged surfaces can he obtained by using equilibrium of a surrounding portion of water. Example 5.10

14 CE297-FA09-Ch5 Page 14

15 CE297-FA09-Ch5 Page 15 Friday, October 23, :15 AM 5.10 Center of Gravity in 3D space; Center of volume The formulas for center of gravity in 2 D can be easily generalized to 3D as follows: 5.11 Composition of Volumes Examples 5.11 and 5.12 in the book.

16 CE297-FA09-Ch5 Page 16 Monday, October 26, :04 AM 5.12 Center of Volume by integration. For complex 3D shapes, triple integrals can be difficult to evaluate exactly. For some special cases one can find the centroid as follows: (i) Bodies of revolution (ii) Volume under a surface Read Example 5.13 Exercise Find the centroid of the volume obtained by rotating the shaded area about the x-axis.

### Copyright 2011 Casa Software Ltd. www.casaxps.com. Centre of Mass

Centre of Mass A central theme in mathematical modelling is that of reducing complex problems to simpler, and hopefully, equivalent problems for which mathematical analysis is possible. The concept of

### m i: is the mass of each particle

Center of Mass (CM): The center of mass is a point which locates the resultant mass of a system of particles or body. It can be within the object (like a human standing straight) or outside the object

### Experiment (2): Metacentric height of floating bodies

Experiment (2): Metacentric height of floating bodies Introduction: The Stability of any vessel which is to float on water, such as a pontoon or ship, is of paramount importance. The theory behind the

### 8 Buoyancy and Stability

Jianming Yang Fall 2012 16 8 Buoyancy and Stability 8.1 Archimedes Principle = fluid weight above 2 ABC fluid weight above 1 ADC = weight of fluid equivalent to body volume In general, ( = displaced fluid

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### 4.2 Free Body Diagrams

CE297-FA09-Ch4 Page 1 Friday, September 18, 2009 12:11 AM Chapter 4: Equilibrium of Rigid Bodies A (rigid) body is said to in equilibrium if the vector sum of ALL forces and all their moments taken about

### 4 Shear Forces and Bending Moments

4 Shear Forces and ending oments Shear Forces and ending oments 8 lb 16 lb roblem 4.3-1 alculate the shear force and bending moment at a cross section just to the left of the 16-lb load acting on the simple

### Section 16: Neutral Axis and Parallel Axis Theorem 16-1

Section 16: Neutral Axis and Parallel Axis Theorem 16-1 Geometry of deformation We will consider the deformation of an ideal, isotropic prismatic beam the cross section is symmetric about y-axis All parts

### EXPERIMENT (2) BUOYANCY & FLOTATION (METACENTRIC HEIGHT)

EXPERIMENT (2) BUOYANCY & FLOTATION (METACENTRIC HEIGHT) 1 By: Eng. Motasem M. Abushaban. Eng. Fedaa M. Fayyad. ARCHIMEDES PRINCIPLE Archimedes Principle states that the buoyant force has a magnitude equal

### This function is symmetric with respect to the y-axis, so I will let - /2 /2 and multiply the area by 2.

INTEGRATION IN POLAR COORDINATES One of the main reasons why we study polar coordinates is to help us to find the area of a region that cannot easily be integrated in terms of x. In this set of notes,

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding

### Announcements. Moment of a Force

Announcements Test observations Units Significant figures Position vectors Moment of a Force Today s Objectives Understand and define Moment Determine moments of a force in 2-D and 3-D cases Moment of

### CENTER OF GRAVITY, CENTER OF MASS AND CENTROID OF A BODY

CENTER OF GRAVITY, CENTER OF MASS AND CENTROID OF A BODY Dr. Amilcar Rincon-Charris, MSME Mechanical Engineering Department MECN 3005 - STATICS Objective : Students will: a) Understand the concepts of

### Stresses in Beam (Basic Topics)

Chapter 5 Stresses in Beam (Basic Topics) 5.1 Introduction Beam : loads acting transversely to the longitudinal axis the loads create shear forces and bending moments, stresses and strains due to V and

