Physics 170 Lecture 2. Phys 170 Lecture 2 1

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1 Physics 170 Lecture 2 Phys 170 Lecture 2 1

2 Phys 170 Lecture 2 2 dministrivia Registration issues? Web page issues? On Connect? Mastering Engineering issues? Course ID MWF300PM

3 Vectors in 2 Dimensions Phys 170 Lecture 2 3

4 Vector rithmetic Rules The order we add vectors doesn t matter: (vector addition is commutative). + B = B + ( ) = c + c B Vector addition and scalar product obey c + B (vector addition and scalar product are distributive). Vector subtraction is defined as: B = + 1 B ( ) Graphically, start with the arrow B Draw the B arrow B, then flip it. then do head-to-tail. D = B = + B ( ) Phys 170 Lecture 2 4

5 Phys 170 Lecture 2 Components and Component Vectors If we know the magnitude and the angle θ from the x-axis, the simplest possible trigonometry gives us the x and y components component is not a vector, it s a scalar. y = sinθ y y θ = cosθ x We sometimes also speak of component vectors, which are the vectors in the x and y directions that add up to the original vector.

6 Phys 170 Lecture 2 6 Notation Review letter with an arrow on top is a vector: Books usually use bold-face with no arrow: dding a subscript of x or y gives a component vector: bsolute value sign means a magnitude: V, V x, V y magnitude is never negative (it might be zero). V V V x, V y If there is an x- or y- subscript but no arrow: V x, V y the symbol is an x- or y- component which is not a vector, and which could be negative

7 Phys 170 Lecture 2 7 X and Y Component Values From Right Triangle The book often specifies direction using a little right triangle. You could work out the angles using arc-sine or arc-cosine or arc-tan, then use sine and cosine to work out the components. y y = x = 4 5 Pro tip: Calculate the components directly from the triangle! It s a waste of time to calculate the angles, then do sin & cos!

8 Phys 170 Lecture 2 Vector ddition Using Components dding vectors is easy if you know the x- and y-components, because the x-components and y-components just add linearly. C y B y y C C x B B x C = + B C x = + B x C y = y + B y

9 Phys 170 Lecture 2 Vector ddition Using Components (2) It even works if the signs are mixed. C y B y y C C x B Bx C = + B C x = + B x C y = y + B y

10 Phys 170 Lecture 2 dd or Subtract Using Components! It s nearly always easier to convert vectors to x-y components, add the components, then convert back to magnitude and/or direction, than to add the vectors using fancy trigonometry! It s more steps, but each step is simple, mechanical, and easy to check. The trigonometry is often tricky and hard to check. When we get to 3 dimensions, the trigonometry gets very hard to even visualize, let alone do it. But components in 3D are not much harder than in 2D.

11 X and Y Components to Magnitude The magnitude is easy to calculate from the components. The components form a right triangle with the original vector y So we just use Pythagoras: = 2 + y 2 y x Phys 170 Lecture 2 11

12 X and Y Components to ngle If y > 0, then θ = tan 1 y y θ y x If y < 0, tan 1 ( y ) x is negative, which means the angle is clockwise. y θ y x Phys 170 Lecture 2 12

13 X and Y Components to ngle (2) If < 0 and y > 0, then θ > +90 But y < 0 so tan 1 y < 0 y < 0 y y > 0 θ x It's exactly 180 wrong, because tan 1 y = tan 1 y x > 0 x θ Phys 170 Lecture 2 13 y

14 X and Y Components to ngle (3) If < 0 and y < 0, then θ = tan 1 y > 0, but it should come out negative! gain it's exactly 180 wrong! < 0 y < 0 y x θ y < 0 y θ x x > 0 Phys 170 Lecture 2 14

15 X and Y Components to ngle (4) If < 0, then tan 1 y (It doesn t matter if y < 0) ( ) is wrong by 180! Scientific calculators typically have a built-in function to convert between (x, y) and (r, θ) that do the right thing with any sign of x and y. If you use this function, you will get the right answer. Or, you can just add or subtract 180 by hand. Phys 170 Lecture 2 15

16 Buzzwords Resultant <----> Vector sum Unit vector <----> Dimensionless vector with magnitude 1 i (or î ) along + x axis Cartesian unit vector <----> j (or ĵ) along + y axis k (or ˆk) along + z axis Cartesian vector <----> Vector written as sum of x, y, and z components times î, ĵ, ˆk = î + y ĵ + z ˆk Phys 170 Lecture 2 16

17 Determine the magnitude and direction measured counterclockwise from the positive x-axis of the resultant force of the three forces acting on the ring. Take F 1 = 500 N and! = 20 o. Phys 170 Lecture 2 17

18 Marks for Tutorial ssignments Write the solution on Engineer s Computation Pad paper 1. Each Tutorial ssignment is marked out of s a general rule, marks are awarded as follows: 3 marks for a correct, clear and informative diagram. Draw straight lines with a ruler. 4 marks for correct and clearly written equations. It is not necessary to include units in the equations. 3 marks for correct numerical answers. Numerical answers are given to three significant figures with correct units. Do not use scientific notation. No marks are given for correctly solving incorrect equations. Phys 170 Lecture 2 18

19 My Drawing Steps ssign names to vectors, also to magnitudes and angles Look at vectors to decide scale, origin, then draw axes Draw long lines at given angles (protractor, 3x4 squares) Measure along lines to draw arrowheads Draw parallel lines in pencil Draw head-to-tail vectors for sum Phys 170 Lecture 2 19

20 My Calculation Steps Write symbolic vector equation Write symbolic x-component equation Write symbolic magnitudes direction-factors x-comp equation Write mag dir-fact x-component equation Write each term s value, many figures; also try keep in memory dd the component terms and write x-component Repeat for y-components Turn x,y comps of sum into magnitude and angle Phys 170 Lecture 2 20

21 The magnitude of the resultant force acting on the bracket is 400 N. Determine the magnitude of! F 1. Take! = 30 o. Disregard the u axis. =30 Phys 170 Lecture 2 21

22 Clicker question We will determine the magnitude of F1 by. Square root of sum of squares of its components B. Subtracting F 2 and F3 from the 400 N bracket force C. Using the fact that the bracket force is at 45 D. Using a really expensive calculator E. I m just hoping for partial credit here Phys 170 Lecture 2 22

23 For Next Week Get Mastering Engineering: Course ID MWF300PM Intro to M.E. assignment, visible now, due next Friday ssignment 1 will appear at 6 PM today, due next Friday Tutorials & Fundamental Problems are optional practice Get textbook, read Chapters 1 & 2 Download, read Notes 2 on the rest of Chapter 2 Download, print Lecture Problems 2, bring to class dditional/extra Problems are optional practice Phys 170 Lecture 2 23