Unformatted text preview: Module 2: Vector Introduc=on
August 27, 2010 Module Content:
1. Scalar and vector quan11es deﬁne many key a7ributes of engineering mechanics problems.
2. Vector opera1ons are important mathema1cal manipula1ons (“addi1on”, “mul1plica1on”) of vectors quan11es.
3. Vectors in sta1cs always have physical meanings. Module Reading, Problems, and Demo:
Reading: Chapter 2
Problems: Fundamental Problem 2.3 (Mastering), Prob. 2.4
Demo: none
Homework PlaPorm: h7p://www.masteringengineering.com
Course Blog: h7p://pages.shan1.virginia.edu/sta1cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 2
1 Concept: What’s a Vector?
• a vector is a quan1ty deﬁned by two pieces of informa1on which are generally considered to be:
• magnitude: how big? • direc1on: which way? • a scalar quan1ty is deﬁned by a single piece of informa1on • so, a “vector” is actually a collec1on of two or more scalars which
together
deﬁne a quan1ty • Examples:
• speed is a scalar (1 m/s), while velocity is a vector (1 m/s heading due east) • accelera1on due to gravity is a vector (9.8 m/s2 in direc1on of center of Earth) • how about force in a cable...scalar or vector? MAE 2300 Sta=cs © E. J. Berger, 2010 2
2 Concept: What’s a Vector?
• a vector is a quan1ty deﬁned by two pieces of informa1on which are generally considered to be:
• magnitude: how big? • direc1on: which way? • a scalar quan1ty is deﬁned by a single piece of informa1on • so, a “vector” is actually a collec1on of two or more scalars which
together
deﬁne a quan1ty • Examples:
• speed is a scalar (1 m/s), while velocity is a vector (1 m/s heading due east) • accelera1on due to gravity is a vector (9.8 m/s2 in direc1on of center of Earth) • how about force in a cable...scalar or vector? MAE 2300 Sta=cs © E. J. Berger, 2010 2
2 Theory: Vector Deﬁni=on MAE 2300 Sta=cs © E. J. Berger, 2010 2
3 Theory: Vector Deﬁni=on MAE 2300 Sta=cs © E. J. Berger, 2010 2
3 Theory: Vector Deﬁni=on MAE 2300 Sta=cs © E. J. Berger, 2010 2
3 Theory: Vector Deﬁni=on MAE 2300 Sta=cs © E. J. Berger, 2010 2
3 Theory: Vector Opera=ons
• there are some quite useful trigonometric ideas that help with vector analysis MAE 2300 Sta=cs © E. J. Berger, 2010 2
4 Theory: Vector Opera=ons
• there are some quite useful trigonometric ideas that help with vector analysis MAE 2300 Sta=cs © E. J. Berger, 2010 2
4 Theory: Vector Opera=ons
• there are some quite useful trigonometric ideas that help with vector analysis MAE 2300 Sta=cs © E. J. Berger, 2010 2
4 Theory: Force Vectors
• let’s introduce an equilibrium concept: MAE 2300 Sta=cs © E. J. Berger, 2010 2
5 Theory: Force Vectors
• let’s introduce an equilibrium concept: Fa + Fb + Fc = 0 MAE 2300 Sta=cs © E. J. Berger, 2010 2
5 Theory: Force Vectors
• let’s introduce an equilibrium concept: Fa + Fb + Fc = 0 Q: what location on the
structure does this
equation represent?
MAE 2300 Sta=cs © E. J. Berger, 2010 2
5 Theory: Cartesian Vectors
• it is oWen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves MAE 2300 Sta=cs © E. J. Berger, 2010 2
6 Theory: Cartesian Vectors
• it is oWen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves MAE 2300 Sta=cs © E. J. Berger, 2010 2
6 Theory: Cartesian Vectors
• it is oWen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves F = Fx i + Fy j
MAE 2300 Sta=cs © E. J. Berger, 2010 2
6 Theory: Cartesian Vectors
• it is oWen useful to deﬁne vectors based upon a Cartesian coordinate system (x,y,z) • as such, we need to think about how we would deﬁne quan==es in speciﬁc direc=ons (like the x,y,z components of a vector) • to achieve this, we use...”unit vectors”, which simply deﬁne the (x,y,z) direc=ons themselves F = Fx i + Fy j
MAE 2300 Sta=cs © E. J. Berger, 2010 2
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 Statics, a00, J.00 Berger, sta=cs, MAE00, vector00

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