### Hydrostatic Force on a Submerged Surface

Experiment 3 Hydrostatic Force on a Submerged Surface Purpose The purpose of this experiment is to experimentally locate the center of pressure of a vertical, submerged, plane surface. The experimental

### General tests Algebra

General tests Algebra Question () : Choose the correct answer : - If = then = a)0 b) 6 c)5 d)4 - The shape which represents Y is a function of is : - - V A B C D o - - y - - V o - - - V o - - - V o - -if

### F N A) 330 N 0.31 B) 310 N 0.33 C) 250 N 0.27 D) 290 N 0.30 E) 370 N 0.26

Physics 23 Exam 2 Spring 2010 Dr. Alward Page 1 1. A 250-N force is directed horizontally as shown to push a 29-kg box up an inclined plane at a constant speed. Determine the magnitude of the normal force,

CHAPTER 2.0 ANSWER 1. A tank is filled with seawater to a depth of 12 ft. If the specific gravity of seawater is 1.03 and the atmospheric pressure at this location is 14.8 psi, the absolute pressure (psi)

### Introduction to Beam. Area Moments of Inertia, Deflection, and Volumes of Beams

Introduction to Beam Theory Area Moments of Inertia, Deflection, and Volumes of Beams Horizontal structural member used to support horizontal loads such as floors, roofs, and decks. Types of beam loads

### Structural Axial, Shear and Bending Moments

Structural Axial, Shear and Bending Moments Positive Internal Forces Acting Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants

### AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 2001 APPLIED MATHEMATICS HIGHER LEVEL

M3 AN ROINN OIDEACHAIS AGUS EOLAÍOCHTA LEAVING CERTIFICATE EXAMINATION, 00 APPLIED MATHEMATICS HIGHER LEVEL FRIDAY, JUNE AFTERNOON,.00 to 4.30 Six questions to be answered. All questions carry equal marks.

### γ [Increase from (1) to (2)] γ (1ft) [Decrease from (2) to (B)]

1. The manometer fluid in the manometer of igure has a specific gravity of 3.46. Pipes A and B both contain water. If the pressure in pipe A is decreased by 1.3 psi and the pressure in pipe B increases

### Name Calculus AP Chapter 7 Outline M. C.

Name Calculus AP Chapter 7 Outline M. C. A. AREA UNDER A CURVE: a. If y = f (x) is continuous and non-negative on [a, b], then the area under the curve of f from a to b is: A = f (x) dx b. If y = f (x)

### Fall 12 PHY 122 Homework Solutions #8

Fall 12 PHY 122 Homework Solutions #8 Chapter 27 Problem 22 An electron moves with velocity v= (7.0i - 6.0j)10 4 m/s in a magnetic field B= (-0.80i + 0.60j)T. Determine the magnitude and direction of the

### Stress and Deformation Analysis. Representing Stresses on a Stress Element. Representing Stresses on a Stress Element con t

Stress and Deformation Analysis Material in this lecture was taken from chapter 3 of Representing Stresses on a Stress Element One main goals of stress analysis is to determine the point within a load-carrying

### CH-205: Fluid Dynamics

CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Solutions of Mid Semester Examination Data Given: Density of water, ρ = 1000 kg/m 3, gravitational acceleration, g

### Section 2.1 Rectangular Coordinate Systems

P a g e 1 Section 2.1 Rectangular Coordinate Systems 1. Pythagorean Theorem In a right triangle, the lengths of the sides are related by the equation where a and b are the lengths of the legs and c is

### Hydrostatic Force on a Curved Surfaces

Hydrostatic Force on a Curved Surfaces Henryk Kudela 1 Hydrostatic Force on a Curved Surface On a curved surface the forces pδa on individual elements differ in direction, so a simple summation of them

### Chapter 11 Equilibrium

11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### Chapter 9 Deflections of Beams

Chapter 9 Deflections of Beams 9.1 Introduction in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along

