Physics 207, Lecture 5, Sept. 17Exercise (2D motion with acceleration) Relative Trajectories: Monkey and HunterSchematic of the problemUniform Circular Motion (UCM) is common so we have specialized termsExample Question (note the commonality with linear motion)Example QuestionRelating rotation motion to linear velocityAngular displacement and velocitySketch of q vs. timeNext part…..Well, if w is linearly increasing …Tangential acceleration?Circular motion also has a radial (perpendicular) componentNon-uniform Circular MotionSlide 15Angular motion, signsCircular MotionMass-based separation with a centrifugeg’s with respect to humansRelative motion and frames of referenceRelative VelocityWhat causes motion? (Actually changes in motion) What are forces ? What kinds of forces are there ? How are forces and motion related ?Slide 30Physics 207: Lecture 5, Pg 1Physics 207, Physics 207, Lecture 5, Sept. 17Lecture 5, Sept. 17Goals:Goals: Solve problems with multiple accelerations in 2-Solve problems with multiple accelerations in 2-dimensions dimensions (including linear, projectile and circular (including linear, projectile and circular motion)motion) Discern different reference frames and understand Discern different reference frames and understand how they relate to particle motion in stationary and how they relate to particle motion in stationary and moving framesmoving frames Recognize different types of forces and know how they Recognize different types of forces and know how they act on an object in a particle representationact on an object in a particle representation Identify forces and draw a Identify forces and draw a Free Body DiagramFree Body DiagramAssignment: HW3, (Chapters 4 & 5, due 9/25, Wednesday)Read through Chapter 6, Sections 1-4Physics 207: Lecture 5, Pg 2Exercise (2D motion with acceleration) Relative Trajectories: Monkey and HunterA. go over the monkey?B. hit the monkey?C. go under the monkey?A hunter sees a monkey in a tree, aims his gun at the monkey and fires. At the same instant the monkey lets go. Does the bullet …Physics 207: Lecture 5, Pg 3Schematic of the problemxB(t) = d = v0 cos tyB(t) = hf = v0 sin t – ½ g t2xM(t) = d yM(t) = h – ½ g t2Does yM(t) = yB(t) = hf? Does anyone want to change their answer ?(x0,y0) = (0 ,0)(vx,vy) = (v0 cos , v0 sin Bullet v0g(x,y) = (d,h)hfWhat happens if g=0 ?How does introducing g change things? MonkeyPhysics 207: Lecture 5, Pg 4Uniform Circular Motion (UCM) is common so we have specialized termsArc traversed s = rTangential velocity vtPeriod, T, and frequency, fAngular position, Angular velocity, Period (T): The time required to do one full revolution, 360° or 2 radiansFrequency (f): 1/T, number of cycles per unit timeAngular velocity or speed = 2f = 2/T, number of radians traced out per unit time (in UCM average and instantaneous will be the same)rvtsPhysics 207: Lecture 5, Pg 5Example Question (note the commonality with linear motion) A horizontally mounted disk 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds.1 What is the period of the initial rotation?2 What is initial angular velocity?3 What is the tangential speed of a point on the rim during this initial period?4 Sketch the angular displacement versus time plot.5 What is the average angular velocity?6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the angular acceleration be?7 What is the magnitude and direction of the acceleration after 10 seconds?Physics 207: Lecture 5, Pg 6Example QuestionExample Question A horizontal turntable 2 meters in diameter spins at constant angular speed such that it first undergoes 10 counter clockwise revolutions in 5 seconds and then, again at constant angular speed, 2 counter clockwise revolutions in 5 seconds.1 What is the period of the turntable during the initial rotationT (time for one revolution) = t /# of revolutions/ time = 5 sec / 10 rev = 0.5 s2 What is initial angular velocity? = angular displacement / time = 2 f = 2 / T = 12.6 rad / s3 What is the tangential speed of a point on the rim during this initial period? We need more…..Physics 207: Lecture 5, Pg 7Relating rotation motion to linear velocityIn UCM a particle moves at constant tangential speed vt around a circle of radius r (only direction changes). Distance = tangential velocity · timeOnce around… 2r = vt Tor, rearranging (2/T) r = vt r = vtDefinition: If UCM then const ant3 So vT = r = 4 rad/s 1 m = 12.6 m/srvts4 A graph of angular displacement () vs. timePhysics 207: Lecture 5, Pg 8Angular displacement and velocity Arc traversed s = rin time t then s = rso s / t = ( / t) rin the limit t 0one gets ds / dt = d/ dt r vt = r = d/ dt if is constant, integrating = d/ dt, we obtain: = + tCounter-clockwise is positive, clockwise is negativervtsPhysics 207: Lecture 5, Pg 9Sketch of Sketch of vs. timevs. timetime (seconds) (radians) = + t = + 5rad = + t = rad + rad5 Avg. angular velocity = / t = 24 /10 rad/sPhysics 207: Lecture 5, Pg 10Next part…..Next part…..6 If now the turntable starts from rest and uniformly accelerates throughout and reaches the same angular displacement in the same time, what must the “tangential acceleration” be?Physics 207: Lecture 5, Pg 11Then angular velocity is no longer constant so d/dt ≠ 0Define tangential acceleration as at = dvt/dt = r d/dtSo s = s0 + (ds/dt)0 t + ½ at t2 and s = r We can relate at to d/dt = o + o t + t2 = o + t Many analogies to linear motion but it isn’t one-to-oneNote: Even if the angular velocity is constant, there is always a radial acceleration.Well, if is linearly increasing …at r12at rPhysics 207: Lecture 5,
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