Statics ENGR 1205 Kaitlin Ford kford@mtroyal.ca B175 Today Introductions Review Course Outline and Class Schedule Course Expectations Chapter 1 1 Review the Course Outline and Class Schedule Go through Handout Also posted on Blackboard Course Format 2 1
Lectures fill-in-the-blank notes posted on Blackboard Read the text and try to fill them in before class I will use the completed slides during lectures (I will not post the completed slides) Tutorials will happen whenever Time for YOU to practice (with help) Assignments Due every Thursday We will mark 2 questions in class (one self 25%, one peer-25%) I will mark one other question (50% of the mark) Quizzes on Tuesdays format will vary 3 My Expectations fair, but I expect a lot tough course, work hard or else need to fairly good at math and physics (vectors) bring text to class read the textbook do all assigned homework 4 2
Keep up with the material. Review math fundamentals. (vectors, trig) Devise and practice problem-solving techniques. Do the recommended practice problems and understand them (don t try to memorize!) Participate. 5 Ask questions WRONG WAY I don t know how to do this question. RIGHT WAY I tried this problem and I think this part is correct, but I m not sure about the next step because 6 3
Leave your cell phone on during class. Request more marks on an assignment when you haven t figured out what you did wrong yet. Send me an e-mail saying I wasn t in class. What did I miss? Fall behind (stay healthy, etc.) 7 the goal of this course is to develop your ability to analyze and solve basic problems we will use vector analysis We will solve two-dimensional (2D) and threedimensional (3D) problems we will use SI units 8 4
Mechanics is a branch of the physical sciences that is concerned with the motion of bodies that are subjected to the action of FORCES (including statics - the special case in which bodies remain at rest) The general principles were first enunciated by Sir Isaac Newton in his Philosophiae Naturalis Principia Mathematica (1687), commonly known as the Principia 9 Mechanics can be subdivided into 3 parts... 1) Rigid Body Mechanics (generally an unrealistic portrayal of situations) - statics (rest) ΣF = 0 (a = 0), ΣM = 0 - dynamics (motion) ΣF 0 (a 0) 2) Deformable-Body Mechanics - includes the mechanics of materials 3) Fluid Mechanics - study of gas and liquids Mechanics is an applied physical science and a key foundation in the engineering sciences 10 5
Historical Development Aristotle Archimedes Galileo NEWTON Euler, d Alembert, Lagrange, g Hamilton then Einstein (quantum or relativistic mechanics) 4 Basic Quantities in Mechanics: SPACE - the position of a point given in terms of three coordinates (x, y, z) measured from a reference point or origin TIME necessary to define an event in addition to spatial coordinates MASS - characterizes and compares bodies e.g. 2 bodies of the same mass are attracted the same amount by the Earth and offer the same resistance to a change in their state of motion 11 The Fourth Basic Quantity is: FORCE a push or pull ; the action of one body on another, either by contact or at a distance (characterized by a point of application, a magnitude, a sense and a line of action; it s a vector) The first 3 quantities are absolute concepts, independent of each other (in Newtonian mechanics) Force is not an independent quantity. It is related to the mass of the body and the variation of its velocity with time (acceleration). d v t F ma m m t t 12 6
We ll study PARTICLES and RIGID BODIES (RB) PARTICLES small amounts of matter occupying one point in space, have mass but size is ignored RIGID BODIES combinations of particles occupying fixed positions in space with respect to one another, positions don t change with added force 13 Scalar quantities those which have only a magnitude Examples: time volume density speed mass Vector quantities those which have magnitude and direction and obey the parallelogram l law of addition i Examples: displacement velocity force 14 7
Free vector one whose action is not confined to or associated with a unique line in space Examples: displacement of a rigid body which is moving without rotation; force couple Sliding vector has a unique line of action in space but not a unique point of application Example: external force on a rigid body Fixed vector a unique point of application is specified Example: force on a deformable body 15 Represent vectors with a representative letter with an arrow above it. E.g. F (or be consistent) For magnitude of vectors, use absolute value notation. E.g. F equal vectors have the same magnitude and direction (but maybe not the same point of application) a negative vector simply has the opposite sense of its positive P P 0 16 8
the parallelogram law for the addition of vectors: the resultant sum of two vectors is the diagonal of the parallelogram formed using the two vectors as adjacent sides 17 parallelogram law note that, in general, P Q P Q since P+ Q= Q+ P (from the parallelogram law), we can conclude that vector addition is commutative R Or, use the triangle rule (arrange vectors tip to tail) and resultant goes from first tail to last tip Find R with scale diagram R Find R using geometry (trigonometry) 18 9