### FLUID FORCES ON CURVED SURFACES; BUOYANCY

FLUID FORCES ON CURVED SURFCES; BUOYNCY The principles applicable to analysis of pressure-induced forces on planar surfaces are directly applicable to curved surfaces. s before, the total force on the

### Physics 210 Q ( PHYSICS210BRIDGE ) My Courses Course Settings

1 of 16 9/7/2012 1:10 PM Logged in as Julie Alexander, Instructor Help Log Out Physics 210 Q1 2012 ( PHYSICS210BRIDGE ) My Courses Course Settings Course Home Assignments Roster Gradebook Item Library

### WebAssign Lesson 3-3 Applications (Homework)

WebAssign Lesson 3-3 Applications (Homework) Current Score : / 27 Due : Tuesday, July 15 2014 11:00 AM MDT Jaimos Skriletz Math 175, section 31, Summer 2 2014 Instructor: Jaimos Skriletz 1. /3 points A

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL 2 SHEAR FORCE AND BENDING MOMENTS IN BEAMS This is the second tutorial on bending of beams. You should judge your progress by completing the self assessment exercises.

### Physics-1 Recitation-7

Physics-1 Recitation-7 Rotation of a Rigid Object About a Fixed Axis 1. The angular position of a point on a wheel is described by. a) Determine angular position, angular speed, and angular acceleration

### 7.3 Volumes Calculus

7. VOLUMES Just like in the last section where we found the area of one arbitrary rectangular strip and used an integral to add up the areas of an infinite number of infinitely thin rectangles, we are

### Higher Technological Institute Civil Engineering Department. Lectures of. Fluid Mechanics. Dr. Amir M. Mobasher

Higher Technological Institute Civil Engineering Department Lectures of Fluid Mechanics Dr. Amir M. Mobasher 1/14/2013 Fluid Mechanics Dr. Amir Mobasher Department of Civil Engineering Faculty of Engineering

### MOHR'S CIRCLE FOR MOMENT OF INERTIA. Wentbridge Viaduct Yorkshire, UK

MOHR'S CIRCLE FOR MOMENT OF INERTIA Wentbridge Viaduct Yorkshire, UK Asymmetrical Sections and Loads To this point, calculation of moment of inertia has been based upon the section being loaded symmetrically

### Mechanics 2. Revision Notes

Mechanics 2 Revision Notes November 2012 Contents 1 Kinematics 3 Constant acceleration in a vertical plane... 3 Variable acceleration... 5 Using vectors... 6 2 Centres of mass 8 Centre of mass of n particles...

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 1 STRESSES IN BEAMS DUE TO BENDING. On completion of this tutorial you should be able to do the following.

MECHANICS OF SOLIDS - BEAMS TUTOIAL 1 STESSES IN BEAMS DUE TO BENDING This is the first tutorial on bending of beams designed for anyone wishing to study it at a fairly advanced level. You should judge

### physics 111N rotational motion

physics 111N rotational motion rotations of a rigid body! suppose we have a body which rotates about some axis! we can define its orientation at any moment by an angle, θ (any point P will do) θ P physics

### MECE 3400 Summer 2016 Homework #1 Forces and Moments as Vectors

ASSIGNED 1) Knowing that α = 40, determine the resultant of the three forces shown: 2) Two cables, AC and BC, are tied together at C and pulled by a force P, as shown. Knowing that P = 500 N, α = 60, and

### EQUILIBRIUM AND ELASTICITY

Chapter 12: EQUILIBRIUM AND ELASTICITY 1 A net torque applied to a rigid object always tends to produce: A linear acceleration B rotational equilibrium C angular acceleration D rotational inertia E none

### Mechanics of Materials. Chapter 4 Shear and Moment In Beams

Mechanics of Materials Chapter 4 Shear and Moment In Beams 4.1 Introduction The term beam refers to a slender bar that carries transverse loading; that is, the applied force are perpendicular to the bar.

