Introduction to Engineering Mechanics Statics October 2009 () Introduction 10/09 1 / 19
Engineering mechanics Engineering mechanics is the physical science that deals with the behavior of bodies under the influence of forces (other bodies). Engineering mechanics can be divided into 3 categoies: 1 Mechanics of Rigid Bodies Statics, which deals with bodies at rest or moving at a uniform velocity Dynamics, which deals with bodies in motion with acceleration 2 Mechanics of Deformable Bodies 3 Fluid Mechanics Compressible - gas Incompressible - liquids We will concentrate on Mechanics of Rigid Bodies, the Statics part In Statics we will assume the bodies to be perfectly rigid, no deformation. () Introduction 10/09 2 / 19
Time: the measure of the succession of events (the interval between two events) Mass: the measure of the inertia of a body, which is its resistance to a change of motion. sometimes called "quantity of matter" Force: the action of one body on another A particle: a body of negligible dimensions Rigid body: a body is considered rigid when the relative deformation between its parts is negligible. () Introduction 10/09 3 / 19 Basic Concepts Space: the geometric region occupied by bodies whose positions are described by linear and angular measurements relative to a coordinate system. Figure: 2D Coordinate System Figure: 3D Coordinate System
Newton s Laws Newton developed the fundamental laws of mechanics. Law I: If the resultant force on a particle is equal to zero, then the particle will remain at rest or move with constant speed Law II: The acceleration of a particle is proportional to the vector sum of forces acting on it, and is in the direction of this vector sum. F = ma Law III: The forces of action between two bodies in contact are equal but opposite in direction. Action Action = Reaction Reaction () Introduction 10/09 4 / 19
Newton s Law of Gravitation This Newton s law can be used to evaluate the weight of a body. This law can be expressed as: F = G m 1m 2 r 2 F = mutual force of attraction between 2 particles G = universal constant known as the constant of gravitation m 1, m 2 = masses of the 2 particles r = distance between the 2 particles One application of the law is the attraction of the earth on a body located on its surface, the weight of the object. () Introduction 10/09 5 / 19
Gravitational Attraction of Earth Let s introduce a constant or an acceleration of gravity at earth s surface g given by: g = G m 1 r 2 m 1 = mass of earth Then, the gravitation force can be expressed as: F = g m 2 m 2 = mass of a body Replacing F with W for weight and m 2 with m for mass of a body results in: W = m g g = 9.81 m/s 2 () Introduction 10/09 6 / 19
Units In mechanics we use four funddamental quantities called dimensions. The four fundamental dimensions and their units and symbols in the two systems are summaried in the following table. Force: Newton (N) 1 N = (1 kg)(1 m/ s 2 ) 1 Newton is the force required to give a mass of 1 kg an acceleration of 1 m/ s 2. () Introduction 10/09 7 / 19
Accuracy, Rounding and Significant Figures The number of significant figures in an answer should be no greater than the number of figures justified by the accuracy of the given data. Accuracy to three significant figures is considered satisfactory for most engineering calculations. 4.25034 4.23911 = 0.01123 0.0112 Rounding numbers to the desired number of significant figures: round to the closer value: 24.57 m 24.6 m (three significant figures) if the number lies at the midpoint of the interval, then round off so that the last digit is even: 6.875 km 6.88 km (three significant figures) () Introduction 10/09 8 / 19
Scalars and Vectors In engineering mechanics statics, we use two quantities, scalars and vectors. 1 A scalar - A mathematical quantity that only has magnitude Examples of some scalars: area, volume, mass energy 2 A vectors - A mathematical quantity that has magnitude and direction. Vectors are used to represent physical quantities that have a magnitude and direction associated with them. The examples of some vectors: forces, velocity, displacement A vector is normally represented pictorially by an arrow (the arrow s length is its magnitude, and it points in its direction) and symbolically by an underlined letter V (in handwritten), using bold type V (in print) or by an arrow symbol over a variable V (in handwritten). The magnitude of the vector V is represented by V or simply by lightface italic type V () Introduction 10/09 9 / 19 V
