普通物理 General Physics 3 – Vectors Quantities 郭艷光Yen-Kuang Kuo 國立彰化師大物理系暨光電科技研究所 電子郵件: ykuo@cc.ncue.edu.tw 網頁: http://ykuo.ncue.edu.tw

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Outline 3-1 What Is Physics? 3-2 Vectors and Scalars 3-3 Adding Vectors Geometrically 3-4 Components of Vectors 3-5 Unit Vectors 3-6 Adding Vectors by Components 3-7 Vectors and the Laws of Physics 3-8 Multiplying Vectors 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-1 What Is Physics? In Physics, we have parameters that can be completely described by a number and are known as “scalars”. Temperature and mass are such parameters. Other physical parameters require additional information about direction and are known as “vectors”. Examples of vectors are displacement, velocity, and acceleration. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-2 Vectors and Scalars Vector: A vector has magnitude and direction. A vector that represents a displacement is called a displacement vector. All three arrows have the same magnitude and direction and thus represent the same displacement. (b) All three paths connecting the two points correspond to the same displacement vector. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-2 Vectors and Scalars Scalar: It doesn’t involve a direction and doesn’t “point” in the spatial sense. Ex: Temperature, pressure, energy, mass, and time. Deal with them by the rules of ordinary algebra. A single value, with a sign, specifies a scalar. Ex: A temperature of 40℉. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-3 Adding Vectors Geometrically As in the vector diagram, The net displacement of these two displacements is a single displacement from A to C. We call AC the vector sum (or resultant) of the vectors AB and BC. AC is the vector sum of the vectors AB and BC. (b) The same vectors relabeled. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-3 Adding Vectors Geometrically Adding to gives the same result as adding to ; that is, 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-3 Adding Vectors Geometrically If we want to add vectors , , and ,we can add and first and then add their vector sum to . We can also add and first and then add that sum to . 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-3 Adding Vectors Geometrically 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-1 In an orienteering class, you have the goal of moving as far (straight-line distance) from base camp as possible by making three straight-line moves. You may use the following displacements in any order: (a) , 2.0 km due east (directly toward the east); (b) , 2.0 km 30° north of east (at an angle of 30° toward the north from due east); (c) 1.0 km due west. Alternatively, you may substitute either for or for . What is the greatest distance you can be from base camp at the end of the third displacement? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-1 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-4 Components of Vectors A component of a vector is the projection of the vector on an axis. The projection of a vector on an x axis is its x component, and similarly the projection on the y axis is the y component. The process of finding the components of a vector is called resolving the vector. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-4 Components of Vectors 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-2 A small airplane leaves an airport on an overcast day and is later sighted 215 km away, in a direction making an angle of 22° east of due north. How far east and north is the airplane from the airport when sighted? Key idea: we are given the magnitude (215 km) and the angle (22° east of due north) of a vector and need to find the components of the vector. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-2 Solutions: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-3 For two decades, spelunking teams sought a connection between the Flint Ridge cave system and Mammoth Cave, which are in Kentucky. When the connection was finally discovered, the combined system was declared the world’s longest cave(more than 200 km long). The team that found the connection had to crawl, climb, and squirm through countless passages, traveling a net 2.6 km westward, 3.9 km southward, and 25 m upward. What was their displacement from start to finish? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-3 Key idea: we have the components of a three-dimensional vector, and we need to find the vector’s magnitude and two angles to specify the vector’s direction. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-3 Solution: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-5 Unit Vectors A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to specify a direction. Right-handed coordinate system: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-5 Unit Vectors The quantities and are vectors, called the vector components of . The quantities and are scalars, called the scalar components of 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-6 Adding Vectors by Components To add vectors and , we must (1) Resolve the vectors into their scalar components. (2) Combine these scalar components , axis by axis, to get the components of the sum . (3) Combine the components of to get itself. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-6 Adding Vectors by Components 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-6 Adding Vectors by Components Subtracting Vectors by Components x O y 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-4 Figure shows the following three vectors: What is their vector sum , which is also shown? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-4 Solutions: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 According to experiments, the desert ant shown in the chapter opening photograph keeps track of its movements along a mental coordinate system. When it wants to return to its home nest, it effectively sums its displacements along the axes of the system to calculate a vector that points directly home. As an example of the calculation, let’s consider an ant making five runs of 6.0 cm each on an xy 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 coordinate system, in the directions shown in Fig. 3-17a, starting from home. At the end of the fifth run, what are the magnitude and angle of the ant’s net displacement vector , and what are those of the home-ward vector that extends from the ant’s final position back to home? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 Key idea: (1) Find the net displacement . (2) Evaluate this sum for the x components alone. (3) We construct from its x and y components. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 Solutions: A search path of five runs. (b) The x and y components of . (c) Vector points the way to the home nest. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-5 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-6 Here is a problem involving vector addition that cannot be solver directly on a vector-capable calculator, using the vector notation of the calculator. A fellow camper is to walk away from you in a straight line (vector ), turn, walk in a second straight line (vector ) and then stop. How far must you walk in a straight line (vector ) to reach her? The three vectors are related by 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-6 has a magnitude of 22.0 m and is directed at an angle of -47.0° (clockwise) from the positive direction of an x axis. has a magnitude of 17.0 m and is directed counterclockwise from the positive direction of the x axis by angle . is in the positive direction of the x axis. What is the magnitude of ? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-6 Solutions: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-6 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

3-7 Vectors and the Laws of Physics The point is that we have great freedom in choosing a coordinate system, because the relations among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes. This is also true of the relations of physics; they are all independent of the choice of coordinate system. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors Multiplying a Vector by a Scalar: If we multiply a vector by a scalar s, we get a new vector. Magnitude of new vector is the product of the magnitude of and the absolute value of s. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors Direction of new vector is the direction of . Multiplying a Vector by a Vector : There are two ways to multiply a vector by a vector: one way produces a scalar (called the scalar product ), and the other produces a new vector (called the vector product ). 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors The Scalar product of two Vectors: The scalar product of the vectors and is written as and defined to be 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors There are only scalars on the right side of equation. is also known as the dot product and is spoken as “a dot b.” A dot product can be regarded as the product of two quantities: (1) The magnitude of one of the vectors. (2) The scalar component of the second vector along the direction of the first vector. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors The commutative law applies to a scalar product, so we can write Two vectors are in unit-vector notation, we write their dot product as 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors The Vector product of two Vectors: The vector product of and , written , produces a third vector whose magnitude is is also known as the cross product, and in speech it is “a cross b.” The direction of is decided by right-hand rule. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors The sense of the vector is given by the right hand rule: (1)Place the vector and tail to tail. (2)Rotate in the plane P along the shortest angle so that it coincides with . 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors (3)Rotate the fingers of the right hand in the same direction. (4)The thumb of the right hand gives the sense of . The vector product of two vectors is also known as the “cross” product. 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 3-8 Multiplying Vectors 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-7 What is the angle between and ? (Caution: Although many of the following steps can be bypassed with a vector-capable calculator, you will learn more about scalar products if, at least here, you use these steps. ) 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-7 Solutions: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-8 Vector lies in the xy plane, has a magnitude of 18 units, and points in a direction 250° from the positive direction of the +x axis. Also, vector has a magnitude of 12 units and points along the positive direction of the +z axis. What is the vector product ? 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-8 Solutions: From the magnitude we write is at an angle of 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 Example 3-9 If and , what is ? Solutions: 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授

普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授 End of chapter 3! 2018/9/20 普通物理講義-3 / 國立彰化師範大學物理系/ 郭艷光教授