### Solid of Revolution - Finding Volume by Rotation

Solid of Revolution - Finding Volume by Rotation Finding the volume of a solid revolution is a method of calculating the volume of a 3D object formed by a rotated area of a 2D space. Finding the volume

### 1 of 7 4/13/2010 8:05 PM

Chapter 33 Homework Due: 8:00am on Wednesday, April 7, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy [Return to Standard Assignment View] Canceling a Magnetic Field

### MECHANICS OF SOLIDS - BEAMS TUTORIAL 3 THE DEFLECTION OF BEAMS

MECHANICS OF SOLIDS - BEAMS TUTORIAL THE DEECTION OF BEAMS This is the third tutorial on the bending of beams. You should judge your progress by completing the self assessment exercises. On completion

### STRAIGHT LINES. , y 1. tan. and m 2. 1 mm. If we take the acute angle between two lines, then tan θ = = 1. x h x x. x 1. ) (x 2

STRAIGHT LINES Chapter 10 10.1 Overview 10.1.1 Slope of a line If θ is the angle made by a line with positive direction of x-axis in anticlockwise direction, then the value of tan θ is called the slope

### Section 10.4 Vectors

Section 10.4 Vectors A vector is represented by using a ray, or arrow, that starts at an initial point and ends at a terminal point. Your textbook will always use a bold letter to indicate a vector (such

Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### Engineering Math II Spring 2015 Solutions for Class Activity #2

Engineering Math II Spring 15 Solutions for Class Activity # Problem 1. Find the area of the region bounded by the parabola y = x, the tangent line to this parabola at 1, 1), and the x-axis. Then find

### Free Response Questions Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom

Free Response Questions 1969-005 Compiled by Kaye Autrey for face-to-face student instruction in the AP Calculus classroom 1 AP Calculus Free-Response Questions 1969 AB 1 Consider the following functions

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Mechanics of Materials. Chapter 6 Deflection of Beams

Mechanics of Materials Chapter 6 Deflection of Beams 6.1 Introduction Because the design of beams is frequently governed by rigidity rather than strength. For example, building codes specify limits on

### Section 6.4. Lecture 23. Section 6.4 The Centroid of a Region; Pappus Theorem on Volumes. Jiwen He. Department of Mathematics, University of Houston

Section 6.4 Lecture 23 Section 6.4 The Centroid of a Region; Pappus Theorem on Volumes Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431 Jiwen

### Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering. Part A

Homework #8 203-1-1721 Physics 2 for Students of Mechanical Engineering Part A 1. Four particles follow the paths shown in Fig. 32-33 below as they pass through the magnetic field there. What can one conclude

### Engineering Mechanics I. Phongsaen PITAKWATCHARA

2103-213 Engineering Mechanics I Phongsaen.P@chula.ac.th May 13, 2011 Contents Preface xiv 1 Introduction to Statics 1 1.1 Basic Concepts............................ 2 1.2 Scalars and Vectors..........................

### Handout 7: Magnetic force. Magnetic force on moving charge

1 Handout 7: Magnetic force Magnetic force on moving charge A particle of charge q moving at velocity v in magnetic field B experiences a magnetic force F = qv B. The direction of the magnetic force is

### THE MAGNETIC FIELD. 9. Magnetism 1

THE MAGNETIC FIELD Magnets always have two poles: north and south Opposite poles attract, like poles repel If a bar magnet is suspended from a string so that it is free to rotate in the horizontal plane,

### CET Moving Charges & Magnetism

CET 2014 Moving Charges & Magnetism 1. When a charged particle moves perpendicular to the direction of uniform magnetic field its a) energy changes. b) momentum changes. c) both energy and momentum

### Buoyancy Problem Set

Buoyancy Problem Set 1) A stone weighs 105 lb in air. When submerged in water, it weighs 67.0 lb. Find the volume and specific gravity of the stone. (Specific gravity of an object: ratio object density

### Applications of the Integral

Chapter 6 Applications of the Integral Evaluating integrals can be tedious and difficult. Mathematica makes this work relatively easy. For example, when computing the area of a region the corresponding

### EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics of a rigid body undergoing rotational motion. In-Class Activities: Applications

