Physics 12. Chapter 1: Vector Analysis in Two Dimensions

Transcription

1 Physics 12 Chapter 1: Vector Analysis in Two Dimensions

2 1. Definitions When studying mechanics in Physics 11, we have realized that there are two major types of quantities that we can measure for the systems of interest. They are, respectively, scalar and vector quantities. What is the difference between them? Chapter 1: Vector Analysis in 2D 2

3 By definition: Scalar quantity is a quantity that has only magnitude. Vector quantity is a quantity that has both magnitude and direction. Chapter 1: Vector Analysis in 2D 3

4 To see more clearly the difference, let s consider the following pair of quantities: distance vs displacement Chapter 1: Vector Analysis in 2D 4

5 Problem: Which path has the largest distance? Which path has the largest displacement? Chapter 1: Vector Analysis in 2D 5

6 A vector can be represented graphically by an arrow. Its tip points toward the direction and its length represents the magnitude of the vector quantity. Vectors can be drawn on typical Cartesian plane or in cardinal orientations. Chapter 1: Vector Analysis in 2D 6

7 For example, to describe the direction of each of the following vectors: N N W 45 E W 60 E S S 45 N of E 60 S of W 45 E of N 30 W of S Chapter 1: Vector Analysis in 2D 7

8 2. Addition of vectors Vectors, like scalars, can be involved in basic arithmetic operations. They can be added, subtracted or multiplied; however, the ways they are done are different from those for real numbers. In physics 11, we have seen briefly that simple arithmetic can be used to add vectors if they are in the same direction. Chapter 1: Vector Analysis in 2D 8

9 However, same method does not work if the two vectors are not along the same line. For example: suppose a person walks 10.0 km east and then walks 5.0 km north. What is the displacement of the person? Chapter 1: Vector Analysis in 2D 9

10 To determine the resultant vector that represents the displacement, the two vectors are drawn on the grids tip-to-tail. As the two vectors are at 90 of each other, the resultant vector can be found by Pythagorean theorem: D 2 R = D D 2 D R = = 11.2 km To determine the direction, we can use trigonometry: tan θ = D 2 D 1 θ = tan = 27 Chapter 1: Vector Analysis in 2D 10

11 The resultant is actually not affected by the order in which the vectors are added. Recall the previous example; assume the man first walks 5 km north and then 10.0 km east. The Pythagorean theorem will yield exactly the same result! Chapter 1: Vector Analysis in 2D 11

12 In general, vectors can be added graphically by means of the tip-totail method which involves the following steps: Step 1: On a diagram, draw the first vector v 1 in scale. Step 2: Next, draw the second vector v 2, to scale, placing its tail at the tip of the first vector and being sure its direction is correct. Step 3: The arrow drawn from the tail of the first vector to the tip of the second vector represents the sum, or resultant, of the two vectors denoted by v = v 1 + v 2. Step 4: Measure the length of the resultant by ruler and the angle with respect to the horizontal by a protractor. They can also be deduced analytically using trigonometry. Chapter 1: Vector Analysis in 2D 12

13 The tip-to-tail method of vector addition can be extended to the cases having three or more vectors. Consider the following case: What is the resultant vector? Chapter 1: Vector Analysis in 2D 13

14 Example: A man walks 250 m due East, then 250 m 60 North of East. Using scale diagrams on graph paper, determine the magnitude and direction of the resultant displacement. Chapter 1: Vector Analysis in 2D 14

15 Another way of adding vectors together is called parallelogram method. In this method, two vectors are drawn starting from a common origin, and a parallelogram is constructed using these two vectors as adjacent sides. The resultant is the diagonal drawn from the common origin. Chapter 1: Vector Analysis in 2D 15

16 3. Subtraction and multiplication of vectors In algebra, subtracting a number is the same as adding the opposite of that number. For example: = = 5 But how about vectors? If subtracting a vector is also the same as adding the opposite of the vector, then we may have to ask what is the opposite of a vector, v? Chapter 1: Vector Analysis in 2D 16

17 Recall that each vector is associated with a magnitude and a direction. Magnitude refers to the size of the vector and is always non-negative. Therefore, the negative sign of the vector must be related to its direction. By definition, the negative of a vector is the vector with its original magnitude but in opposite direction. Chapter 1: Vector Analysis in 2D 17

18 With this definition, we would be able to perform vector subtraction. The difference between two vectors can be interpreted as: v 2 v 1 = v 2 + ( v 1 ) Graphically, using the tip-to-tail method, we can determine the resultant vector: Chapter 1: Vector Analysis in 2D 18

19 Example: Given the following vectors: A is 6.3 cm in the direction 18 N of E B is 5 cm in the direction 53 N of W Draw the diagrams and find R for: (i) R = A + B (ii) R = A B (iii) R = B A Chapter 1: Vector Analysis in 2D 19

20 When a vector is multiplied by a scalar c, the vector is either stretched (if c > 1), compressed (if 0 < c < 1), or reversed (if c < 0). The magnitude of the resultant is changed by a factor c. Chapter 1: Vector Analysis in 2D 20

21 4. Vector addition by components As shown previously, any given vector can be written as a sum of two vectors, and there exist infinite number of such pairs. Therefore, it is always possible to resolve a vector into a pair of perpendicular vectors, each of which is lying along a coordinate axis. This process is called a resolution, and the resulting two vectors are called the components of the original vector. Chapter 1: Vector Analysis in 2D 21

