St_Mod4_F10

St_Mod4_F10 - Module 4: Yes, Even More Vectors...

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Unformatted text preview: Module 4: Yes, Even More Vectors September 1, 2010 Module Content: 1. Dot and cross product opera=ons also have specific physical meanings in sta=cs. 2. Par=cle equilibrium analysis rests upon free body diagrams. Module Reading, Problems, and Demo: Reading: Chapters 2, 3 Problems: Prob. 2.90 (posi=on vectors/coordinate angles), Fund. Prob. 2.29 (dot product) Demo: none Homework PlaWorm: hXp://www.masteringengineering.com Course Blog: hXp://pages.shan=.virginia.edu/sta=cs2010 MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 1 Applica'on: Posi'on Vectors • posi=on vectors are oZen useful for defining the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 2 Applica'on: Posi'on Vectors • posi=on vectors are oZen useful for defining the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 2 Applica'on: Posi'on Vectors • posi=on vectors are oZen useful for defining the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 2 Applica'on: Posi'on Vectors • posi=on vectors are oZen useful for defining the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 2 Applica'on: Posi'on Vectors • posi=on vectors are oZen useful for defining the direc=on of a force along a line of ac=on, such as in a cable, chair, or rope: = magnitude of the posi1on vector MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 2 Theory: Dot Products • the dot product opera=on is defined as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x ­axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 3 Theory: Dot Products • the dot product opera=on is defined as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x ­axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 3 Theory: Dot Products • the dot product opera=on is defined as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x ­axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 3 Theory: Dot Products • the dot product opera=on is defined as the product of the magnitudes of two vectors mul=plied by the cosine of the angle between the vectors • in plain English, by example: project a vector F in 2D space onto the x ­axis • and in general: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 3 Remarks: Dot Products • there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68 ­69 • the few key points to remember: • • the dot product of two vectors can always be determined from the following basic rela=ons: • • dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 4 Remarks: Dot Products • there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68 ­69 • the few key points to remember: • • the dot product of two vectors can always be determined from the following basic rela=ons: • • dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 4 Remarks: Dot Products • there are lots of interes=ng mathema=cal rules about dot products...read about them on pages 68 ­69 • the few key points to remember: • • the dot product of two vectors can always be determined from the following basic rela=ons: • • dot products produce scalar quan==es (NOT vectors) and similarly for the other combina=ons and there is also an inverse formula=on for the dot product which allows us to calculate the angle between two vectors: MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 4 Introducing Chapter 3: Par'cle Equilibrium • par=cle equilibrium can be expressed as a specific case of Newton’s Law: • note that this is a vector equa=on • the way we actually write this equa=on is an explicit func=on of how we draw the diagram represen=ng the forces ac=ng on the par=cle: this diagram is the free body diagram • the usual way we approach this is to: 1. draw the par=cle 2. define a coordinate system 3. draw the forces MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 5 Let’s Examine FBDs Together... MAE 2300 Sta=cs © E. J. Berger, 2010 4 ­ 6 ...
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