Units, Physical Quantities, and Vectors. 8/29/2013 Physics 208

Transcription

1 Chapter 1 Units, Physical Quantities, and Vectors 1

2 Goals for Chapter 1 To learn three fundamental quantities of physics and the units to measure them To keep track of significant figures in calculations To understand vectors and scalars and how to add vectors graphically To determine vector components and how to use them in calculations To understand unit vectors and how to use them with components to describe vectors To learn two ways of multiplying vectors Review of the basic calculus that you will need in the course 2

3 Reminders SmartPhysics pre-lectures and check points are due before lecture. (as of this time I see that there are 113/156 students signed up for SP.) iclicker? 1- yes, 2-not yet. Mastering Physics homework assignments are due Monday s at 8 am for the preceding week. (There is an introduction to MP that is due Saturday at noon. So far 88 of you have claimed your MP access. If you special issues with MP access, let me know.) iclicker? 1-yes, 2-not yet. 3

4 More. WebAssign labs begin next week. Before your recitation meets, you should read the lab description in the on-line lab manual on WebAssign and take the pre-lab quiz. At this time I see 116 of you have claimed your WA access. Those still having issues getting access, let me know. iclicker? 1-yes, 2-not yet. 4

5 Lastly.. iclicker2. As mentioned in the previous class meeting, we will be using the iclicker throughout the term to get you input on various questions that may come up. If you haven t been able to obtain your iclicker yet please work to have this sorted out by next week. 5

6 Estimates and orders of magnitude An order-of-magnitude estimate of a quantity gives a rough idea of its magnitude. 6

7 Vectors and scalars A scalar quantity can be described by a single number. A vector quantity has both a magnitude and a direction in space. In this book, a vector quantity is represented in boldface italic type with an arrow over it: A. The magnitude of A is written as A or A. 7

8 Drawing vectors Draw a vector as a line with an arrowhead at its tip. The length of the line shows the vector s magnitude. The direction of the line shows the vector s direction. Figure 1.10 shows equal-magnitude vectors having the same direction and opposite directions. 8

9 Adding two vectors graphically Two vectors may be added dgraphically using either the parallelogram l method or the head-to-tail method. 9

10 Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order. 10

11 Subtracting vectors Figure 1.14 shows how to subtract vectors. 11

12 Multiplying a vector by a scalar If c is a scalar, the product ca has magnitude c A. Figure 1.15 illustrates multiplication of a vector by a positive scalar and a negative 8/29/2013 scalar. Physics

13 Components of a vector Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors. Any vector can be represented by an x-component p A x and a y- component A y. Use trigonometry to find the components of a vector: A x =Acos θ and A y = Asin i θ, where θ is measured dfrom the +x-axis toward dth the +y-axis. 13

14 Positive and negative components Figure 1.18 The components of a vector can be positive or negative numbers, as shown in the figure. 14

15 Finding components We can calculate the components of a vector from its magnitude and direction. Follow Example

16 Calculations using components 2 2 A A= Ax+ Ay and tanθ = A y We can use the components of a vector to find its magnitude and direction: We can use the components of a set of vectors to find the components of their sum: R = A + B + C + L, R = A + B + C + L x x x x y y y y Refer to Problem-Solving Strategy 1.3. x 16

17 Adding vectors using their components 17

18 Unit vectors A unit vector has a magnitude of 1 with no units. The unit vector î points in the j +x-direction, jpoints in the +y- k direction, and kpoints in the +z-direction. Any vector can be expressed in terms of its components as j k A =A x î+ A y j + A z. y k Follow Example

19 The scalar product r r AB= AB cos φ. The scalar product (l (also called the dot product ) of two vectors is Figures 1.25 and 1.26 illustrate the scalar product. 19

20 Calculating a scalar product In terms of components, r r AB= A B + A B + A B x x y y z z. Example 1.10 shows how to calculate a scalar product in two ways. [Insert figure 1.27 here] 20

21 Finding an angle using the scalar product Example 1.11 shows how to use components to find the angle between two vectors. 21

22 The vector product The vector product ( cross product ) of two vectors has magnitude r r A B = ABsinφ and dthe righthand rule gives its direction. See Figures 1.29 and

23 Calculating the vector product Use ABsinφ to find the magnitude and the right-hand rule to find the direction. Refer to Example

24 Calculus Summary The Derivative of lim Δx 0 f ( x2) x 2 1 a f ( x x function, 1 ) = df dx f(x) is defined as follows: where Δx = x 2 x 1 Graphically this corresponds to the slope of the tangent to the curve f(x) () at the point x. 24

25 Figure 2.3

26 For a simple polynomial function like f ( x ) = kx we can carry out this limiting process and we find df kx2 kx1 x2 x1 = limδ x 0 = k = k dx x x x x for a general polynomial of df dx = knx n f ( x) = kx n we find 26

27 The integral of a function the definite integral We can define a second operation on these polynomial functions investigating the area under an arbitrary curve. will call the integral of note it with the following symbol. b a the function f ( x) dx This is what we f ( x) from a to b, and we will 27

28 Figure 2.28

29 The area under a simple polynomial curve like f ( x ) = kx + b, a "straight line"can be worked out easily. b a f ( x) dx = ka( b a) + Then for a general polynomial of b a f ( x) dx b = a kx n dx = ( kb ka)( b a) = k( b a 2 2 the form, 1 n+ 1 1 n+ 1 n+ 1 ka n+ 1 kb f ( x) = 2 ) n kx for n any positive integer, 29

30 The indefinite integral f n+ 1 kx n 1 n+ 1 ( x ) dx = kx dx = 1 kx + constant of integration n 30

31 The connection between differentiation and integration The operations of integration and differentiation of function when we applied in series es to the tesame function ucto return you to that tatfunction. ucto. d dx df ( x) f ( x) dx = f ( x) and the dx = f ( x) dx + c a 31

32 Calculus summary The derivative of a polynomial f ( x) = df ( x ) n 1 = knx dx The integral of a 1 n+ 1 f ( x) dx = n + 1kx + c and df ( x) dx = f ( x ) + dx polynomial f c ( x ) = 32 kx kx n n