Chapter 2 Vectors and Scalars PDF

Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. Chapter 2 Vectors and Scalars Some physical quantities, such as time, temperature, mass, and density, can be described completely by a single number with a unit. But many...

Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. Chapter 2 Vectors and Scalars Some physical quantities, such as time, temperature, mass, and density, can be described completely by a single number with a unit. But many other important quantities in physics have a direction associated with them and cannot be described by a single number. A simple example is the motion of an airplane: We must say not only how fast the plane is moving but also in what direction. The speed of the airplane combined with its direction of motion constitute a quantity called velocity. Another example is force, which in physics means a push or pull exerted on a body. Giving a complete description of a force means describing both how hard the force pushes or pulls on the body and the direction of the push or pull. When a physical quantity is described by a single number, we call it a scalar quantity. In contrast, a vector quantity has both a magnitude (the “how much” or “how big” part) and a direction in space. Calculations that combine scalar quantities use the operations of ordinary arithmetic. For example, 6kg + 3kg = 9kg, or 4 * 2s = 8s. However, combining vectors requires a different set of operations. To understand more about vectors and how they combine, we start with the simplest vector quantity, displacement. Displacement is a change in the position of an object. Displacement is a vector quantity because we must state not only how far the object moves but also in what direction. Walking 3 km north from your front door doesn’t get you to the same place as walking 3 km southeast; these two displacements have the same magnitude but different directions. Coordinate system Many aspects of physics involve a description of a location in space. For example, we saw that the mathematical description of an object’s motion requires a method for describing the object’s position at various times. In two dimensions, this description is accomplished with the use of the Cartesian coordinate system, in which perpendicular axes intersect at a point defined as the origin (Figure 2-1). Cartesian coordinates are also called rectangular coordinates. Sometimes it is more convenient to represent a point in a plane by its plane polar coordinates (r, ) as shown in Figure 2-2a. In this polar coordinate system, r is the distance from the origin to the point having Cartesian coordinates (x, y) and  is the angle between a fixed axis and a line drawn from the origin to the point. The fixed axis is often the positive x axis, and  is usually measured counterclockwise from it. From the right triangle in Figure 2-2b, we find that sin = y/r and that cos= x/r. Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian coordinates by using the equations 15 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. 𝑥 = 𝑟 cos 𝜃 Eq. 2-1 𝑦 = 𝑟 sin 𝜃 Eq. 2-2 Furthermore, the definitions of trigonometry tell us that 𝑦 Eq. 2-3 tan 𝜃 = 𝑥 𝑟= 𝑥 +𝑦 Eq. 2-4 The latest equation is known as Pythagorean theorem. These four expressions relating the coordinates (x, y) to the coordinates (r, ) apply only when  is defined as shown in Figure 2-2a—in other words, when positive  is an angle measured counterclockwise from the positive x axis. (Some scientific calculators perform conversions between Cartesian and polar coordinates based on these standard conventions). If the reference axis for the polar angle  is chosen to be one other than the positive x axis or if the sense of increasing  is chosen differently, the expressions relating the two sets of coordinates will change. Figure 2-1 Designation of points in a Cartesian coordinate system. 16 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. Figure 2-2 Designation of points in a Polar coordinate system. Example: 1. The Cartesian coordinates of a point in the xy plane are (x, y) = (-3.50, -2.50) m. Find the polar coordinates of this point. Ans(r=4.3 m, =216o) o 2. The polar coordinates of a point are r = 5.5m and θ =240. What are the Cartesian coordinates of this point? Ans(x=-2.75 m, y=-4.76 m) Some Properties of Vectors In this section, we shall investigate general properties of vectors representing physical quantities. We also discuss how to add and subtract vectors using both algebraic and geometric methods. Equality of Two Vectors For many purposes, two vectors 𝑨⃗ and 𝑩⃗ may defined to be equal if they have the same magnitude and if they point in the same direction. That