Lecture 4Coordinate Systems and vectorsVectors look like...Vectors act like…ScalarsVectors and 2D vector addition2D Vector subtractionSlide 13Unit VectorsVector addition using components:Example Vector AdditionConverting Coordinate Systems (Decomposing vectors)Decomposing vectors into components A mass on a frictionless inclined planeSlide 19Motion in 2 or 3 dimensionsInstantaneous VelocityAverage AccelerationTangential ║ & radial accelerationKinematicsSpecial CaseAnother trajectorySlide 30Exercises 1 & 2 Trajectories with accelerationExercise 1 Trajectories with accelerationExercise 2 Trajectories with accelerationSlide 34Lecture 4Physics 207: Lecture 4, Pg 1Lecture 4Today: Ch. 3 (all) & Ch. 4 (start) Perform vector algebra (addition and subtraction) Interconvert between Cartesian and Polar coordinates Work with 2D motion• Deconstruct motion into x & y or parallel & perpendicular• Obtain velocities• Obtain accelerationsDeduce components parallel and perpendicular to the trajectory pathPhysics 207: Lecture 4, Pg 5Coordinate Systems and vectorsIn 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-In 2 dimensions there are two commonly used systems, Cartesian Coordinates (x,y) Circular Coordinates (r,)In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)Physics 207: Lecture 4, Pg 7Vectors look like...There are two common ways of indicating that something is a vector quantity:Boldface notation: A“Arrow” notation:A orAAPhysics 207: Lecture 4, Pg 8Vectors act like… Vectors have both magnitude and a direction Vectors: position, displacement, velocity, acceleration Magnitude of a vector A ≡ |A|For vector addition or subtraction we can shift vector position at will (NO ROTATION)Two vectors are equal if their directions, magnitudes & units match. ABCA = CA ≠B, B ≠ CPhysics 207: Lecture 4, Pg 9ScalarsA scalar is an ordinary number. A magnitude ( + or - ), but no direction May have units (e.g. kg) but can be just a number No bold face and no arrow on top.The product of a vector and a scalar is another vector in the same “direction” but with modified magnitudeABA = -0.75 BPhysics 207: Lecture 4, Pg 11Vectors and 2D vector additionThe sum of two vectors is another vector.A = B + CBCABCPhysics 207: Lecture 4, Pg 122D Vector subtractionVector subtraction can be defined in terms of addition.B - CBC= B + (-1)CPhysics 207: Lecture 4, Pg 132D Vector subtractionVector subtraction can be defined in terms of addition.B - CB-CB-CB - C= B + (-1)CB+CDifferent direction and magnitude !Physics 207: Lecture 4, Pg 14Unit VectorsA Unit Vector points : a length 1 and no unitsGives a direction. Unit vector uu points in the direction of UU Often denoted with a “hat”: u = û U = |U| û û xyziijjkkUseful examples are the cartesian unit vectors [ i, j, k ] or Point in the direction of the x, y and z axes.R = rx i + ry j + rz k orR = x i + y j + z k]ˆ,ˆ,ˆ[ zyxPhysics 207: Lecture 4, Pg 15Vector addition using components:Consider, in 2D, C = A + B. (a) CC = (Ax ii + Ay jj ) + (Bx i i + By jj ) = (Ax + Bx )ii + (Ay + By ) (b) CC = (Cx ii + Cy jj )Comparing components of (a) and (b): Cx = Ax + Bx Cy = Ay + By |C| =[ (Cx)2+ (Cy)2 ]1/2 CCBxAAByBBAxAyPhysics 207: Lecture 4, Pg 16ExampleVector AdditionA.A.{{3,-4,2} B. {4,-2,5}C. {5,-2,4}D. None of the above Vector A = {0,2,1} Vector B = {3,0,2} Vector C = {1,-4,2}What is the resultant vector, D, from adding A+B+C?Physics 207: Lecture 4, Pg 17Converting Coordinate Systems (Decomposing vectors)In polar coordinates the vector R = (r,)In Cartesian the vector R = (rx,ry) = (x,y)We can convert between the two as follows:•In 3D tan-1 ( y / x )22yxr yx(x,y)rryrxjˆ iˆ sincosyxrryrrxryx222zyxr Physics 207: Lecture 4, Pg 18Decomposing vectors into componentsA mass on a frictionless inclined planeA block of mass m slides down a frictionless ramp that makes angle with respect to horizontal. What is its acceleration a ?maPhysics 207: Lecture 4, Pg 19Decomposing vectors into componentsA mass on a frictionless inclined planeA block of mass m slides down a frictionless ramp that makes angle with respect to horizontal. What is its acceleration a ?mg = - g j g sin -g cos xˆyˆPhysics 207: Lecture 4, Pg 20Motion in 2 or 3 dimensionsPosition DisplacementVelocity (avg.)Acceleration (avg.)ffiitrtr , and , ifrrrtravg.vtvaavg.Physics 207: Lecture 4, Pg 22Instantaneous VelocityBut how we think about requires knowledge of the path.The direction of the instantaneous velocity is along a line that is tangent to the path of the particle’s direction of motion.vPhysics 207: Lecture 4, Pg 23Average AccelerationThe average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). aInstantaneous accelerationPhysics 207: Lecture 4, Pg 25Tangential ║ & radial accelerationAcceleration outcomes:If parallel changes in the magnitude of (speeding up/ slowing down)Perpendicular changes in the direction of (turn left or right)aaa=+22 Traaa aar||aaTavvvPhysics 207: Lecture 4, Pg 26KinematicsIn 2-dim. position, velocity, and acceleration of a particle: r = x i + y j v = vx i + vy j (i , j unit vectors ) a = ax i + ay jAll this complexity is hidden away inr = r(t) v = dr / dt a = d2r / dt222dtxdax22dtydaydtdxvxdtdyvy)( txx )( tyy 22102210 v)( :accel.y constant if with, v)( :accel. constant x if with,00tatytytatxtxyyxxPhysics 207: Lecture 4, Pg 28Special CaseThrowing an object with x along the horizontal and y along the vertical. x and y motion both coexist and t is common to bothLet g act in the –y direction, v0x= v0 and v0y= 0 yt04xt = 04ytx04x vs ty vs tx vs yPhysics 207: Lecture 4, Pg 29Another trajectoryx vs yt = 0t =10Can you identify the dynamics in this picture?How many distinct regimes are there?Are vx or vy = 0 ? Is vx >,< or = vy ?yxPhysics 207: Lecture 4, Pg 30Another trajectoryx vs yt = 0t =10Can
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