UCF PHY 2048 - Vectors - D2914508

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UCF PHY 2048 - Vectors

School name University of Central Florida

Course Phy 2048- General Physics Using Calculus I

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Chapter 3 - VectorsI. DefinitionII. Arithmetic operations involving vectorsA) Addition and subtraction - Graphical method- Analytical method  Vector componentsB) MultiplicationReview of angle reference systemOrigin of angle reference systemθ10º<θ1<90º90º<θ2<180ºθ2180º<θ3<270ºθ3θ4270º<θ4<360º90º180º270º0ºΘ4=300º=-60ºAngle originI. DefinitionVector quantity: quantity with a magnitude and a direction. It can be represented by a vector.Examples: displacement, velocity, acceleration.Same displacementDisplacement  does not describe the object’s path.Scalar quantity: quantity with magnitude, no direction.Examples: temperature, pressureRules:)1.3()( lawecommutativabba)2.3()()()( laweassociativcbacbaII. Arithmetic operations involving vectors- Geometrical methodabbasVector addition:basVector subtraction:)3.3()( babadVector component: projection of the vector on an axis.sin)4.3(cosaaaayxxyyxaaaaatan)5.3(22Vector magnitudeVector directionaofcomponentsScalarUnit vector: Vector with magnitude 1.No dimensions, no units.axeszyxofdirectionpositiveinvectorsunitkji ,,ˆ,ˆ,ˆ)6.3(ˆˆjaiaayxVector component- Analytical method: adding vectors by components.Vector addition:)7.3(ˆ)(ˆ)( jbaibabaryyxxVectors & Physics:-The relationships among vectors do not depend on the location of the origin of the coordinate system or on the orientation of the axes.- The laws of physics are independent of the choice of coordinate system.')8.3(''2222yxyxaaaaaMultiplying vectors:- Vector by a scalar:- Vector by a vector:Scalar product = scalar quantityasf)9.3(coszzyyxxbababaabba (dot product))90(0cos0)0(1cosbaabbaRule:)10.3(abba090cos1110cos11jkkjikkiijjikkjjiiMultiplying vectors:- Vector by a vectorVector product = vectorsinˆ)(ˆ)(ˆ)(abckabbajbaabiabbacbayxyxxzxzzyzy(cross product)MagnitudeAngle between two vectors:babacos)12.3()( baabRule:)90(1sin)0(0sin0abbabaDirection  right hand rulebacontainingplanetolarperpendicuc,1) Place a and b tail to tail without altering their orientations.2) c will be along a line perpendicular to the plane that contains a and bwhere they meet.3) Sweep a into b through the smallest angle between them.Vector productRight-handed coordinate systemxyzijkLeft-handed coordinate systemyxzijk00sin11 kkjjiijkiikijkkjkijjikkjjii)()()(0P1: If B is added to C = 3i + 4j, the result is a vector in the positive direction of the y axis, with amagnitude equal to that of C. What is the magnitude of B?ˆˆ2.319ˆˆ3ˆ5)ˆ4ˆ3(543ˆ)ˆ4ˆ3(22BjiBjjiBDCjDDjiBCBMethod 1Method 22.32sin22/2sin9.36)4/3(tanDBDBDCBθIsosceles triangleP2: A fire ant goes through three displacements along level ground: d1for 0.4m SW, d20.5m E, d3=0.6m at60º North of East. Let the positive x direction be East and the positive y direction be North. (a) What are thex and y components of d1,d2and d3? (b) What are the x and the y components, the magnitude and the directionof the ant’s net displacement? (c) If the ant is to return directly to the starting point, how far and in what directionshould it move?NEd3d245ºd1Dmdmddmdmdmdyxyxyx52.060sin6.030.060cos6.005.028.045sin4.028.045cos4.0332211(a)(b)EastofNorthmDmjijijiddDmjiijiddd8.2452.024.0tan57.024.052.0)ˆ24.0ˆ52.0()ˆ52.0ˆ3.0()ˆ28.0ˆ22.0()ˆ28.0ˆ22.0(ˆ5.0)ˆ28.0ˆ28.0(12234214d4(c) Return vector  negative of net displacement,D=0.57m, directed 25º South of WestP2kjidkjidkjidˆ2ˆ3ˆ4ˆ3ˆ2ˆˆ6ˆ5ˆ4321?,)(?)(?)(?)(212121321ddofplaneinanddtolarperpendicudofComponentddalongdofComponentczandrbetweenAnglebdddraθd1d2kjikjikjikjidddraˆ7ˆ6ˆ9)ˆ2ˆ3ˆ4()ˆ3ˆ2ˆ()ˆ6ˆ5ˆ4()(321mrrkrb88.1276912388.127cos7cos1ˆ)(2221 mdmdddddddddddddddc74.33212.374.312coscoscos1218104)(2222212111//121212121d1//d1perpmdmdddddperpperp77.865416.82.377.8)(2221221212//11P3kjidkjidIfˆˆ2ˆ5ˆ4ˆ2ˆ321?)4()(2121ddddTip: Think before calculate !!!04090cos),(4)(4)4(),()(2121212121babtolarperpendicuaplaneddtolarperpendicubddddplaneddincontainedaddyxAB130º1405090)(  AandybetweenAnglea90)(),(ˆ,ˆ)(,)(xyBAplanelarperpendicuCbecausekjangleCBAyAnglebP4:Vectors A and B lie in an xy plane. A has a magnitude 8.00 and angle 130º; B hascomponents Bx=-7.72,By= -9.20. What are the angles between the negative direction ofthe y axis and (a) the direction of A, (b) the direction of AxB, (c) the direction of Ax(B+3k)?ˆkjikjiEADkjikBEDkBADirectioncˆ61.94ˆ42.15ˆ39.18320.972.7013.614.5ˆˆˆˆ3ˆ2.9ˆ72.7ˆ3)ˆ3()(9961.9742.151ˆcos42.15)ˆ61.94ˆ42.15ˆ39.18(ˆˆ61.9761.9442.1539.18222DDjkjijDjDP5: A wheel with a radius of 45 cm rolls without sleeping along a horizontal floor. At time t1the dot P paintedon the rim of the wheel is at the point of contact

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