is, 𝑨⃗ = 𝑩⃗ only if A = B and if 𝑨⃗ and 𝑩⃗ point in the same direction along parallel lines. For example, all the vectors in Figure 2-3 are equal even though they have different starting points. This property allows us to move a vector to a position parallel to itself in a diagram without affecting the vector. Figure 2-3 Example of equal vectors 17 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. Adding Vectors The rules for adding vectors are conveniently described by a graphical method. To add vector 𝑩⃗ to vector 𝑨⃗, first draw vector 𝑨⃗ on graph paper, with its magnitude represented by a convenient length scale, and then draw vector 𝑩⃗ to the same scale, with its tail starting from the tip of 𝑨⃗. as shown in Figure 2-4. The resultant vector 𝑹⃗ = 𝑨⃗ +𝑩⃗ is the vector drawn from the tail of 𝑨⃗ to the tip of 𝑩⃗. A geometric construction can also be used to add more than two vectors as is shown in Figure 2-5 for the case of four vectors. The resultant vector 𝑹⃗ = 𝑨⃗ + 𝑩⃗ + 𝑪⃗ + 𝑫⃗ is the vector that completes the polygon. In other words, R is the vector drawn from the tail of the first vector to the tip of the last vector. This technique for adding vectors is often called the “head to tail method.” Figure 2-4 Adding two vectors Figure 2-5 Geometric construction for summing four vectors  Commutative Law of Addition When two vectors are added, the sum is independent of the order of the addition (Figure 2-6a).  Associative Property of Addition When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped (Figure 2-6b). 18 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. a) commutative law b) associative property Figure 2-6 Commutative and associative property law In summary, a vector quantity has both magnitude and direction and also obeys the laws of vector addition. When two or more vectors are added together, they must all have the same units and they must all be the same type of quantity. It would be meaningless to add a velocity vector (for example, 60 km/h to the east) to a displacement vector (for example, 200 km to the north) because these vectors represent different physical quantities. The same rule also applies to scalars. For example, it would be meaningless to add time intervals to temperatures. Negative of a Vector The negative of the vector 𝑨⃗ is defined as the vector that when added to 𝑨⃗ gives zero for the vector sum. That is, 𝑨⃗+ (-𝑨 ⃗)=0. The vectors 𝑨⃗ and -𝑨⃗ have the same magnitude but point in opposite directions. Subtracting Vectors The operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation 𝑨⃗- 𝑩⃗ as vector -𝑩⃗ added to vector 𝑨⃗: 𝑨⃗ − 𝑩⃗ = 𝑨⃗ + −𝑩⃗ Eq. 2-5 The geometric construction for subtracting two vectors in this way is illustrated in Figure 2-7a. Another way of looking at vector subtraction is to notice that the difference 𝑨⃗-𝑩⃗ between two vectors 𝑨⃗ and 𝑩⃗ is what you must add to the second vector to obtain the first. In this case, as Figure 2-7b shows, the vector 𝑨⃗-𝑩⃗ points from the tip of the second vector to the tip of the first. Figure 2-7 Vector subtractions Multiplying a Vector by a Scalar The result of the multiplication or division of a vector by a scalar is a vector. The magnitude of the vector is multiplied or divided by the scalar. If the scalar is positive, the direction of the 19 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. result is the same as of the original vector. If the scalar is negative, the direction of the result is opposite that of the original vector. For example, the vector 5𝑨⃗ is five times as long as 𝑨⃗ and points in the same direction as 𝑨⃗; the vector -1/3 𝑨⃗ is one-third the length of 𝑨⃗ and points in the direction opposite 𝑨⃗. Ex:- A car travels 20.0 km due north and then 35.0 km in a direction 60.0° west of north as shown in figure below. Find the magnitude and direction of the car’s resultant displacement. Sol:- - Geometrical solution (H.W) - Algebraic solution From the law of cosines: 𝑹= 𝑨𝟐 + 𝑩𝟐 − 𝟐𝑨𝑩𝐜𝐨 𝐬 𝜽 𝑹= 𝟐𝟎. 𝟎𝟐 + 𝟑𝟓. 𝟎𝟐 − 𝟐(𝟐𝟎)(𝟑𝟓)𝐜𝐨 𝐬 𝟏𝟐𝟎𝒐 = 𝟒𝟖. 