Introduction to Mechanics Vectors in 2 Dimensions

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1 Introduction to Mechanics Vectors in 2 Dimensions Lana heridan De Anza College Jan 29, 2018

2 Last time inertia freel falling objects acceleration due to gravit

3 verview vectors in 2 dimensions some trigonometr review

4 Vectors and calars scalar A scalar quantit indicates an amount. It is represented b a real number. (Assuming it is a phsical quantit.)

5 Vectors and calars scalar A scalar quantit indicates an amount. It is represented b a real number. (Assuming it is a phsical quantit.) vector A 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.

6 How can we write vectors? - with angles Bearing angles Eample, a plane flies at a bearing of 70 N E Generic reference angles A baseball is thrown at 10 m s 1 at 30 above the horizontal.

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

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

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

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

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

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

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

14 Representing Vectors: Unit Vectors We can write a vector in the, -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 ˆ, ŷ).

15 Representing Vectors: Unit Vectors We can write a vector in the, -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 ˆ, ŷ). A 2 dimensional vector can be written as v = (2, 1) = 2i + j.

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

17 Vectors Properties and perations Equalit Vectors A = B if and onl if the magnitudes and directions are the same. (Each component is the same.)

18 Vectors Properties and perations Equalit Vectors Commutative A = Blaw ifof and addition onl if the magnitudes and directions A are 1 B the 5 B 1 A same. (Each component is the same.) Addition 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 A + B A R A B B D B C A A D B C Figure 3.6 When vector B is Figure 3.7 Geometric construction for summing four vectors. The added to vector A, 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 A to the tip of... B. but how? the one that completes the polgon.

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

20 Trigonometr, E 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. At 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.

21 ase, we can find the height rom Harding to the top of b>d Trigonometr, for d. E 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 At 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 component 37 ft. r r = ft r = 337 ft nd if he had 1 Walker, walked 4th 6.00 ed, * 10 page 2 ft from 60. the cliff to make his measurement?

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

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

24 b = ft Trigonometr, E 3.1 an h = b tan u = * 10 2 ft2 tan 34.0 = 337 ft d = (a) how high is the cliff? b cos u = 5.00 * 102 ft = 603 ft cos 34.0 tan θ = r r Multipl both sides b r : the dient r r = 337 ft olve: r = r tan θ r = ft r = (500 ft) tan(34.0 ) = 337 ft (to 3 s.f.) ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

25 the top of b = Trigonometr, E 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 r = ft ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

26 the top of b = Trigonometr, E ft an d = (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 b cos u = 5.00 * 102 ft = 603 ft cos 34.0 Pthagorean theorem r = r 2 + r 2 the dient r r = 337 ft r = ft ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

27 the top of b = Trigonometr, E ft an d = (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 b cos u = 5.00 * 102 ft = 603 ft cos 34.0 Pthagorean theorem r = r 2 + r 2 the dient r r = 337 ft olve: r = (500 ft) 2 + (337 ft) 2 = 603 ft (to 3 s.f.) r = ft ked 6.00 * 10 2 ft from the cliff to make his measurement? 1 Walker, 4th ed, page 60.

28 ummar free fall vectors in 2 dimensions trigonometr review First Test this Thursda, Feb 1. Homework Tomorrow please bring a ruler and a protractor. Walker Phsics: read Chapter 3 Ch 3, onward from page 76. Questions: 2, 4, 11. Problems: 1, 5, 7, 11, 13

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