Statics and Vectors. ME 202 Topics. Statics. Engineering Mechanics

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1 Engineering Mechanics Statics and Vectors ME 202 Mechanical System: One whose behavior can be completely described in terms of force, mass, distance, time and temperature. Engineering mechanics: Branch of engineering/physics/applied mathematics dealing with mechanical systems. 1 2 Statics Statics is the branch of engineering mechanics based on the approximation that nothing depends on time. An equivalent approximation is to say that time stops (freeze frame, snapshot) while we analyze. In this class, we also assume that nothing depends on temperature. ME 202 Topics Statics Vectors, forces, moments (6) Static equilibrium (6) Trusses, frames (6) First mass moment, mass center (2) Second mass moment (2) Second moment of area (2) Exams (4) 3 4

2 Fundamental Ideas The basic task in statics problems is to use Newton's laws to find the forces and moments in a structure. Newton s laws are statements about forces. Moments are defined in terms of forces. The fundamental idea is force. Force When you step onto a scale to be weighed, your body exerts a force on the scale. When you prop a ladder against a wall, the ladder exerts a force on the wall. What are the fundamental properties of forces? 5 6 Model of a Force Properties of a Force In engineering and physics, we pretend that a force is confined to a line called the line of action of the force. This is an approximation (a model) because the net push or pull exerted by one body on another is not due to something that acts along a line. (Contact between bodies always occurs over a finite area, not at a point.) The magnitude of a force is the amount of force in some units such as pounds, ounces, newtons, tons, etc. The other fundamental property of a force is its direction. Forces can be represented as vectors. 7 8

3 Vectors We represent a vector as a line segment with an arrowhead on one end. The end with the arrowhead is called the head of the vector. The other end is called the tail of the vector. head 9 tail Forces as Vectors Let the length of the vector represent the magnitude of the force. (The scale can be whatever is convenient. An inch of length could represent any amount of force.) Let the direction of the vector represent the direction of the force. 10 Vector Expressions Magnitude and direction Direction by unit vector Direction by angles Direction by direction cosines Rectangular (Cartesian) components y F = F û = F û F = F ĵ î x θ û a b û θ (measured CCW from x axis) F = F cos( θ x )î + cos θ y = F x î + F x ĵ ( ) ĵ

4 3-D (Cartesian) Vectors A = A x î + A y ĵ + A z ˆk A = A = A ( ) 2 + A z 2 = A x 2 + A y 2 + A z 2 ( ) A = A û = A cosαî + cosβ ĵ + cosγ ˆk Arbitrary 3-D Position Vectors Given: any two points in space (say A & B). r A = x A î + y A ĵ + z A ˆk r B = x B î + y B ĵ + z B ˆk Calculate a position vector from point A to point B. By inspection, = r B r A r B = r A + = ( x B x A )î + y B y A r A/B = B x r B ( ) ĵ + ( z B z A ) ˆk î A r A z ˆk ĵ y Arbitrary 3-D Unit Vectors Given any two points in space (say A & B). Calculate a unit vector in the direction from point A to point B. Call this unit vector û B/A. û B/A = vector from A to B magnitude of vector from A to B = ( û B/A = x x B A )î + y y B A x B x A B ( ) ĵ + ( z z B A ) ˆk ( ) 2 + ( y B y A ) 2 + ( z B z A ) 2 x û B/A î A ˆk z ĵ = û B/A y 3-D Vector Components Suppose that A and B are anywhere on the line of action of F. Then F = F û B/A where F and û B/A are in the same direction. F = F û B/A = F But ( x x B A )î + y y B A B ( ) ĵ + ( z z B A ) ˆk ( x B x A ) 2 + ( y B y A ) 2 + ( z B z A ) = F 2 x F A z û B/A î ĵ F = ( F x î + F ĵ + F ˆk y z ) = F( cosθ x î + cosθ ĵ + cosθ ˆk y z ) cosθ x = x A x B, cosθ y = y A y B, cosθ z = z A z B ˆk y 15 16

5 Statics and Vectors 2 The behavior of a purely mechanical system does not depend on electrical, electronic, nuclear, biological, chemical or magnetic principles. Specific subjects that are part of engineering mechanics include statics, dynamics, stress analysis, fluid mechanics, heat transfer, etc. We begin with statics. 3 Nothing in this world is truly static. But the principles of statics are mostly sufficient for designing things that are not expected to move significantly, such as bridges, buildings, roads, pipelines, etc. For many structures, designers must refine their static analyses to account for dynamic effects, such as thermal cycles, vibration, winds, floods, etc. But the design begins with static analysis. 4 Of the 30 meetings that we will have in this course, one was an introduction, four will be for exams and one will be for review prior to the final exam. As you can see from this list of the topics to be covered, 18 of the other 24 meetings will address topics in statics. The final six regular meetings will address topics needed for subsequent courses in other areas of engineering mechanics, such as ME 231 Dynamics and ME 321 Mechanics of Materials. 1 of 5

6 5 Relativistic effects, which are governed by Einstein s law of special relativity, are important only for objects moving at very high speeds. For static analysis, Einstein s equations reduce to those of Newton s laws. Hence, Newton s laws are sufficient for static analysis. The fundamental quantity in Newton s laws is force. 6 We are all familiar with the forces encountered in our everyday interactions with other things. Whether you stand, sit or recline, there is a force between you and the floor, chair, bed or other structure that supports you. 7 Because we use the simple model of a force described on this slide, we will find it convenient to represent forces as vectors. 8 The fundamental properties of a force, which are magnitude and direction, are the same as the fundamental properties of a vector. 2 of 5

7 9 As shown here, the symbol for a vector is a line segment with an arrowhead on one end. The negative of a vector points in the opposite direction of the vector. So, if the arrowhead were on the other end of the vector shown, we would have the negative of the vector shown. 10 Anything that has both magnitude and direction can be represented as a vector. This includes forces and moments, both of which will be discussed a bit later in the course. 11 There are many ways to describe the direction of a vector, and we will consider several of them. There are also many ways to express a vector as the sum of multiple components, but we will discuss only Cartesian components, which are also referred to as rectangular components. 12 Any force can be expressed as the product of its magnitude and a unit vector pointing in its direction. A unit vector is dimensionless and its magnitude is one. The only information in a unit vector is its direction. Any two unit vectors that have the same direction are actually the same unit vector. In two dimensions, two of the many acceptable ways of expressing a unit vector are shown at the upper right. The expression on the second line of the slide is also acceptable. On the third line, the expression in square brackets is also a unit vector. It is expressed in Cartesian components and its scalar 3 of 5

8 components are called direction cosines. Since the magnitude of the unit vector is one, the square root of the sum of the squares of the direction cosines must be one. 13 In three dimensions, the only practical way of expressing a unit vector is with Cartesian components. As it is in two dimensions, so it is in three: the scalar components of a unit vector are its direction cosines. 14 Suppose that we know the positions of two points, A and B, and would like to know the vector that gives the position of B relative to A. We label the desired vector as and the vector. The position vectors of A and B form a vector triangle. The equation that relates the three vectors states that going directly from the origin to B gives the same change in position as first going from the origin to A and then going from A to B. Solving this equation for and substituting the known expressions for the positions of A and B give the desired result. Note that the position of A relative to B is the negative of the position of B relative to A. 4 of 5

9 15 A unit vector pointing from A to B is found by dividing by its own magnitude. The three scalar components of the unit vector are the direction cosines of. 16 A force of magnitude F with direction from A to B, can be written as the product of F and the unit vector pointing from A to B. The unit vector is derived on slides 14 and of 5

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