Statics ENGR 1205 Kaitlin Ford kford@mtroyal.ca B175 Today Introductions Review Course Outline and Class Schedule Course Expectations Start Chapter 1 1
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 2
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 3
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 4
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 5
We will learn to model real-life situations using math Common in design to answer questions like: Is it strong enough? How big does it need to be? Mathematical models are approximations of the physical situations that help to predict behaviour. 6
Newton s First Law: if ΣF = 0, a body stays at rest or doesn t accelerate (constant velocity or constant speed with unchanging direction) 7
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. 8
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) 9
Newton s Law of Gravitation: F Mm G r 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 10
4 Basic Properties in Statics: 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 Property 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). v F ma m m t d t t 12
SI UNITS 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 )) 13
SI UNITS cont know metric prefixes (m, µ, n) and (k, M, G) for time: s, min, hr note 200 000 has no comma since commas mean decimal points in Europe Other rules in BOX 1.2 on page 7 US CUSTOMARY UNITS Feet, Pounds, Slugs SI unit conversions in front cover of text & table 1.3 14
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 law of addition Examples: displacement velocity force 15
In text (and some slides) vector symbols are in bold, F By hand, represent vectors with a representative letter with an arrow or half arrow above it. BE CONSISTENT F F For magnitude of vectors text uses Euclidean norm bars, by hand can use absolute value notation F 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
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 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 Find R using geometry (trigonometry) R 18
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 vectors, add the first and second, and then keep adding one vector at a time P Q S P Q S P Q S P P 2P such that the direction of 2P is the direction of P and the magnitude of 2P is 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 20
Vectors can be mathematically represented as: Magnitude and direction - direction in reference to some origin - in 3D use space angles θ x, θ y, θ z Rectangular components - a sum of vectors along perpendicular axes - generally along x, y, z axes 21
In three dimensions a Cartesian system is made up of three mutually perpendicular planes. A 3-D cartesian system can be left or right handed. In a right-handed system you can find the positive z-axis by pointing the fingers of your right hand in the positive x direction and curling them into the positive y direction. The direction of your thumb is the direction of the positive z-axis. 22
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We will also define vectors in terms of components and unit vectors unit vectors are vectors with a magnitude of 1. Unit vectors î & ĵ are along the x & y axes such that: F x F x i and F y F y j 24
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. Diagrams and Drawings (see section 1.5): Create separate diagrams for each of the bodies involved with a clear indication of all forces acting on each body. 25
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 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. 26
Accuracy, Limits and Approximations 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 27
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 28
As a general rule for engineering problems, the data are seldom known with an accuracy greater than 0.2%. As a practical rule FOR YOUR FINAL ANSWER, use 4 significant digits if the lead digit is 1 and 3 significant digits, e.g. 27.0 and 15.00. Keep more digits for intermediate steps. DON T WRITE DOWN EVERY DIGIT YOUR CALCULATOR GIVES YOU! 29