Presentation on theme: "ENGR-1100 Introduction to Engineering Analysis Section 4"— Presentation transcript:
1ENGR-1100 Introduction to Engineering Analysis Section 4 Instructor: Professor Suvranu DeOffice: JEC 5002Office Ph: x6096Office hours: Tuesday and Friday 2:00-3:00 pmCourse Coordinator: Mohamed Aboul-Seoud x2317
2Teaching assistants: TA Ademola Akinlalu (Graduate)Office hours: M 3P-5PM;T 10AM -12Noon; W: 12Noon-2PM; R 3PM-5PM; F 10AM-12 NoonOffice: JEC 1022Ji Ming Hong (Undergraduate)
3ALL CLASSES ARE CLOSED-LAPTOP unless otherwise stated. Some Important PointsStudio course (combined lesson & problem session)ALL CLASSES ARE CLOSED-LAPTOP unless otherwise stated.Bring your relevant textbook, calculators, pencil, engineering computation paper to class EVERYDAYImportance of laptop/MATLAB to solve complicated problemsImportant tools: syllabus, 2 textbooks (listed in syllabus), laptop, pencil and, engineering computation paper
4Course websites Course official web site: My website for the course:McGraw-Hill Connect website (for Home works) for this section:McGraw-Hill Connect help:
5Course handouts Course syllabus: download from Supplementary information: download fromConnect quick steps: download fromMatlab tutorial: download from
6Course format Mini lectures Daily Class Activities (CA) 5% (drop 4 lowest grades). NO makeup for CA.Daily Homeworks (HW). 15% (drop 2 lowest grades). HWs due next day of class 12 noon. NO LATE SUBMISSIONS!Three mid term exams (2/15, 3/21, 4/18) in SAGE 3510: + total 55%Exam times: Wednesday 8 – 9:50 amMake-up exams (2/22 , 3/28, 4/25) in TBD 5:00-6:50pmGrade challenges must be within a week (6-8pm Exam 1: 2/20, 2/21; Exam 2: 3/26, 3/27; Exam 3: 4/23, 4/24)No make-ups for missed make-ups!1 final exam (time: TBA) : 25%
7Course objectivesFormulation and solution of static equilibrium problems for particles and rigid bodies.A bit of linear algebra: solution of sets of linear equations as they arise in mechanics and matrix operations.Use your laptop (running Matlab) for the manipulation of vector quantities and solution of systems of equations (only as an aid to completely solve “realistic” problems)
8Lecture outlineNewton’s lawsUnits of measurementVectors
9MechanicsMechanics is the branch of science that deals with the state of rest or motion of bodies under the action of forcesMechanicsMechanics ofrigid bodiesdeformable bodiesfluidsIn this class we will exclusively deal with the mechanics of rigid bodies.Few basic principles but exceedingly wide applications
10Very large Very small Mechanics Statics Dynamics Net force=0 In this class we will deal with the statics of rigid bodies.
11Modeling Physical Problem Physical Model Mathematical model (set of equations)Does answermake sense?Physical idealizations: particles,rigid body, concentrated forces, etc.Physical laws: Newton’s lawsApplied to each interacting body(free body diagram)Solution of equations: Using pen+paper/own code/ canned software like MatlabHappy YES!No!
12Physical Idealizations Continuum: For most engineering applications assume matter to be a continuous distribution rather than a conglomeration of particles.Rigid body: A continuum that does not undergo any deformation.Particle: No dimensions, only has mass. Important simplifying assumption for situation where mass is more important than exactly how it is distributed.Point force: A body transmits force to another through a finite area of contact. But it is sometimes easier to assume that a finite force is transmitted through an infinitesimal area.
13Newton’s Laws of Motion Law I: (Principle of equilibrium of forces) A particle remains at rest or continues to move in a straight line with uniform velocity (this is what we mean by being “in equilibrium”) if there is no unbalanced force acting on it.Inertial reference frameNecessary condition for equilibriumFoundation of StaticsVector equation
14Newton’s Laws of Motion Law II: (Nonequilibrium of forces) The acceleration of a particle is proportional to the resultant force acting on it and is in the direction of this force.is the resultant force acting on a particle of mass ‘m’.Foundation of DynamicsNecessary condition for equilibrium corresponding toVector equation
15Newton’s Laws of Motion Law III: (Principle of action and reaction) If one body exerts a force on a second body, then the second body exerts a force on the first body that is (1) equal in magnitude, (2) opposite in direction and (3) collinear (same line of action).EXTREMELY IMPORTANT to keep this in mind when working out problems!!
16Newton’s Laws of Motion Force (F)Force (F)PencilW=mg weight of pencilTableR (force actingon the pencil)R (force actingon the table)Need to isolate the bodies and consider the forces acting on them (Free Body Diagram).Be careful about which force in the pair we are talking about!
