General Physics I, Spring Vectors

Transcription

1 General Physics I, Spring 2011 Vectors 1

2 Vectors: Introduction A vector quantity in physics is one that has a magnitude (absolute value) and a direction. We have seen three already: displacement, velocity, acceleration. The value of the vector quantity must be accompanied by its unit. (For example, The velocity of the car is 3.5 m/s east. ) Vectors are represented graphically by arrows. The magnitude of the vector is represented by the length of the arrow. The direction of the vector is indicated by the direction of the arrow. Two vectors are equal if their magnitudes are equal and their directions are the same. The vectors do not have to be in the same place! We add, subtract, and multiply vectors according to laws of vector algebra, which are different from those of ordinary algebra. 2

3 Vector Addition: Graphical Vectors are added graphically using the tip-to-tail rule. To add two vectors, place the tail of the second vector at the tip (arrowhead) of the first. The sum vector, or resultant, is the vector whose tail is at the tail of the first vector and whose tip is at the tip of the second vector. Any number of vectors may be added using the tip-to-tail rule. Vectors may also be added using the parallelogram rule. The two vectors are placed tip-to-tail as before. Then a vector equal to the first and another equal to second are drawn such that the four vectors form a parallelogram. The sum or resultant vector is the diagonal of the parallelogram going from the tail of the first vector to the tip of the second. 3

4 Vector Addition: Tip-to-Tail Rule 4

5 Vector Addition: Parallelogram Rule From: University Physics, 11 th edition (Pearson Addison-Wesley) 5

6 Negative of a Vector The negative of a vector is a vector having the same magnitude but pointing in the opposite direction. The sum of a vector and its negative must be zero (i.e., the zero vector); this is easily shown by the tip-to-tail method. 6

7 Vector Subtraction Subtracting the vector from the vector is equivalent to adding B to A. Thus, A B= A+ ( B). B A 7

8 Scalar Times a Vector A scalar quantity in physics is one that is completely specified by one number and its unit. As you recall, speed is a scalar quantity; it has only a numerical value (along with its unit). A scalar times a vector gives a vector whose magnitude is the product of the magnitudes of the scalar and the vector. If the scalar is positive, the product vector points in the same direction as the original vector. If the scalar is negative, the product vector points in the opposite direction to the original vector. V = 3.5A Example: The vector has 3.5 times the magnitude of and points in the opposite direction to. A A 8

9 Scalar Times a Vector 9

10 Two vectors F and G are added to give a new vector H. Vectors F and G have equal magnitudes. Nothing else is known or may be assumed about them. Which statement(s) must be TRUE? 1. The magnitude of H can never be smaller than the magnitude of either F or G. 2. The magnitude of H can never be greater than the F G magnitude of either or. 3. The magnitude of always depends on the angle between and. 4. The magnitude of is always twice the magnitude of (or ). G F G H H F 10

11 Workbook: Chapter 3, Exercises

12 Vector Components: Introduction Using the graphical method for vector algebra quickly becomes tedious. Using vector components allows one to precisely do vector algebra using calculations. Let us consider motion in two dimensions (2D). An x-y coordinate system allows us to specify points in 2D space. The x-y coordinate system has 4 quadrants as shown in the figure. 12

13 Component Vectors Unlike in the one-dimensional case, there is now an infinite number of different directions in which a vector can point. To specify the direction of a vector, we can use the angle that the vector makes with the x-axis or y- axis. Consider a vector in 2D space, which we will describe by an x-y system. As the picture shows, there are two vectors, one parallel to the x-axis ( A x ) and one parallel to the y-axis ( A y ), which add together to give A. These vectors are called the component vectors of. A A θ The angle θ specifies The direction of A. 13

14 Vector Components Since a component vector is always along the x or y axis, we need only its magnitude (absolute value) and sign to specify it. The magnitude of a component vector along with a sign (positive or negative) to specify its direction along the axis is called a component. Since the component is just a number, its symbol is not written with an arrow over it. Thus, if the component vector is A x, the corresponding component is A x, which is called the x-component. Similarly, the y-component (A y ) of the vector A is the magnitude of the component vector A y along with a sign to indicate direction along the y-axis. The next slide has some examples. 14

15 Vector Components 15

16 Workbook: Chapter 3, Exercises

17 Review: Trigonometry sin θ = opposite. hypotenuse cos θ = adjacent. hypotenuse tan θ = opposite. adjacent Calculating Components 17

18 Calculating Components A The vector and its component vectors form a right triangle. The Pythagorean theorem can be used to find the magnitude of A (the hypotenuse) and its direction if the components are known. The components can be calculated using trigonometry if the magnitude and direction are known. We call the process of calculating components resolving a vector into its components. 18

19 Calculating Components To specify the direction of a vector, we need to find which quadrant the vector is in. Then we specify the direction as an angle above or below the positive or negative x-axis. y II I Quadrant A x A y A y A x θ θ A x A y I + + II - + A y A x θ A x θ A y x III - - IV + - III IV θ = A y tan 1. A x 19

20 Calculating Components: Summary If the magnitude and direction of a vector are known, calculate the components using the formulas: A = Acosθ A x y = Asin θ. Use quadrants to get sign of the component correct. If the components are known, calculate the magnitude and angle using the following formulas: A= A2 2 x + Ay θ = tan 1 Specify the direction as θ degrees above or below the positive or negative x-axis, depending on the quadrant. A A y x 20

21 Workbook: Chapter 3, Exercises

22 Adding Vector With Components If a number of vectors are added then the x-component of the resultant (sum) vector is simply the sum of the individual x-components. The same rule applies for the y component. The proof is illustrated below for the sum of two vectors. Components: Cx = Ax + Bx C = A + B y y y 22

23 Textbook: Chapter 3 Homework Questions and Problems Q 4; P 11, 13 23