General Physics (PHY 170) Vectors. Tuesday, January 8, 13

Transcription

1 General Physics (PHY 170) Vectors 1

2 Scalar and Vector Quantities Scalar quantities are completely described by magnitude only (temperature...) Vector quantities need both magnitude (size) and direction to completely describe them (force, displacement, velocity, ) Represented by an arrow, the length of the arrow is proportional to the magnitude of the vector Head of the arrow represents the direction Indicated by the name with an arrow on top of it. 2

3 Scalars Versus Vectors Scalar: quantity is a number with units Vector: quantity has both a magnitude (with units) and a direction How to get to the Library: you need to know how far (0.5 miles) and which way. 3

4 The Components of a Vector Even though you know how far and in which direction the library is, you may not be able to walk there in a straight line 4

5 Math Review: Trigonometry Pythagorean Theorem 5

6 Example: how high is the building? Known: angle and one side Find:! another side Key: tangent is defined via two sides! α 6

7 The Components of a Vector Length, angle, and components can be calculated from each other using trigonometry: Convention: We measure counterclockwise from the +x axis 7

8 2D Cartesian and Polar Coordinate Representations 8

9 The Components of a Vector We can resolve vector into perpendicular components using two-dimensional coordinate systems: Polar Coordinates Cartesian Coordinates 9

10 Example: Height of a Cliff In Jules Vern s Mysterious Island, Capt. Cyrus Harding wants to find the height of a cliff. He stands with his back to the base of the cliff and marches straight away from it for 500 ft. At this point, he lies on the ground and measures the angle from horizontal to the top of the cliff. (a) If the angle is 34.0, how high is the cliff? (b) What is the straight line distance d from Capt. Harding to the top of the cliff? 10

11 The Components of a Vector Signs of vector components and range of : 0 θ θ st quadrant 2 nd quadrant 180 θ rd quadrant 270 θ th quadrant 11

12 Question 1 A vector has a negative x-component and a positive y-component. The angle that specifies its direction is measured counterclockwise from the x axis. In what range is? 12

13 Combining Vectors Graphically 13

14 Adding Vectors 14

15 Adding Vectors Graphically Adding vectors graphically: Place the tail of the second at the head of the first. The sum points from the tail of the first to the head of the last. 15

16 Subtracting Vectors The negative of a vector is a vector of the same magnitude pointing in the opposite direction. Here, D = A B. 16

17 Subtracting Vectors 17

18 Adding Vectors by Components A = (Ax, Ay) B = (Bx, By) C = A + B = (Cx, Cy) Cx = Ax + Bx 18 Cy = Ay + By

19 Adding Vectors by Components 1) Find the components of each vector to be added. 2) Add the x- and y-components separately. 3) Find the resultant vector. 19

20 Multiplication by a Scalar Multiplication by a scalar: B=cA Vector B points in the same direction of A B has the magnitude of ca 20

21 Properties of Vectors 21

22 Properties of Vectors Vectors are equal if they have the same magnitude and direction. 22

23 Vector Summary Scalar: number, with appropriate units Vector: quantity with magnitude (positive and with units) and direction Vector components: A x = A cos, B y = B sin Magnitude: A = (A 2 x + A y2 ) 1/2 Direction: = tan-1 (A y / A x ) Graphical vector addition: Place tail of second at head of first; sum points from tail of first to head of last or parallelogram method placing vectors to the same origin. 23

24 Unit Vectors Unit vectors are dimensionless vectors of unit length. For example, the vector A = A x X ^ has magnitude (with units) A x and points along the +x axis. A 24

25 Unit Vectors A = A^ x + A^ y + A^ z = A x x ^ + A y y ^ + A z ^z = (Ax, Ay, Az) Example: A = 4x ^ -2y ^ + 5z ^ = (4,-2,5) An alternative notation is: x ^ = ^i, ^y = j ^ and z ^ =k ^ The module of A is: 25

26 Sum of vectors and product with a scalar 26