17Law of gravitation M.m F=G F Two bodies of mass M and m are mutually attracted to each other with equal and opposite forces F and –F of magnitude F given by the formula:M.mr2F=Gwhere r is the distance between the center of mass of the two bodies; and G is the Universal Gravitational Constant.G=3.439(10-8)ft3/(slug*s2) in the U.S customary system of units.G=6.673(10-11)m3/(kg*s2) in SI system of unitsMmrF
18Mass and weight Me.m F=G =mg Me The mass m of a body is an absolute quantity.The weight W of a body is the gravitational attraction exerted on the body by the earth or by another massive body such as another planet.At the surface of the earth:Me.mre2F=G=mgWhere: Me is the mass of the earth.re is the mean radius of the earthMere2g=GAt sea level and latitude 450g =32.17 ft/s2 = m/s2
19Units of measurementThe U.S customary system of units (the British gravitational system)Base units are foot (ft) for length, the pound (lb) for force, and the second (s) for timePound is defined as the weight at sea level and altitude of 450 of a platinum standardThe international system of units (SI)Three class of units(1) base units(2) derived units(3) derived units with special name
20Derived units with special name Base unitsQuantityUnitSymbolLengthmetermMasskilogramkgtimesecondsDerived unitsQuantityUnitSymbolAreaSquare meterm2VolumeCubic meterm3Linear velocityMeter per secondm/sDerived units with special nameQuantityUnitSymbolPlane angleradianradSolid anglesteradiansr
21SI / U.S. customary units conversion QuantityU.S. customary to SISI to U.S. customaryLength1 ft = m1 m = ftVelocity1 ft/s = m/s1 m/s = ft/sMass1 slug = kg1 kg = slug
22Scalar and vectorsA scalar quantity is completely described by a magnitude (a number).-Examples: mass, density, length, speed, time, temperature.A vector quantity has1. Magnitude2. Direction (expressed by the line of action + sense)3. Obey parallelogram law of addition-Examples: force, moment, velocity, acceleration.We will represent vectors by bold face symbols (e.g., F) in the lecture. But, when you write, you can use the symbol with an arrow on top (e.g., )
23Vectors: geometric representation A vector is geometrically represented as a line segment with an arrow indicating directionLine of actionFLength represents magnitude (F)HeadTailDirection of arrowdirection of vectorLength of arrowmagnitude of vectorQuestion: What is a vector having the same magnitude and line of action, but opposite sense?
24Operations on Vectors: Multiplication by scalars magnitude (2F)magnitude (F)2FFmagnitude (2F)magnitude (F)-2F-Fn can be a fraction less than 1, can n be 0?
25Operations on Vectors: Adding vectors using Parallelogram Rule Task: Add two vectors ( P and Q ) to obtain a “resultant” vector (R) that has the same effect as the original vectorsVectors are added using the Parallelogram lawPP+RQQR=P+Q=Q+PTo obtain the resultant, add two vectors using parallelogram lawAddition of vectors is commutative (order does not matter)
26Vectors in rectangular coordinate systems- two dimensional xy(v1,v2)vv2v1OIf the tail of the vector (v) is at the origin, then the coordinates of the terminal point (head) (v1,v2) are called the Cartesian components of the vector.v=(v1,v2)V = v1 i + v2 jOr,
27Vectors in rectangular coordinate systems- multiplication by a scalar xy(2v1,2v2)2v2v22v1OThe components of the vector 2v are (2v1, 2v2)
28The sum of two vectors – by adding components (two dimensional ) xy(v1+w1,v2+w2)v2w1(w1,w2)w2wv(v1,v2)v1Just add the x- and y-componentsv+w=(v1+w1,v2+w2)Or,v + w = (v1 + w1 )i + (v2 + w2 ) j
29Vectors in rectangular coordinate systems- Three dimensional z(v1,v2,v3)vv2v2v3yOv1v2x(v1,v2,v3) are the coordinates of the terminal point (head) of vector v
30The sum of two vectors – rectangular components (Three dimensional ) zxy(a1,a2,a3)(b1,b2,b3)abOa+b=(a1 +b1,a2+b2, a3 +b3)
31Vectors with initial point not at the origin (VERY IMPORTANT!!) zuP2(x2 ,y2 ,z2)P1(x1 ,y1 ,z1)wvOxCoordinates of head minus coordinates of tailHence
32Vectors with initial point not at the origin (VERY IMPORTANT!!) zz2-z1uP2(x2 ,y2 ,z2)P1(x1 ,y1 ,z1)y2-y1x2-x1OxThe components are the projections of the vector along the x-, y- and z-axes
33ExampleFind the components of the vector having initial point P1 and terminal point P2P1(-1,0,2), P2(0,-1,0)Solution:v= (0-(-1),-1-0,0-2)=(1,-1,-2)Head (P2) minus tail (P1)
34Vector arithmeticIf u,v,w are vectors in 2- or 3-space and k and l are scalar, then the following relationship holds:u+v=v+uu+0=0+u=uk(lu)=(kl)u(k+l)u=ku+lu(u+v)+w=u+(v+w)u+(-u)=0k(u+v)= ku+ kv1u=u
35Class assignment: (on a separate piece of paper with your name and RIN on top) please submit to TA at the end of the lecture1. Find the component of the vector having initial point P1 and terminal point P2(a) P1 = (-5,0), P2 = (-3,1)2. Let u = (-3,1,2), v = (4,0,-8) and w = (6,-1,-4). Find the x, y and z components of:(a) 6u + 2v(b) -3(v – 8w)
36IEA wisdom “Success in IEA is proportional to the number of problems solved.”