Introduction to Mechanics Vectors and Trigonometry

Transcription

1 Introduction to Mechanics Vectors and Trigonometr Lana heridan De nza College Oct 16, 2017

2 Last time order of magnitude calculations vectors and scalars

3 Overview vectors and trigonometr how to solve problems

4 Vectors vector vector quantit indicates both an amount (magnitude) and a direction. It is represented b a real number for each possible direction, or a real number and (an) angle(s). In the lecture notes vectors are represented using bold variables.

5 How can we write vectors? - as a list vector in the x, -plane could be written ( ( ) ( ) 2 2, 1 or 2 1 or 1 ) (In some textbooks it is written 2, 1, but there are reasons not to write it this wa.)

6 along coordinate axes. These projections are called the components of the vector or its rectangular components. n vector can be completel described b its components. Consider vector a invector the x, -plane ling in could the x be plane written and making an arbitrar angle u with the positive x axis as shown in Figure 3.12a. This ( vector ) can be expressed as the sum of two other component ( vectors ) ( ) 2 2, 1 x, which is parallel to the x axis, and or 2 1, which is parallel to the axis. From Figure 3.12b, we orsee that 1 the three vectors form a right triangle and that 5 x 1. We shall often refer to the components of a vector, written x and (without the boldface notation). The component (In x some represents textbooks the projection it is written of 2, along 1, the but x there axis, are and reasons the component not to represents the projection of along the axis. These components can be positive or write negative. it this The wa.) component x is positive if the component vector x points in the positive x direction and is negative if x points in the negative x direction. similar When statement drawn is inmade the x, for -plane the component it looks like:. How can we write vectors? - as a list 1 O a u x 2 x 1 O b u x 2 x F l r v r v v f

7 along coordinate axes. These projections are called the components of the vector or its rectangular components. n vector can be completel described b its components. Consider vector a invector the x, -plane ling in could the x be plane written and making an arbitrar angle u with the positive x axis as shown in Figure 3.12a. This ( vector ) can be expressed as the sum of two other component ( vectors ) ( ) 2 2, 1 x, which is parallel to the x axis, and or 2 1 or, which is parallel to the axis. From Figure 3.12b, we see that 1 the three vectors form a right triangle and that 5 x 1. We shall often refer to the components of a vector, written x and (without the boldface notation). The component x represents the projection of along the x axis, and the component represents We sathe that projection 2 is theof x-component along the axis. of the These vector components (2, 1) and can 1be ispositive the or negative. The component -component. x is positive if the component vector x points in the positive x direction and is negative if x points in the negative x direction. similar statement is made for the component. Vector Components 1 O a u x 2 x 1 O b u x 2 x F l r v r v v f

8 Consider a vector ling in the x plane and making an arbitrar angle ith Components the positive x axis as shown in Figure 3.12a. This vector can be expressed as th m of two other component vectors x, which is parallel to the x axis, and Consider the 2 dimensional vector., whic parallel to the axis. From Figure 3.12b, we see that the three vectors form ght triangle ince the and two that vectors 5 add x 1 together. We bshall attaching often the refer head to the of one component to a vector the tail, of written the other, x and which is (without the same the as boldface adding the notation). components, The compo nt x represents the projection of along the x axis, and the component we can alwas write a vector in the x, -plane as the sum of two presents the projection of along the axis. These components can be positiv negative. component The component vectors. x is positive if the component vector x points i e positive x direction and is negative if x points in the negative x direction. ilar statement is made for the component = x +. O a u x x O b u x x

9 m of two other component vectors x, which is parallel to the x axis, and, whic parallel to the axis. From Figure 3.12b, we see that the three vectors form Components ght triangle and that 5 x 1. We shall often refer to the component a vector, written x and (without the boldface notation). The compo nt x represents the projection of along the x axis, and the component presents For the example, projection of ( along ) the ( axis. ) ( These ) components can be positiv negative. The component 2, 1 = 2, 0 + 0, 1 x is positive if the component vector x points i e positive We then x direction sa thatand 2 is is the negative x-component if x points of the in vector the negative (2, 1) and x direction. 1 ilar statement is the -component. is made for the component. O a u x x O b u x x

10 Representing Vectors: Unit Vectors We can write a vector in the x, -plane as the sum of two component vectors. To indicate the components we define unit vectors.

11 Representing Vectors: Unit Vectors We can write a vector in the x, -plane as the sum of two component vectors. To indicate the components we define unit vectors. Unit vectors have a magnitude of one unit.

