Physics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN

Transcription

1 Phsics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN EMU Phsics Department

2 Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem Januar 21, 2015

3 qfrom Cartesian to Polar coordinate sstem qfrom Polar to Cartesian sstem Direction: Magnitude Answer: (4.30 m, 216) Januar 21, 2015

4 Vector vs. Scalar Review A 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 All phsical quantities encountered in this tet will be either a scalar or a vector q A vector quantit has both magnitude (number value + unit) and direction q A scalar is completel specified b onl a magnitude (number value + unit) Januar 21, 2015

5 q Vectors Vector and Scalar n Displacement Quantities n Velocit (magnitude and direction) n Acceleration 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 A is A n Or an arrow above the vector: n In the pictures, we will alwas show vectors as arrows n Arrows point the direction n To describe the magnitude of a vector we will use absolute value sign: or just A, A n Magnitude is alwas positive, the magnitude of a vector is equal to the length of a vector. A Januar 21, 2015

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 An 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 A directions) A = B; A + ( A) = 0 B Januar 21, 2015

8 Describing Vectors Algebraicall 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. Januar 21, 2015

9 Components of a Vector θ q The -component of a vector is the projection along the -ais A cosθ = A = Acosθ A q The -component of a vector is the projection along the -ais A sinθ = A = Asinθ A q Then, A = A + A A = A + A Januar 21, 2015

10 More About Components q The components are the legs of the right triangle whose hpotenuse is A A = A = A = tan Acos( θ ) Asin( Aθ A A and ( θ ) A = ) + θ = tan A A ( A ) ( ) = A A or θ = tan 1 A A θ Or, Januar 21, 2015

11 Unit Vectors z θ k j i q Components of a vector are vectors q Unit vectors i-hat, j-hat, k-hat iˆ ˆj k ˆ z q Unit vectors used to specif direction q Unit vectors have a magnitude of 1 q Then A = A + A A = A = A + A A iˆ + Magnitude + Sign A ˆj Unit vector Januar 21, 2015

12 Adding Vectors Algebraicall q Consider two vectors q Then q If A = B = A + B A iˆ + B iˆ + = ( A iˆ + A B ˆ) j + ) ˆj ( A + B i + ( A + B ) j A = A + A )ˆ C = A + q so ˆj ˆj A = ( A + B )ˆ i + ( A C = A+ B = C A + B + B ( B iˆ + B ˆ) j = B ˆ Januar 21, 2015

13 Eample 1: Operations with Vectors q C = Vector A 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 A and B. A = 3 iˆ + 5 ˆj B = 4iˆ 2 ˆj A + B = ( 3+ 4)ˆ i + (5 2) ˆj = 1ˆ i + 3 ˆj C =1 = 3 2 C 1/ 2 C = ( C + C ) = C 1 θ = tan ( ) = tan C 2 1 (1 3 2 = ) 1/ = 3.16 Januar 21, 2015

14 Eample 2 : Januar 21, 2015

15 Eample 3 : Januar 21, 2015

16 Multipling Vectors Januar 21, 2015

17 Scalar Product Januar 21, 2015

18 q q Cross Product The cross product of two vectors sas something about how perpendicular the are. Magnitude: n n n n C = A B = C AB sinθ A B θ 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 sinθ j k i j B θ i k A A sinθ Februar 13, 2017

19 Cross Product q q Direction: C perpendicular to both A and B (right-hand rule) n Place A and B tail to tail n n n Right hand, not left hand Four fingers are pointed along the first vector A sweep from first vector A into second vector B through the smaller angle between them n Your outstretched thumb points the direction of C First practice A B = B A? A B = B A? r r r r A B= B A Februar 13, 2017

20 More about Cross Product q q q q q q The quantit ABsinθ is the area of the parallelogram formed b A and B The direction of C is perpendicular to the plane formed b A and B Cross product is not commutative r r r r A B= B A The distributive law The derivative of cross product obes the chain rule Calculate cross product A B = A ( B + C) = d dt A B + ( A B) A C = da B + dt ( A B A B )ˆ i + ( A B A B ) ˆj + ( A B z z z z A A B db dt ) kˆ Februar 13, 2017

21 E. 4: Find: Eamples of Cross Products Solution: r r A B? A B = 0 + 4ˆ i Where: E.5: Calculate r F given a force and its location F = ( 2ˆ i + 3 ˆ) j N r = (4ˆ i + 5 ˆj ) m r r Solution: F = (4iˆ+ 5 ˆj) (2iˆ+ 3 ˆj) = 4iˆ 2iˆ+ 4iˆ 3ˆj+ 5ˆj 2iˆ+ 5ˆj 3ˆj A = (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 r r A B= i Februar 13, 2017 k iˆ ˆj kˆ

22 Eample 6: Januar 21, 2015

23 Eample 7: Januar 21, 2015

24 Summar q Polar coordinates of vector A (A, θ) q Cartesian coordinates (A, A ) q Relations between them: q Beware of tan 180-degree ambiguit q Unit vectors: q Addition of vectors: q Scalar multiplication of a vector: A A = Acos( θ ) = Asin( θ ) 2 A = A + A A A ( θ) A = Aiˆ ˆ ˆ + Aj+ Ak z C = A+ B = ( A + B)ˆ i + ( A + B C A + B C = A + B = 2 ( ) ( ) 1 tan = or θ = tan A A aa = aaiˆ+ aa ˆj ) ˆj Januar 21, 2015

25 Problem 1: A particle undergoes three consecutive displacements: = , = and = Find the components of the resultant displacement and its magnitude. Ans: = 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. Januar 21, 2015

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32 Problem 5 Problem 6 Problem 7 Januar 21, 2015

33 Problem 8 Januar 21, 2015

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