### Applications of Integration Day 1

Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under

### PROBLEM SET. Practice Problems for Exam #1. Math 1352, Fall 2004. Oct. 1, 2004 ANSWERS

PROBLEM SET Practice Problems for Exam # Math 352, Fall 24 Oct., 24 ANSWERS i Problem. vlet R be the region bounded by the curves x = y 2 and y = x. A. Find the volume of the solid generated by revolving

### p atmospheric Statics : Pressure Hydrostatic Pressure: linear change in pressure with depth Measure depth, h, from free surface Pressure Head p gh

IVE1400: n Introduction to Fluid Mechanics Statics : Pressure : Statics r P Sleigh: P..Sleigh@leeds.ac.uk r J Noakes:.J.Noakes@leeds.ac.uk January 008 Module web site: www.efm.leeds.ac.uk/ive/fluidslevel1

### VIII. Magnetic Fields - Worked Examples

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.0 Spring 003 VIII. Magnetic Fields - Worked Examples Example : Rolling rod A rod with a mass m and a radius R is mounted on two parallel rails

### Mechanics Lecture Notes. 1 Notes for lectures 12 and 13: Motion in a circle

Mechanics Lecture Notes Notes for lectures 2 and 3: Motion in a circle. Introduction The important result in this lecture concerns the force required to keep a particle moving on a circular path: if the

### Figure 1- Different parts of experimental apparatus.

Objectives Determination of center of buoyancy Determination of metacentric height Investigation of stability of floating objects Apparatus The unit shown in Fig. 1 consists of a pontoon (1) and a water

### Section 10.2 The Parabola

243 Section 10.2 The Parabola Recall that the graph of f(x) = x 2 is a parabola with vertex of (0, 0) and axis of symmetry of x = 0: x = 0 f(x) = x 2 vertex: (0, 0) In this section, we want to expand our

### 3-D Dynamics of Rigid Bodies

3-D Dynamics of Rigid Bodies Introduction of third dimension :: Third component of vectors representing force, linear velocity, linear acceleration, and linear momentum :: Two additional components for

### 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration

Ch 9 Rotation 9.1 Rotational Kinematics: Angular Velocity and Angular Acceleration Q: What is angular velocity? Angular speed? What symbols are used to denote each? What units are used? Q: What is linear

### Center of Mass/Momentum

Center of Mass/Momentum 1. 2. An L-shaped piece, represented by the shaded area on the figure, is cut from a metal plate of uniform thickness. The point that corresponds to the center of mass of the L-shaped

### Chapter 10: Topics in Analytic Geometry

Chapter 10: Topics in Analytic Geometry 10.1 Parabolas V In blue we see the parabola. It may be defined as the locus of points in the plane that a equidistant from a fixed point (F, the focus) and a fixed

### FLUID MECHANICS. Problem 2: Consider a water at 20 0 C flows between two parallel fixed plates.

FLUID MECHANICS Problem 1: Pressures are sometimes determined by measuring the height of a column of liquid in a vertical tube. What diameter of clean glass tubing is required so that the rise of water

### ( ) where W is work, f(x) is force as a function of distance, and x is distance.

Work by Integration 1. Finding the work required to stretch a spring 2. Finding the work required to wind a wire around a drum 3. Finding the work required to pump liquid from a tank 4. Finding the work

### Solution Derivations for Capa #11

Solution Derivations for Capa #11 1) A horizontal circular platform (M = 128.1 kg, r = 3.11 m) rotates about a frictionless vertical axle. A student (m = 68.3 kg) walks slowly from the rim of the platform

### Chapter 1: Statics. A) Newtonian Mechanics B) Relativistic Mechanics

Chapter 1: Statics 1. The subject of mechanics deals with what happens to a body when is / are applied to it. A) magnetic field B) heat C ) forces D) neutrons E) lasers 2. still remains the basis of most

### WEDNESDAY, 2 MAY 1.30 PM 2.25 PM. 3 Full credit will be given only where the solution contains appropriate working.

C 500/1/01 NATIONAL QUALIFICATIONS 01 WEDNESDAY, MAY 1.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator. Answer as many questions as you can. Full

