Physics 203 College Physics I Fall 2012

1. Physics 203College Physics IFall 2012 S. A. Yost Chapter 3 Motion in 2 Dimensions – Part 1

2. Today’s Topics • Vectors • We will introduce the concept of vectors, which have many applications throughout physics, and are the most important new mathematical concept used in the course.

3. Thursday’s Assignment • Read Ch. 3, except section 8. • A problem set on HW3 on Ch. 3 will be due next Tuesday. • The first exam is now scheduled for Thursday, Sept. 20. The calendar in the syllabus posted on CitLearn has been updated. • You do not need to memorize equations: the essential ones will be provided for the exam.

4. Quiz: Question 2 • Which of the equations gives the correct relation between the vectors in the figure? • A.A + B + C = 0 • B. A = B + C • C. B = A + C • D. C = A + B • E. None of these → → → → → → → A → B → → → → → → → C

5. Quiz: Question 1 • Which of the following is a vector? • A. Mass • B. Temperature • C. Distance • D. Displacement • E. Speed

6. Quiz: Question 3 → → → • Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B? • A. Always • B. When A and B point in opposite directions. • C. Never • D. When A and B are parallel. • E. When A and B are perpendicular. → → → → → → → → →

7. Quiz: Question 4 → • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? • A. – 20, 60o • B. 20, 240o • C. 20, – 30o • D. – 20, 240o • E. – 20, – 30o → → A y θ = 60o x

8. Vectors and Scalars • Scalars are quantities described entirely by a number, with no need to specify a direction – the temperature, for example. • Vectorsrequire both a magnitude and direction to be fully specified. • Describing motion in 2 or more dimensions requires vectors. • Also forces, which must act in some direction, are described by vectors.

9. Quiz: Question 1 • Which of the following is a vector? • A. Mass • B. Temperature • C. Distance • D. Displacement • E. Speed

10. Displacement Vectors • The position of a point Brelative to a point Ais given by a displacement vector Dpointing from AtoB. • This vector tells you how to get from point Ato point B. → B → D A

11. Cartesian Components Cartesian coordinates are used to label points in a plane. The lengths of a vector along the two axes are called its Cartesian components. Dx = 2, Dy = 5. y → Dy D x 0 Dx

12. Polar Coordinates A vector can also be specified by giving its magnitudeanddirection. The magnitude is the length of the vector: D = |D|. The direction can be given by an angle relative to an axis. The angle in polar coordinates is measured counterclockwise from the xaxis. y → → D θ x 0

13. Mathematical Review: Right Triangle • The sides of a right triangle satisfy the Pythagorean Theorem: • a2 + b2 = c2 c b a

14. Mathematical Review: Trigonometry • The ratios of sides of a right triangle define the trigonometric functions. • sin θ = b/c cosθ = a/c tan θ = b/a • cscθ = c/b sec θ = c/a cot θ = a/b • Inverses: θ = asin (b/c) = acos(a/c) = atan(b/a) c b θ a

15. Polar Coordinates → Find the magnitude and direction of D. Dx = 2, Dy = 5 D = √ Dx2 + Dy2 = √29 = 5.4 tan θ = 5/2 = 2.5 θ = tan-1 (2.5) = 68o y → D θ x 0

16. Vector Addition • Geometrically, two vectors are added by following one to the end, then following the second from that point, and finding the net displacement. • Components: • = + Cx = Ax + Bx Cy = Ay + By → → → → → → A B C C B A

17. Quiz: Question 2 • Which of the equations gives the correct relation between the vectors in the figure? • A.A + B + C = 0 • B. A = B + C • C. B = A + C • D. C = A + B • E. None of these → → → → → → → A → B → → → → → → → C

18. Vectors • Two vectors, AandB, of length 5and3respectively, lie in a plane, but the directions are unspecified. • What is the maximum magnitude of A + B? • |A+B| = 8 • What is the minimum magnitude of A + B? • |A+ B|=2 → → → → → → → → → → → → → → C B A A B C → →

19. Scalar Multiple • Vectors can be multiplied by scalars (numbers). • Multiplying by a positive number changes the length, not the direction: • Multiplying by a negative number also changes the direction by 180o: → → → → A 2A A – A

20. Quiz: Question 4 → • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? • A. – 20, 60o • B. 20, 240o • C. 20, – 30o • D. – 20, 240o • E. – 20, – 30o → → A y θ = 60o x

21. Quiz: Question 4 → • Vector A has a magnitude of 10 and a direction angle θ = 60omeasured counter-clockwise from the +x axis. What are the magnitude and direction angle of the vector – 2A? • A. – 20, 60o • B. 20, 240o • C. 20, – 30o • D. – 20, 240o • E. – 20, – 30o → → y A θ = 240o 10 θ = 60o x 20 → – 2A

22. Vector Difference → → • The vector difference A–B can be formed by adding the vector – B to the vector A. • A–B can be interpreted as the displacement that takes you from B to A. → → → → A – B → A → B → → → → → – B

23. Quiz: Question 3 → → → • Suppose C = A – B. Under what circumstances is the length of C equal to the sum of the lengths of A and B? • A. Always • B. When A and B point in opposite directions. • C. Never • D. When A and B are parallel. • E. When A and B are perpendicular. → → → → → → → → → → C A B → →