sy04_sept17_07hc - Physics 207 Lecture 4 Physics 207...

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Page 1 Physics 207 – Lecture 4 Physics 207: Lecture 4, Pg 1 Physics 207, Lecture 4, Sept. 17 Agenda Assignment: Read Chapter 5 circle6 MP Problem Set 2 due Wednesday (should have started) circle6 MP Problem Set 3, Chapters 4 and 5 (available soon) circle6 Chapter 3, Chapter 4 (forces) xrhombus Vector addition, subtraction and components xrhombus Inclined plane xrhombus Force xrhombus Mass xrhombus Newton Newton’ s 1 st and 2 and 2 nd Laws xrhombus Free Body Diagrams Physics 207: Lecture 4, Pg 2 Vector addition circle6 The sum of two vectors is another vector. A = B + C B C A B C Physics 207: Lecture 4, Pg 3 Vector subtraction Vector subtraction circle6 Vector subtraction can be defined in terms of addition. B - C B C B - C B - C = B + (-1) C A Different direction and magnitude ! Physics 207: Lecture 4, Pg 4 Unit Vectors circle6 A Unit Vector is a vector having length 1 and no units circle6 It is used to specify a direction. circle6 Unit vector u points in the direction of U xrhombus Often denoted with a “hat”: u = û U = |U| U = |U| û û x y z i j k circle6 Useful examples are the cartesian unit vectors [ i, j, k i, j, k ] xrhombus Point in the direction of the x , y and z axes. R = r x i + r y j + r z k Physics 207: Lecture 4, Pg 5 Vector addition using components: Vector addition using components: circle6 Consider C = A + B . (a) C = (A x i + A y j ) + (B x i + B y j ) = (A x + B x ) i + (A y + B y ) (b) C = (C x i + C y j ) circle6 Comparing components of (a) and (b): xrhombus C x = A x + B x xrhombus C y = A y + B y C B x A B y B A x A y Physics 207: Lecture 4, Pg 6 Lecture 4, Exercise 1 Exercise 1 Vector Addition circle6 Vector A = {0,2,1} circle6 Vector B = {3,0,2} circle6 Vector C = {1,-4,2} What is the resultant vector, D , from adding A + B + C ? A) {3, {3,- 4,2} B) {4, {4,- 2,5} C) {5, {5,- 2,4}
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Page 2 Physics 207 – Lecture 4 Physics 207: Lecture 4, Pg 7 Lecture 4, Lecture 4, Exercise 1 Exercise 1 Vector Addition Vector Addition A. {3, {3,- 4,2} B. B. {4, {4,- 2,5} C. {5, {5,- 2,4} D. None of the above circle6 Vector A = {0,2,1} circle6 Vector B = {3,0,2} circle6 Vector C = {1,-4,2} What is the resultant vector, D , from adding A + B + C ? Physics 207: Lecture 4, Pg 8 Converting Coordinate Systems circle6 In polar coordinates the vector R = (r, θ ) circle6 In Cartesian the vector R = (r x ,r y ) = (x,y) circle6 We can convert between the two as follows: In 3D cylindrical coordinates ( r , θ, z ), r is the same as the magnitude of the vector in the x-y plane [ sqrt (x 2 +y 2 )] θ = tan -1 ( y / x ) 2 2 y x r + = y x (x,y) θ r r y r x j ˆ i ˆ sin cos y x R r y r r x r y x + = = = = = θ θ Physics 207: Lecture 4, Pg 9 Exercise: Frictionless inclined plane Exercise: Frictionless inclined plane circle6 A block of mass m slides down a frictionless ramp that makes angle θ with respect to horizontal.
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