Physics 207, Lecture 4, Sept. 15Slide 2Vector additionVector subtractionUnit VectorsVector addition using components:Exercise 1 Vector AdditionConverting Coordinate SystemsExercise: Frictionless inclined planeResolving vectors, little g & the inclined planeDynamics II: Motion along a line but with a twist (2D dimensional motion, magnitude and directions)Instantaneous VelocityAverage AccelerationInstantaneous AccelerationMotion along a path ( displacement, velocity, acceleration )Generalized motion with non-zero acceleration:Examples of motion: ChemotaxisKinematicsSpecial CaseAnother trajectorySlide 21Exercise 2 & 3 Trajectories with accelerationExercise 2 Trajectories with accelerationExercise 3 Trajectories with accelerationTrajectory with constant acceleration along the verticalSlide 26Slide 27Home Exercise The PendulumHome Exercise The Pendulum SolutionSlide 30Physics 207: Lecture 4, Pg 1Physics 207, Physics 207, Lecture 4, Sept. 15Lecture 4, Sept. 15Goals for Chapts. 3 & 4Goals for Chapts. 3 & 4 Perform vector algebraPerform vector algebra (addition & subtraction) graphically or by addition & subtraction) graphically or by x,y & z components x,y & z components Interconvert between Interconvert between CartesianCartesian and and Polar Polar coordinatescoordinates Distinguish Distinguish position-timeposition-time graphs from graphs from particle trajectoryparticle trajectory plots plots Obtain particle velocities and accelerations from trajectory plotsObtain particle velocities and accelerations from trajectory plots Determine both Determine both magnitudemagnitude and and accelerationacceleration parallel and parallel and perpendicular to the trajectory path perpendicular to the trajectory path Solve problems with multiple accelerations in 2-dimensions Solve problems with multiple accelerations in 2-dimensions (including linear, projectile and circular motion)(including linear, projectile and circular motion) Discern different reference frames and understand how they relate Discern different reference frames and understand how they relate to particle motion in stationary and moving framesto particle motion in stationary and moving framesPhysics 207: Lecture 4, Pg 2Physics 207, Physics 207, Lecture 4, Sept. 15Lecture 4, Sept. 15Assignment: Read Chapter 5 (sections 1- 4 Assignment: Read Chapter 5 (sections 1- 4 carefully)carefully)MP Problem Set 2 MP Problem Set 2 due Wednesdaydue Wednesday (should have (should have started)started)MP Problem Set 3, Chapters 4 and 5 (available MP Problem Set 3, Chapters 4 and 5 (available today)today)Physics 207: Lecture 4, Pg 3Vector additionVector additionThe sum of two vectors is another vector.A = B + CBCABCPhysics 207: Lecture 4, Pg 4Vector subtractionVector subtractionVector subtraction can be defined in terms of addition.B - CBC= B + (-1)CB-CB - CADifferent direction and magnitude !A = B + CPhysics 207: Lecture 4, Pg 5Unit VectorsUnit VectorsA Unit Vector Unit Vector is a vector having length 1 and no unitsIt is used to specify a direction. Unit vector uu points in the direction of UUOften denoted with a “hat”: uu = û U = |U| U = |U| ûû û û xyziijjkkUseful examples are the cartesian unit vectors [ i, j, ki, j, k ] Point in the direction of the x, y and z axes.R = rx i + ry j + rz kPhysics 207: Lecture 4, Pg 6Vector addition using components:Vector addition using components:Consider, in 2D, CC = AA + BB. (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 7Exercise 1Exercise 1Vector AdditionVector AdditionA.A.{3,-4,2}{3,-4,2} B.B.{4,-2,5}{4,-2,5}C.C.{5,-2,4}{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 8Converting Coordinate SystemsConverting Coordinate SystemsIn 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 cylindrical coordinates (r,z), r is the same as the magnitude of the vector in the x-y plane [sqrt(x2 +y2)]tan-1 ( y / x )22yxr yx(x,y)rrryrxjˆ iˆ coscosyxRryrrxryxPhysics 207: Lecture 4, Pg 9Exercise: Frictionless inclined planeExercise: 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 10Resolving vectors, little g & the inclined planeg (bold face, vector) can be resolved into its x,y or x’,y’ components g = - g j g = - g cos j’ + g sin i’ The bigger the tilt the faster the acceleration…..along the inclinex’y’xygPhysics 207: Lecture 4, Pg 11Dynamics II: Motion along a line but with a twist(2D dimensional motion, magnitude and directions)Particle motions involve a path or trajectoryRecall instantaneous velocity and accelerationThese are vector expressions reflecting x, y & z motionrr = rr(t)vv = drr / dtaa = d2rr / dt2Physics 207: Lecture 4, Pg 12Instantaneous VelocityInstantaneous 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.vThe magnitude of the instantaneous velocity vector is the speed, s. (Knight uses v)s = (vx2 + vy2 + vz )1/2Physics 207: Lecture 4, Pg 13Average AccelerationAverage AccelerationThe average acceleration of particle motion reflects changes in the instantaneous velocity vector (divided by the time interval during which that change occurs). aThe average acceleration is a vector quantity directed along ∆v ( a vector! )Physics 207: Lecture 4, Pg 14Instantaneous AccelerationInstantaneous AccelerationThe instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zeroThe instantaneous acceleration is a vector with components parallel (tangential) and/or perpendicular (radial) to the tangent of the path Changes in a particle’s path may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change(Even if
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