Phsics 101 Lecture 2 Vectors Dr. Ali ÖVGÜN EMU Phsics Department www.aovgun.com
Coordinate Sstems qcartesian coordinate sstem qpolar coordinate sstem Januar 21, 2015
qfrom Cartesian to Polar coordinate sstem qfrom Polar to Cartesian sstem Direction: Magnitude Answer: (4.30 m, 216) Januar 21, 2015
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
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
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
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
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 25 0. Can be described b components? For eample: our displacement is 1.36 m in the positive direction and 0.634 m in the positive direction. Januar 21, 2015
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
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 2 2 1 = ) + θ = tan A 2 2 + A ( A ) ( ) = A A or θ = tan 1 A A θ Or, Januar 21, 2015
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
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
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 = + 3 2 ) 1/ 2 71.56 = 3.16 Januar 21, 2015
Eample 2 : Januar 21, 2015
Eample 3 : Januar 21, 2015
Multipling Vectors Januar 21, 2015
Scalar Product Januar 21, 2015
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
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
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
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 4 5 0 2 3 0
Eample 6: Januar 21, 2015
Eample 7: Januar 21, 2015
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
Problem 1: A particle undergoes three consecutive displacements: = 15 + 30 + 12, = 23 14 5 and = 13 + 15. Find the components of the resultant displacement and its magnitude. Ans: = 25 + 31 + 7 and = 40 Problem 2: The polar coordinates of a point are = 5.5 and = 240. What are the Cartesian coordinates of this point? Sln: = "#$% = 5.5 "#240 = 5.5 0.5 = 2.75 = "#$% = 5.5 sin 240 = 5.5 0.866 = 4.76 (-2.75, -4.76)m Problem 3: If = (3 4 + 4) ve = 2 + 4 8); (a) Epress in unit vector notation, = (2 ). (b) Find the magnitude and direction of. Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Problem 5 Problem 6 Problem 7 Januar 21, 2015
Problem 8 Januar 21, 2015
Januar 21, 2015
Januar 21, 2015
Januar 21, 2015