### Chapter 3. Technical Measurement and Vectors

Chapter 3. Technical Measurement and Vectors Unit Conversions 3-1. soccer field is 100 m long and 60 m across. What are the length and width of the field in feet? 100 cm 1 in. 1 ft (100 m) 328 ft 1 m 2.54

### Physics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd

Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle

### Composite Sections and Steel Beam Design. Composite Design. Steel Beam Selection - ASD Composite Sections Analysis Method

Architecture 324 Structures II Composite Sections and Steel Beam Design Steel Beam Selection - ASD Composite Sections Analysis Method Photo by Mike Greenwood, 2009. Used with permission University of Michigan,

### Magnetism Conceptual Questions. Name: Class: Date:

Name: Class: Date: Magnetism 22.1 Conceptual Questions 1) A proton, moving north, enters a magnetic field. Because of this field, the proton curves downward. We may conclude that the magnetic field must

### SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Exam Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 1) A lawn roller in the form of a uniform solid cylinder is being pulled horizontally by a horizontal

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

### Physics 201 Homework 8

Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

### EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS

EQUATIONS OF MOTION: ROTATION ABOUT A FIXED AXIS Today s Objectives: Students will be able to: 1. Analyze the planar kinetics In-Class Activities: of a rigid body undergoing rotational motion. Check Homework

### Tutorial 4. Buoyancy and floatation

Tutorial 4 uoyancy and floatation 1. A rectangular pontoon has a width of 6m, length of 10m and a draught of 2m in fresh water. Calculate (a) weight of pontoon, (b) its draught in seawater of density 1025

### 2 A Dielectric Sphere in a Uniform Electric Field

Dielectric Problems and Electric Susceptability Lecture 10 1 A Dielectric Filled Parallel Plate Capacitor Suppose an infinite, parallel plate capacitor with a dielectric of dielectric constant ǫ is inserted

### PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

### MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA. Define and calculate 1st. moments of areas. Define and calculate 2nd moments of areas.

MECHANICAL PRINCIPLES HNC/D MOMENTS OF AREA The concepts of first and second moments of area fundamental to several areas of engineering including solid mechanics and fluid mechanics. Students who are

### Problem Set 1. Ans: a = 1.74 m/s 2, t = 4.80 s

Problem Set 1 1.1 A bicyclist starts from rest and after traveling along a straight path a distance of 20 m reaches a speed of 30 km/h. Determine her constant acceleration. How long does it take her to

### Mechanics Cycle 2 Chapter 13+ Chapter 13+ Revisit Torque. Revisit Statics

Chapter 13+ Revisit Torque Revisit: Statics (equilibrium) Torque formula To-Do: Torque due to weight is simple Different forms of the torque formula Cross product Revisit Statics Recall that when nothing

### Version PREVIEW Practice 8 carroll (11108) 1

Version PREVIEW Practice 8 carroll 11108 1 This print-out should have 12 questions. Multiple-choice questions may continue on the net column or page find all choices before answering. Inertia of Solids

### Multiple Reflections. What You Need For each 2-3 students: 1 set of hinged mirrors 1 protractor Copy of The Pirate Handout

Multiple Reflections Overview We know that when light reflects off a plane mirror, the image appears left/right reversed. Once you bring in another mirror and change the angle between them, it is much

### 5 Solutions /23/09 5:11 PM Page 377

5 Solutions 44918 1/23/09 5:11 PM Page 377 5 71. The rod assembly is used to support the 250-lb cylinder. Determine the components of reaction at the ball-andsocket joint, the smooth journal bearing E,

### 3 rd International Physics Olympiad 1969, Brno, Czechoslovakia

3 rd International Physics Olympiad 1969, Brno, Czechoslovakia Problem 1. Figure 1 shows a mechanical system consisting of three carts A, B and C of masses m 1 = 0.3 kg, m 2 = 0.2 kg and m 3 = 1.5 kg respectively.