12 Representing Vectors: Unit Vectors We can write a vector in the x, -plane as the sum of two component vectors. To indicate the components we define unit vectors. Unit vectors have a magnitude of one unit. In two dimensions, a pair of perpendicular unit vectors are usuall denoted i and j (or sometimes ˆx, ŷ).

13 Representing Vectors: Unit Vectors We can write a vector in the x, -plane as the sum of two component vectors. To indicate the components we define unit vectors. Unit vectors have a magnitude of one unit. In two dimensions, a pair of perpendicular unit vectors are usuall denoted i and j (or sometimes ˆx, ŷ). 2 dimensional vector can be written as v = (2, 1) = 2i + j.

14 m of two other component vectors x, which is parallel to the x axis, and, whic parallel Components to the axis. From Figure 3.12b, we see that the three vectors form ght triangle and that 5 x 1. We shall often refer to the component a vector, written Vector is the sum x and of a piece (without the boldface notation). The compo along x and a piece along : nt x represents the projection of along the x axis, and the component = x i + j. presents the projection of along the axis. These components can be positiv negative. The component x is the i-component (or x is positive if the component vector x-component) of and x points i e positive x direction and is negative if x points in the negative x direction. is the j-component (or -component) of. ilar statement is made for the component. O a u x x O b u x x

15 Vectors Properties and Operations Equalit Vectors = B if and onl if the magnitudes and directions are the same. (Each component is the same.)

16 Vectors Properties and Operations Equalit Vectors Commutative = Blaw ifof and addition onl if the magnitudes and directions are 1 B the 5 B 1 same. (Each component is the same.) ddition tion. (This fact ma seem trivial, but as ou will important when vectors are multiplied. Procedure cussed in Chapters 7 and 11.) This propert, which construction in Figure 3.8, is known as the commu + B R B B D B C D B C Figure 3.6 When vector B is Figure 3.7 Geometric construction for summing four vectors. The added to vector, the resultant R is To calculate the addition of vectors, we usuall break them into the vector that runs from the tail of resultant vector R is b definition components to the tip of... B. but how? the one that completes the polgon.

17 en Trigonometr the magnitude and direction of a tor, find its components: x x x = cos u = sin u sin θ = ; cos θ = x ; tan θ = x

18 Trigonometr, Ex 3.1 Captain Crus Harding wants to find the height of a cliff. He stands with his back to the base of the cliff, then marches straight awa from it for ft. t this point he lies on the ground and measures the angle from the horizontal to the top of the cliff. If the angle is 34.0, (a) how high is the cliff? (b) What is the straight-line distance from Captain Harding to the top of the cliff? 1 Walker, 4th ed, page 60.

19 ase, we can find the height rom Harding to the top of b>d Trigonometr, for d. Ex 3.1 f the cliff, h: e distance e cliff: b = ft Captain Crus Harding wants to find the height of a cliff. He stands with his back to the base of the cliff, then marches straight awa fromh it= for b tan 5.00 u = * 2 10 ft. 2 ft2 t tan this 34.0 point = 337 ft he lies on the ground and measures the angle from the horizontal to the top of the cliff. If the angle is 34.0 b d =, (a) how high is the cliff? (b) What is the cos u straight-line distance = 5.00 * 102 ft = 603 ft cos 34.0 from Captain Harding to the top of the cliff? se the Pthagorean 10 2 ft2 2 = 603 ft rus Harding to the e is 603 ft and its dil, the x component 37 ft. O r r x = ft r = 337 ft x nd if he had 1 Walker, walked 4th 6.00 ed, * 10 page 2 ft from 60. the cliff to make his measurement?

20 Trigonometr, Ex 3.1 d = b = ft h = b tan u = * 10 2 ft2 tan 34.0 = 337 ft (a) how high is the cliff? b cos u = 5.00 * 102 ft = 603 ft cos 34.0 an the dient r r = 337 ft O x r x = ft ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

21 the top of b = Trigonometr, Ex ft (b) What is the straight-line distance from Captain Harding to the top of the cliff? h = b tan u = * 10 2 ft2 tan 34.0 = 337 ft d = b cos u = 5.00 * 102 ft = 603 ft cos 34.0 an the dient r r = 337 ft O x r x = ft ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

22 Using Vectors: Example nd runs 100 m south then turns west and runs another 50.0 m. ll this takes him 15.0 s. What is his displacement from his starting point?