Types of Vectors 1 Fixed (or bound) vectors:a vector for which a unique point of application is specified and thus cannot be moved without modifying the conditions of the problem. (A force on a deformable body) 2 Free vector: a vector whose action is not confined to or associated with a unique line in space. (A translation of a rigid body) 3 Sliding vector: a vector for which a unique line in space (line of action) must be maintained. (A force in a rigid body) Two vectors are equal if they have the same magnitude and direction. A negative vector of a given vector has the same magnitude but the opposite direction of the vector. () Introduction 10/09 10 / 19
Vector Addition All vectors obey the parallelogram law of addition. arallelogram Law: The sum of 2 vectors can be obtained by attaching the tails of 2 vectors to the same point and constructing a parallelogram R = +, where R is the resultant vector. In general, R = + unless the vectors are collinear. Triangle Rule: The sum of 2 vectors can be obtained by attaching the tail of one vector to the head of the other vector R = + Vector addition is communitative, + = + R () Introduction 10/09 11 / 19 R Tail R R V Head
Addition of 3 Vectors arallelogram Law R1 R1 R S S S R1 = + R = R 1 + S = + + S Triangle Rule S R1 R1 R S S R1 = + R = R 1 + S = + + S olygon Rule: Successive applications of triangle rule S R S R= + + S Vector addition is associative, + + S = ( + ) + S = + ( + S) () Introduction 10/09 12 / 19
Vector Subtraction The resultant of the difference between two vectors and of the same type may be expressed as R = = + ( ) Subtraction is thus defined as a special case of addition, so the rules of vector addition also apply to vector subtraction. R () Introduction 10/09 13 / 19
Resolution of A vector into Its Components Sometimes it is needed to resolve a vector into its components. There are infinite number of ways to resolve a vector as shown in Figure below. 1 2 1 2 = 1 + 2 To have definite answer of the vector components, either one of the followings must be case: 1 Except one component, the other components are knowns. For example, if the 2 component is known, then we can obtain the definite result of the 1 component. 2 Lines of actions of the components are known. For example, when you are given a coordinate system. () Introduction 10/09 14 / 19
Resolution of A vector into Its Components Y y θ x X What are the x and y components of if = 1000 N, θ = 30 x = cos(30 ) = 866 N y = sin(30 ) = 500 N On the other hand, given x and y, what is? = 2 x + 2 y = 866 2 + 500 2 = 1000 N () Introduction 10/09 15 / 19
Example 1.1 The fixed structure shown below in which = 500 N and T = 200 N. Combine and T into a single force R B T B θ 5 m T α θ R α 75 ο A 3 m C D tan α = BD AD The law of cosines: = 5 sin 75 3+5 cos 75 = α = 48.4 c 2 =a 2 + b 2 2ab cos(θ c ) R 2 = 200 2 + 500 2 2(200)(500) cos(48.4 ) R = 396.5 N The law of sines: 200 sin θ = 396.5 sin 48.4 = θ = 22.2 () Introduction 10/09 16 / 19
Example 1.2 A barge is pulled by 2 tugboats. The resultant of the forces exerted by the tugboats is a 1000 N directed along the center axis of the barge. Find: (a) Tension in each rope if α = 45 and (b) The value of α such that the tension in rope 2 is minimum. 1 30 ο α 2 1000 N 45 ο 30 ο T 2 T 1 ossible direction of T (a) 2 Direction of T1 1000 N ο 30 T 2 (b) (a) T 1 sin 45 = T 2 sin 30 = 1000 sin 105 = T 1=732 N T 2 =517 N (b) T 2 1000 sin 30 = sin(180 30 α) = T 2 sin 30 = 1000 sin(150 α) Considering the above Eq., sin(150 α) = 1 for T 2 to be minimum. Thus, α = 60 = T 1=1000 sin 60 =866 N T 2 =1000 sin 30 =500 N () Introduction 10/09 17 / 19
Example 2.3 The vertical force F of 80 N acts downward at A on the two-membered frame. Determine the magnitudes of the two components of F directed along AB and AC. B A 45 ο FAB 45 ο 75 ο ο 60 F AC 80 N F F ο 30 C F AB sin 60 = F AC sin 45 = 80 sin 75 = F AB =72 N F AC =59 N () Introduction 10/09 18 / 19
A Unit Vector () Introduction 10/09 19 / 19