Physics 101. Vectors. Lecture 2. h0r33fy. EMU Physics Department. Assist. Prof. Dr. Ali ÖVGÜN

Transcription

1 Phsics 101 Lecture 2 Vectors ssist. Prof. Dr. li ÖVGÜN EMU Phsics Department h0r33f

2 Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem

3 qfrom Cartesian to Polar coordinate sstem qfrom Polar to Cartesian sstem Direction: Magnitude nswer: (4.30 m, 216)

4 Vector vs. Scalar Review librar is located 0.5 mi from ou. Can ou point where eactl it is? You also need to know the direction in which ou should walk to the librar q ll phsical quantities encountered in this tet will be either a scalar or a vector q vector quantit has both magnitude (number value + unit) and direction q scalar is completel specified b onl a magnitude (number value + unit)

5 q Vectors Vector and Scalar n Displacement Quantities n Velocit (magnitude and direction) n cceleration n Force n Momentum n Weight q Scalars: n Distance n Speed (magnitude of velocit) n Temperature n Mass n Energ n Time To describe a vector we need more information than to describe a scalar Therefore vectors are more Januar 21, comple 2015

6 Important Notation q To describe vectors we will use: n The bold font: Vector is n Or an arrow above the vector: n In the pictures, we will alwas show vectors as arrows n rrows point the direction n To describe the magnitude of a vector we will use absolute value sign: or just, n Magnitude is alwas positive, the magnitude of a vector is equal to the length of a vector.

7 Properties of Vectors q Equalit of Two Vectors n Two vectors are equal if the have the same magnitude and the same direction q Movement of vectors in a diagram n n vector can be moved parallel to itself without being affected q Negative Vectors n Two vectors are negative if the have the same magnitude but are 180 apart (opposite directions) = - B; + ( - ) = 0 B

8 Describing Vectors lgebraicall Vectors: Described b the number, units and direction Vectors: Can be described b their magnitude and direction. For eample: Your displacement is 1.5 m at an angle of Can be described b components? For eample: our displacement is 1.36 m in the positive direction and m in the positive direction.

9 Components of a Vector q q The -component of a vector is the projection along the -ais cosq = = cosq q The -component of a vector is the projection along the -ais sinq = = sinq q Then, = + = +

10 More bout Components q The components are the legs of the right triangle whose hpotenuse is ì = í î = ì = ï í ïtan ïî cos( q ) sin( q and ( q ) - æ ö = ) + q = tan ç è ø ( ) ( ) = or q = tan -1 æ ç è ö ø q Or,

11 Unit Vectors z q k j i q Components of a vector are vectors q Unit vectors i-hat, j-hat, k-hat i ˆ ˆj k ˆ z q Unit vectors used to specif direction q Unit vectors have a magnitude of 1 q Then = + = = + iˆ + Magnitude + Sign ˆj Unit vector

12 dding Vectors lgebraicall q Consider two vectors q Then q If = B = + B iˆ + B iˆ + = ( iˆ + B ˆ) j + ) ˆj ( + B i + ( + B ) j = + )ˆ q so ˆj ˆj = ( + B )ˆ i + ( C = + B = C + B + B ( B iˆ + B ˆ) j = C = + B ˆ

13 Eample 1: Operations with Vectors q C = Vector is described algebraicall as (-3, 5), while vector B is (4, -2). Find the value of magnitude and direction of the sum (C) of the vectors and B. = -3 iˆ + 5 ˆj B = 4iˆ - 2 ˆj + B = (-3+ 4)ˆ i + (5-2) ˆj = 1ˆ i + 3 ˆj C =1 = 3 2 C 1/ 2 C = ( C + C ) = C -1 - q = tan ( ) = tan C 2 1 (1 3 2 = ) 1/ = 3.16

14 Eample 2 :

15 Eample 3 :

16 Multipling Vectors

17 Scalar Product

18 q q Cross Product The cross product of two vectors sas something about how perpendicular the are. Magnitude: n n n n C = B = C B sinq B q is smaller angle between the vectors Cross product of an parallel vectors = zero Cross product is maimum for perpendicular vectors = Cross products of Cartesian unit vectors: iˆ ˆj = kˆ; iˆ iˆ = 0; iˆ kˆ = - ˆ; j ˆj ˆj = 0; ˆj kˆ = iˆ kˆ kˆ = 0 z B sinq j k i j B q i k sinq Februar 25, 2019

19 Cross Product q q Direction: C perpendicular to both and B (right-hand rule) n n n n Place and B tail to tail Right hand, not left hand Four fingers are pointed along the first vector sweep from first vector into second vector B through the smaller angle between them n Your outstretched thumb points the direction of C First practice B = B? B = B? B= - B Februar 25, 2019

20 More about Cross Product q q q q q q The quantit Bsinq is the area of the parallelogram formed b and B The direction of C is perpendicular to the plane formed b and B Cross product is not commutative B= - B The distributive law The derivative of cross product obes the chain rule Calculate cross product B = ( B z - ( B + C) B z = B + d dt )ˆ i + ( B z ( B) - B z C = d B dt ) ˆj + ( B + - B db dt ) kˆ Februar 25, 2019

21 E. 4: Find: Eamples of Cross Products Solution: B? B = 0 + 4ˆ i Where: E.5: Calculate # given a force and its location F = ( 2ˆ i + 3 ˆ) j N r = (4ˆ i + 5 ˆj ) m Solution: r F = (4iˆ+ 5 ˆj) (2iˆ+ 3 ˆj) = 4iˆ 2iˆ+ 4iˆ 3ˆj+ 5ˆj 2iˆ+ 5ˆj 3ˆj = (2ˆ i + 3 ˆ) j (-iˆ + 2 ˆ) j ˆj - 3 ˆj iˆ + 0 = 2 iˆ + 3 ˆj B = -iˆ + 2 ˆj = 2ˆ i (-iˆ) + 2ˆ i 2 ˆj + 3 ˆj (-iˆ) + 3 ˆj 2 ˆj = 4kˆ + 3kˆ = 7kˆ = 0 + 4iˆ 3 ˆj+ 5 ˆj 2iˆ+ 0 = 12kˆ- 10kˆ= 2 kˆ (Nm) j B= i Februar 25, 2019 k iˆ ˆj kˆ

22 Eample 6:

23 Eample 7:

24 Summar q Polar coordinates of vector (, q) q Cartesian coordinates (, ) q Relations between them: q Beware of tan 180-degree ambiguit q Unit vectors: q ddition of vectors: q Scalar multiplication of a vector: ìï í ïî = cos( q ) = sin( q ) ì 2 = + ï í æ ö ( q) = iˆ ˆ ˆ + j+ k z C = + B = ( + B)ˆ i + ( + B C + B C = + B = 2 ( ) ( ) -1 ï tan = or q = tan ç ïî è ø a = aiˆ+ a ˆj ) ˆj

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27 Problem 5 Problem 6 Problem 7

28 Problem 8

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32 Problem 1: particle undergoes three consecutive displacements: = , = and = Find the components of the resultant displacement and its magnitude. ns: = and = 40 Problem 2: The polar coordinates of a point are = 5.5 and = 240. What are the Cartesian coordinates of this point? Sln: = "#$% = 5.5 "#240 = = 2.75 = "#$% = 5.5 sin 240 = = 4.76 (-2.75, -4.76)m Problem 3: If = ( ) ve = ); (a) Epress in unit vector notation, = (2 ). (b) Find the magnitude and direction of.

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