23 Vectors Properties and Operations Equalit Vectors Commutative = Blaw ifof and addition onl if the magnitudes and directions are 1 B the 5 B 1 same. (Each component is the same.) ddition When two vectors are added, the sum is indep tion. (This fact ma seem trivial, but as ou will important when vectors are multiplied. Procedure cussed in Chapters 7 and 11.) This propert, which construction in Figure 3.8, is known as the commu + B R B B D B C D B C Figure 3.6 When vector B is added to vector, the resultant R is the vector that runs from the tail of Figure 3.7 Geometric construction for summing four vectors. The resultant vector R is b definition

24 e bookkeeping ents Vectors separatel. Properties and Operations Doing addition: To add vectors, break each vector into components and sum each component independentl. (3.14) nt vector are (3.15) R B R B, we add all the tor and use the mponents with x B x x tained from its R x Figure 3.16 This geometric

25 ght triangle and that 5 x 1. We shall often refer to the component a Components vector, written x and (without the boldface notation). The compo nt x represents the projection of along the x axis, and the component presents the projection of along the axis. These components can be positiv negative. The component Vector is the sum of a x is positive if the component vector piece along x and a piece along : x points i e positive x direction and is negative if x points in the negative x direction. = x i + j. ilar statement is made for the component. O a u x Notice that x = cos θ and = sin θ. x O b u x x

26 Vectors Properties and Operations Doing addition: lmost alwas the right answer is to break each vector into components and sum each component independentl. Example w = 5 m at 36.9 above the horizontal. u = 17 m at 28.1 above the horizontal. w + u =?

27 Vectors Properties and 1 B Operations 5 B 1 Properties of ddition addition construction in Figure 3.8, is known as the commutative law of addition: Draw B, then add. (3.5) B ector B is e resultant R is rom the tail of + B = B + (commutative) D B C ( + B) + C = + (B + C) (associative) Figure 3.7 Geometric construction for summing four vectors. The shows that 1 B 5 B 1 or, in Figure 3.8 This construction 3.3 ome Properties of Vectors resultant vector R is b definition other words, that vector addition is the one that completes the polgon. dd B and C commutative. ; dd and B Figure 3.9 Geo ; then add the then add C tions for verifin to law of addition. result to. the result. ( B C ) B C D C B B C ( ) B C B B R B B C B B Draw, then add B.

28 Thinking about Vectors What can ou sa about two vectors that add together to equal zero?

29 Thinking about Vectors What can ou sa about two vectors that add together to equal zero? When can a nonzero vector have a zero horizontal component?

30 The negative of the vector is defined as the vector that when added to gives zero for the vector sum. That is, The vectors and 2 have the Vectors same Properties magnitude but point in and opposite Operations directions. Negation ubtracting Vectors The operation of vector subtraction makes use of the definition of the negative of a vector. We define the operation 2 B as vector 2 B added to vector : If u = v then u has the same magnitude as v but points in the opposite direction. Figure 3.10a. ubtraction B = + ( B) 2 B B 2 (3.7) The geometric construction for subtracting two vectors in this wa is illustrated in nother wa of looking at vector subtraction is to notice that the difference 2 B between two vectors and B is what ou have to add to the second vector would draw We B here if we were adding it to. B Vector C B is the vector we must add to B to obtain. B a B dding B to is equivalent to subtracting B from. b B C B Figure 3.10 vector B fro tor 2 B is eq vector B an site directio looking at ve

31 Vectors Properties and Operations There are several different multiplicative operations on vectors. For right now, we will onl talk about how to multipl a vector b a scalar.

32 Vectors Properties and Operations There are several different multiplicative operations on vectors. For right now, we will onl talk about how to multipl a vector b a scalar. Multiplication b a scalar uppose we want to multipl a scalar, like the number 5, b the vector: v = 2 i + 1 j The result is: 5v = (5 2) i + (5 1) j = 10 i + 5 j Each component is multiplied b the scalar. The direction of the vector doesn t change, but its magnitude increases b a factor of 5.

33 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibl be a valid equation? a = b () es (B) no

34 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibl be a valid equation? a = n () es (B) no

35 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibl be a valid equation? a = n () es (B) no

36 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibl be a valid equation? a = b + n () es (B) no

37 Quick review of Vector Expressions Let a, b, and c be vectors. Let l, m, and n be scalars. Could this possibl be a valid equation? a = n b () es (B) no

38 ummar vectors and trigonometr how to solve problems First Test this Thursda, Oct 19. Homework (will not be collected) Walker Phsics: PREV: Ch 3, onward from page 76. Questions: 2, 4, 11. Problems: 1, 5, 7, 11, 13 NEW: Ch 3, Questions: 7, 8, 9. Problems: 15, 25, 77