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9/3/2013PHY 113 C Fall Lecture 31 PHY 113 C General Physics I 11 AM-12:15 PM TR Olin 101 Plan for Lecture 3: Chapter 3 – Vectors 1.Abstract notion of vectors 2.Displacement vectors 3.Other examples

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9/3/2013PHY 113 C Fall Lecture 32

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9/3/2013 PHY 113 C Fall Lecture 33

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9/3/2013PHY 113 C Fall Lecture 34 iclicker question A.Have you attended a tutoring session yet? B.Have you attended a lab session yet? C.Have you attended both tutoring and lab sessions?

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9/3/2013PHY 113 C Fall Lecture 35 Mathematics Review -- Appendix B Serwey & Jewett iclicker question A.Have you used this appendix? B.Have you used the appendix, and find it helpful? C.Have you used the appendix, but find it unhelpful? iclicker question Have you changed your webassign password yet? A.yes B.no

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9/3/2013PHY 113 C Fall Lecture 36 Question from Webassign #

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9/3/2013PHY 113 C Fall Lecture 37 Mathematics Review -- Appendix B Serwey & Jewett a b c

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9/3/2013PHY 113 C Fall Lecture 38 Definition of a vector 1.A vector is defined by its length and direction. 2.Addition, subtraction, and two forms of multiplication can be defined 3.In practice, we can use trigonometry or component analysis for quantitative work involving vectors. 4.Abstract vectors are useful in physics and mathematics.

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9/3/2013PHY 113 C Fall Lecture 39 Vector addition: a b a – b Vector subtraction: a -b a + b

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9/3/2013PHY 113 C Fall Lecture 310 Some useful trigonometric relations (see Appendix B of your text) c b a Law of cosines: a 2 = b 2 + c 2 - 2bc cos b 2 = c 2 + a 2 - 2ca cos c 2 = a 2 + b 2 - 2ab cos Law of sines:

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9/3/2013PHY 113 C Fall Lecture 311 Some useful trigonometric relations -- continued (from Appendix B of your text) c b a Law of cosines: a 2 = b 2 + c 2 - 2bc cos b 2 = c 2 + a 2 - 2ca cos c 2 = a 2 + b 2 - 2ab cos

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9/3/2013PHY 113 C Fall Lecture 312 Some useful trigonometric relations -- continued Example: c=? 15 10 Law of cosines: a 2 = b 2 + c 2 - 2bc cos b 2 = c 2 + a 2 - 2ca cos c 2 = a 2 + b 2 - 2ab cos

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9/3/2013PHY 113 C Fall Lecture 313 Possible realization of previous example: c=? 15m 10m Start South East A pirate map gives directions to buried treasure following the indicated arrows. A wily physics students decides to take the easterly direct route after computing the distance c. treasure

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9/3/2013PHY 113 C Fall Lecture 314 Quantitative representation of a vector Cartesian coordinates A AxAx AyAy

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9/3/2013PHY 113 C Fall Lecture 315 Quantitative representation of a vector reference direction Note: can be specified in degrees or radians; make sure that your calculator knows your intentions! Polar coordinates

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9/3/2013PHY 113 C Fall Lecture 316 Quantitative representation of a vector Polar & cartesian coordinates

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9/3/2013PHY 113 C Fall Lecture 317 Vector components: axax ayay

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9/3/2013PHY 113 C Fall Lecture 318 axax ayay a = 1 m Suppose you are given the length of the vector a as shown. How can you find the components? A.a x =a cos a y =a sin B.a x =a sin a y =a cos C.Neither of these D.Both of these iclicker question

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9/3/2013PHY 113 C Fall Lecture 319 Vector components; using trigonometry An orthogonal coordinate system A

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9/3/2013PHY 113 C Fall Lecture 320 Vector components: axax ayay byby bxbx

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9/3/2013PHY 113 C Fall Lecture 321 Examples VectorsScalars Position rTime t Velocity vMass m Acceleration aVolume V Force FDensity m/V Momentum pVector components

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9/3/2013PHY 113 C Fall Lecture 322 Vector components Vector multiplication “Dot” product “Cross” product

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9/3/2013PHY 113 C Fall Lecture 323 Example of vector addition: a b a + b

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9/3/2013PHY 113 C Fall Lecture 324 a b a + b

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9/3/2013PHY 113 C Fall Lecture 325 Webassign version: Note: In this case the angle is actually measured as north of east.

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9/3/2013PHY 113 C Fall Lecture 326 Another example:

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9/3/2013PHY 113 C Fall Lecture 327 iclicker question A.Because physics professors like to confuse students B.Because physics professors like to use beautiful mathematical concepts if at all possible C.Because all physical phenomena can be described by vectors. D.Because there are some examples in physics that can be described by vectors Why are we spending 75 minutes discussing vectors

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9/3/2013PHY 113 C Fall Lecture 328 Example: Vector addition of velocities VbVb VwVw V total

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9/3/2013PHY 113 C Fall Lecture 329 Example: Displacement in two dimensions (0,0) (8,5)

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