22 When a vector is resolved into two perpendicular components, their magnitudes can be found easily using the Pythagorean theorem and trigonometry. sin θ = V y V V y = V sin θ cos θ = V x V V x = V cos θ tan θ = V y V x V 2 = V x 2 + V y 2 Chapter 1: Vector Analysis in 2D 22

23 Example: A cannon is shot at a muzzle velocity of 1500 m/s at an angle of 60 to the horizontal. What are the vertical and horizontal components of the velocity? Chapter 1: Vector Analysis in 2D 23

24 Example: A boy pulls a wagon with a force of 100 N at 40 to the horizontal. Find the pulling force and the lifting force. [64 N, 77 N] Chapter 1: Vector Analysis in 2D 24

25 Note that components are not always drawn as horizontal and vertical components. Sometimes it is more convenient, when dealing with systems in mechanics, to define the two components as parallel and perpendicular to the plane on which the object of interest is perched. For example: Consider an object that rests on a slope of angle θ. Chapter 1: Vector Analysis in 2D 25

26 What are W and W, and how to determine them? The parallel component W is the pulling force down the plane due to the gravitational pull W. It is calculated by W = W sin θ The perpendicular component W is the press on the object due to the gravitational pull W. It is calculated by W = W cos θ Chapter 1: Vector Analysis in 2D 26

27 Example: A ball rests on a slope. The ball weighs 600 N and the slope is 40 to the horizontal. What is the press on the ball to the surface and what is the pulling force that drags the ball down the slope? Chapter 1: Vector Analysis in 2D 27

28 Resolving vectors into components offers an alternative way of finding the resultant in a vector addition. Consider the addition of two vectors, v 1 and v 2 in the following diagram: Chapter 1: Vector Analysis in 2D 28

29 We can obtain the resultant by using the tip-to-tail method. However, from the diagram, we see that same answer can be obtained by adding respectively the components of the two vectors. That is, v Rx = v 1x + v 2x v Ry = v 1y + v 2y The magnitude and direction can be calculated in the usual way: v R = v x 2 + v y 2 θ = tan 1 v y v x Chapter 1: Vector Analysis in 2D 29

30 Example: A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction. She then drives in a direction 60.0 south of east for 47.0 km. What is her displacement from the post office? [30.0 km, 38.5 S of E] Chapter 1: Vector Analysis in 2D 30

31 Example: An airplane trip involves three legs, with two stopovers as shown. The first leg is due east for 620 km; the second leg is southeast for 440 km; and the third leg is 53 south of west, for 550 km. What is the plane s total displacement? [960 km, 51 S of E] Chapter 1: Vector Analysis in 2D 31

32 Example: What is the resultant force of the following system? Chapter 1: Vector Analysis in 2D 32

33 5. Relative velocity We have encountered the situation when learning relativity in Physics 11 that observations on an event from different frames of reference may result in different, or sometimes even apparently contradicting conclusions. Chapter 1: Vector Analysis in 2D 33

34 For example, consider two trains approaching one another, each with a speed of 80 km/h with respect to the Earth. What will an observer measure for the speed of the trains if he is on Earth or on either one of the train? 80 km/h 80 km/h Chapter 1: Vector Analysis in 2D 34

35 When the two velocity vectors are along the same direction, the relative velocity can be easily obtained by simple vector addition or subtraction. Assume that there are two objects A and B moving colinearly with the velocity v A and v B, respectively. By definition, the velocity of A relative to B is given by v AB = v A v B Pay attention on the sign when computing relative velocities; they are vectors and have directions! Chapter 1: Vector Analysis in 2D 35

36 If, however, the two velocity vectors point in different directions, then we will have to use vector addition to find the resultant. To avoid the problem of adding or subtracting wrong velocities, it is recommended that each vector be specified by two labels, the first one being the object, and the second one being the reference frame in which it has this velocity. v AB The magnitude of the velocity is v A is the object The velocity is measured in the reference frame B Chapter 1: Vector Analysis in 2D 36

37 Consider the following example: Chapter 1: Vector Analysis in 2D 37

38 Example: Consider a boat heads north at the velocity of 1.85 m/s directly across a river whose westward current is 1.20 m/s. What is the velocity of the boat relative to the shore? If the river is 110 m wide, how long will it take to cross and how far downstream will the boat be then? Chapter 1: Vector Analysis in 2D 38

39 Another common scenario we may encounter involves directing a boat in a correct direction to deal with water stream. Consider the following example: In order to make sure the boat goes straight North, at what upstream angle must it head? sin θ = v WS v BW If v BW = 1.85 m/s and v WS = 1.20 m/s, then θ = sin 1 v WS v BW = sin = 40.4 Chapter 1: Vector Analysis in 2D 39

40 Example: There is a 15.0 m/s wind blowing due East and you start riding your bike North at 9.0 m/s. What is the velocity of the wind on your face? [17.5 m/s at 59 E of S] Chapter 1: Vector Analysis in 2D 40

41 Example: A plane is capable of travelling at 120 m/s in still air. Where must the pilot head the plane in order to end up going due North when there is a 35 m/s West wind? Chapter 1: Vector Analysis in 2D 41