𝟐 𝒌𝒎 sin 𝛽 sin 𝜃 = →. sin 𝛽 = 0.629 → 𝛽 = 38.9 𝐵 𝑅 Components of a vector The graphical method of adding vectors is not recommended whenever high accuracy is required or in three-dimensional problems. In this section, we describe a method of adding vectors that makes use of the projections of vectors along coordinate axes. These projections are called the components of the vector or its rectangular components. Any vector can be completely described by its components. The vector 𝑨⃗ can be expressed as the sum of two other ( x and y) component vectors Ax and Ay. From Figure 2-8b, we see that the three vectors form a right triangle and that 𝑨⃗ = 𝑨⃗x+𝑨⃗y. The components of 𝑨⃗ can be written as Ax and Ay (without the boldface notation). The component Ax represents the projection of 𝑨⃗ along the x axis, and the component Ay represents 20 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. the projection of 𝑨⃗ along the y axis. These components can be positive or negative. The component Ax is positive if the component vector 𝑨⃗x points in the positive x direction and is negative if 𝑨⃗x points in the negative x direction. The same is true for the component Ay. It can be seen that the y-component is moved to the end of the x-component. This is due to the fact that any vector can be moved parallel to itself without being affected. Figure 2-8 Vector's components The magnitude of these components are the lengths of the two sides of a right triangle with hypotenuse of length A. Therefore, the magnitude and direction of 𝑨⃗ are related to its components: 𝐴= 𝐴 +𝐴 𝐴 𝜃 = tan 𝐴 𝐴 = 𝐴 cos 𝜃 𝑎𝑛𝑑 𝐴 = 𝐴 sin 𝜃 This assumes the angle θ is measured with respect to the x-axis. Notice that the signs of the components Ax and Ay depend on the angle . For example, if θ = 120°, Ax is negative and Ay is positive. If θ =225°, both Ax and Ay are negative. Figure 2-9 summarizes the signs of the components when 𝑨⃗ lies in the various quadrants. 21 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. Figure 2-9 Sign of vector's components Unit Vector A unit vector is a dimensionless vector with a magnitude of exactly 1. Unit vectors are used to specify a direction and have no other physical significance. The symbols i, j and k are used to represent unit vectors in the positive x, y, and z directions, respectively. The unit vectors i, j, and k form a set of mutually perpendicular vectors in a right-handed coordinate system as shown in Figure 2-10. The magnitude of these vectors is 1, |𝑖| = |𝑗| = |𝑘| = 1. Figure 2-10 Unit vectors Consider a vector 𝑨⃗ lying in the xy plane as shown in Figure 2-10. The product of the component Ax and the unit vector I is the component vector 𝑨⃗𝒙 = 𝐴 𝒊, which lies on the x axis and has magnitude |𝐴 |. Likewise, 𝑨⃗𝒚 = 𝐴 𝒋 is the component vector of magnitude 𝐴 lying on the y axis. Therefore, the unit-vector notation for 𝑨⃗ is 𝑨⃗ = 𝐴 𝒊 + 𝐴 𝒋 Eq. 2-6 22 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. A point lies in the xy plane as shown in figure below and has Cartesian coordinates of (x, y). The point can be specified by the position vector. 𝒓⃗ = 𝑥 𝒊 + 𝑦 𝒋 Eq. 2-7 This gives the components of the vector and its coordinates. Now, suppose we wish to add vector 𝑩⃗ to vector 𝑨⃗ from previous equation, where vector has components Bx and By. The resultant vector 𝑹⃗ = 𝑨⃗ + 𝑩⃗ is: 𝑹⃗ = 𝐴 𝒊 + 𝐴 𝒋 + 𝐵 𝒊 + 𝐵 𝒋 = (𝐴 + 𝐵 )𝒊 + 𝐴 + 𝐵 𝒋 Eq. 2-8 Because 𝑹⃗ = 𝑅 𝒊 + 𝑅 𝒋, we see that the components of the resultant vector are 𝑅 =𝐴 +𝐵 Eq. 2-9 𝑅 =𝐴 +𝐵 Eq. 2-10 The magnitude of 𝑹⃗ and the angle it makes with the x axis from its components are obtained using the following relationships: Eq. 2-11 = 𝑅 +𝑅 = (𝐴 + 𝐵 ) + 𝐴 + 𝐵 23 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. 𝑅 𝐴 +𝐵 Eq. 2-12 𝑅 tan 𝜃 = = 𝑅 𝐴 +𝐵 The same method can be extended to adding three or more vectors. Assume 𝑹⃗ = 𝑨⃗ + 𝑩⃗ + 𝑪⃗ Eq. 2-13 And 𝑹⃗ = (𝐴 + 𝐵 + 𝐶 )𝒊 + 𝐴 + 𝐵 + 𝐶 𝒋 + (𝐴 + 𝐵 + 𝐶 )𝒌 Eq. 2-14 We should notice that the latest equation was extended to three-dimensional vectors. Eq. 2-15 𝑅= 𝑅 +𝑅 +𝑅 𝑅 𝑅 𝑅 Eq. 2-16 𝜃 = ,𝜃 = ,𝜃 = 𝑅 𝑅 𝑅 Examples:- 1- Find the sum of two vectors 𝑨⃗ and 𝑩⃗ lying in the xy plane and given by: 𝑨⃗ = (𝟐. 𝟎 𝒊 + 𝟐. 𝟎 𝒋) and 𝑩⃗ = (𝟐. 𝟎 𝒊 − 𝟒. 𝟎 𝒋) 2- A particle undergoes three consecutive displacements: ∆𝒓𝟏⃗ = (𝟏𝟓 𝒊 + 𝟑𝟎𝒋 + 𝟏𝟐𝒌)𝑐𝑚, ∆𝒓𝟐⃗ = (𝟐𝟑 𝒊 − 𝟏𝟒𝒋 − 𝟓𝒌)𝑐𝑚, and ∆𝒓𝟑⃗ = (−𝟏𝟑 𝒊 + 𝟏𝟓𝒋)𝑐𝑚. Find the components of the resultant displacement and its magnitude. 3- A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. (A) Determine the components of the hiker’s displacement for each day. (B) Determine the components of the hiker’s resultant displacement 𝑹⃗ for the trip. Find an expression for 𝑹⃗ in terms of unit vectors. H.W: 1. Yes or no: Is each of the following quantities a vector? (a) force (b) temperature (c) the volume of water in a can (d) the ratings of a TV show (e) the height of a building (f) the velocity of a sports car (g) the age of the Universe 2. A book is moved once around the perimeter of a tabletop with dimensions 1.0 m X 2.0 m. If the book ends up at its initial position, what is its displacement? What is the distance traveled? 24 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. 