BASIC TRIGONOMETRY RATIOS (for right angle triangles): SINE LAW: COSINE LAW: 19 subtraction is the addition of a negative sense vector e.g. - = + = to sum 3 or more coplanar vectors, add the first and second, and then keep adding one vector at a time (repetitively applying the triangle rule) i.e. P Q S P Q S P Q S i e vector addition is associative (grouping doesn t matter) 20 10
OR, apply the tip to tail method in one fast step this extension of the parallelogram law is called the polygon rule e.g. P P 2P such that the direction of 2P is the direction of P and the magnitude of is 2P 2 P For kp, the direction of kp is the same as P if k>0 and it s the opposite of P if k<0, while the magnitude is always kp 21 Vectors can be mathematically represented in two ways: Rectangular components - a sum of vectors along perpendicular axes - generally along x, y, z axes Magnitude and direction - direction in reference to some origin 22 11
Newton s First Law: if ΣF = 0, a body stays at rest or doesn t accelerate (constant velocity or constant speed with unchanging direction) 23 Newton s Second Law: if ΣF 0, F = ma The acceleration of a particle is proportional to the vector sum of the forces acting on it, and is in the same direction of this vector sum. 24 12
Newton s Third Law: the forces of action and reaction between interacting bodies are equal in magnitude and opposite in direction (same line of action, but opposite sense) 25 We will use S.I. Units (Système International d Unités) SI units are absolute i.e. they mean the same thing everywhere The base units are length (m), mass (kg) and time (s). the 3 are independent units, defined arbitrarily Force (newtons, N) is a derived unit Defined as the force that gives an acceleration of 1 m/s 2 to a mass of 1 kg. N = kg*m/s 2 (F = ma = kg (m/s 2 )) 26 13
know metric prefixes (m, µ, n) and (k, M, G) for time: s, min, hr for area: m 2 (1 m x 1 m) for volume: m 3 (1 m x 1 m x 1 m) -for liquids, dm 3 = 1 L note the SI unit conversions in front cover of text note 200 000 has no comma since commas mean decimal points in Europe 27 Mm F G r Newton s Law of Gravitation: 2 where: F = the Force exerted by one object (M) on another (m) G = the universal gravitational constant r = the distance between the masses The Force of Gravitation exerted by the Earth on an object (at the surface) is: M W mg g G r earth, 2 earth g varies by location on the Earth we will use g = 9.81 m/s 2 28 14
The mathematical formulation of a physical problem represents an ideal description, or model, which approximates but never quite matches the actual physical condition. Examples of assumptions/simplifications: neglect small distances, angles, and forces Rigid bodies, force distribution area 29 The accuracy of a solution depends on: accuracy of the given data, and accuracy of the computations performed. The solution cannot be more accurate than the less accurate (worst) of these two. The use of calculators generally makes the accuracy of the computations much greater than the accuracy of the data. Hence, the solution accuracy is usually limited by the data accuracy. e.g. given 75 000 ± 100 N ± 0.13% so an answer of 14 322 N, is really 14 322 ± 20 N 30 15
As a general rule for engineering problems, the data are seldom known with an accuracy greater than 0.2%. As a practical rule, use 4 significant digits if the lead digit is 1 and 3 significant digits, e.g. 27.0 and 15.00 DON T WRITE DOWN EVERY DIGIT YOUR CALCULATOR GIVES YOU! 31 The ability to clearly communicate a solution is vital Your solutions must be clear (presentation, layout, handwriting, logical progression) as well as complete and correct Include (mandatory minimum): Problem Statement: Includes given data, specification of what is to be determined, and a figure showing all quantities involved. Free-Body Diagrams: Create separate diagrams for each of the bodies involved with a clear indication of all forces acting on each body. 32 16
Fundamental Principles: Newton s Laws (and relevant equations) are applied to express the conditions of rest or motion of each body. The rules of algebra are applied to solve the equations for the unknown quantities. Could be an equation or a statement. Solution Check: -Test for errors in reasoning by verifying that the units of the computed results are correct, -test for errors in computation by substituting i given data and computed results into previously unused equations based on the six principles, -always apply experience and physical intuition to assess whether results seem reasonable. 33 The screw is subjected to two forces F 1 and F 2. Determine the magnitude and direction of the resultant force. a) Using a scale diagram b) Using trigonometry c) Using components 10 o F 2 = 150 N 15 o F 1 = 100 N 34 17
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