3. Let 𝑨⃗ represent a velocity vector pointing from the origin into the second quadrant. (a) Is its x component positive, negative, or zero? (b) Is its y component positive, negative, or zero? Let 𝑩⃗ represent a velocity vector pointing from the origin into the fourth quadrant. (c) Is its x component positive, negative, or zero? (d) Is its y component positive, negative, or zero? (e) Consider the vector 𝑨⃗ + 𝑩⃗. What, if anything, can you conclude about quadrants it must be in or cannot be in? (f) Now consider the vector 𝑩⃗ - 𝑨⃗. What, if anything, can you conclude about quadrants it must be in or cannot be in? 4. The polar coordinates of a point are r = 5.50 m and  = 240°. What are the Cartesian coordinates of this point? 5. Two points in a plane have polar coordinates (2.50 m, 30.0°) and (3.80 m, 120.0°). Determine (a) the Cartesian coordinates of these points and (b) the distance between them. 6. A fly lands on one wall of a room. The lower left-hand corner of the wall is selected as the origin of a two-dimensional Cartesian coordinate system. If the fly is located at the point having coordinates (2.00, 1.00) m, (a)how far is it from the corner of the room? (b) What is its location in polar coordinates? 7. The rectangular coordinates of a point are given by (2, y), and its polar coordinates are (r, 30°). Determine y and r. 8. A plane flies from base camp to lake A, 280 km away in the direction 20.0° north of east. After dropping off supplies it flies to lake B, which is 190 km at 30.0° west of north from lake A. Graphically determine the distance and direction from lake B to the base camp. 9. A surveyor measures the distance across a straight river by the following method: starting directly across from a tree on the opposite bank, she walks 100 m along the river-bank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is 35.0°. How wide is the river? 10. A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she travels? 11. A map suggests that Atlanta is 730 miles in a direction of 5.00° north of east from Dallas. The same map shows that Chicago is 560 miles in a direction of 21.0° west of north from Atlanta. Modeling the Earth as flat, use this information to find the displacement from Dallas to Chicago. 25 Physics for Engineers (C.E 112) University of Al-Qadisiyah- College of Engineering - Civil Eng. Dept. 12. A novice golfer on the green takes three strokes to sink the ball. The successive displacements of the ball are 4.00 m to the north, 2.00 m northeast, and 1.00 m at 30.0° west of south. Starting at the same initial point, an expert golfer could make the hole in what single displacement? 13. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0° north of west with a speed of 41.0 km/h. Three hours later the course of the hurricane suddenly shifts due north, and its speed slows to 25.0 km/h. How far from Grand Bahama is the eye 4.50 h after it passes over the island? 14. A ferryboat transports tourists among three islands. It sails from the first island to the second island, 4.76 km away, in a direction 37.0° north of east. It then sails from the second island to the third island in a direction 69.0° west of north. Finally, it returns to the first island, sailing in a direction 28.0° east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands. Chapter 3 Motion in one dimension In this section, only motion in one dimension, that is, motion of an object along a straight line will be considered. From everyday experience we recognize that motion of an object represents a continuous change in the object’s position. In physics, we can categorize motion into three types: translational, rotational, and vibrational. A car traveling on a highway is an example of translational motion, the Earth’s spin on its axis is an example of rotational motion, and the back-and-forth movement of a pendulum is an example of vibrational motion. For now, only the motion along a straight line will be considered. In our study of translational motion, we use what is called the particle model and describe the moving object as a particle regardless of its size. In general, a particle is a point-like object, that is, an object that has mass but is of infinitesimal size. For example, if we wish to describe the motion of the Earth around the Sun, we can treat the Earth as a particle and obtain reasonably accurate data about its orbit. Position, Velocity, and Speed The motion of a particle is completely known if the particle’s position in space is known at all times. A particle’s position is the location of the particle with respect to a chosen reference point that we can consider to be the origin of a coordinate system. Consider a car moving back and forth along the x axis as in Figure 3-1a. When we begin collecting position data, the car is 30 m to the right of a road sign, which we will use to identify the reference position x = 0. We 26