Physics 207, Lecture 8, Oct. 1, by Prof. Pupa GilbertRotation requires a new lexiconRelating rotation motion to linear velocityAngular displacement and velocityPowerPoint PresentationSlide 6And if w is increasing…Newton’s Laws and Circular Motion (Chapter 7)Examples of central forces are:Circular MotionMass-based separation with a centrifugeSlide 12g-forces with respect to humansAcceleration has its limitsA bad day at the lab….Non-uniform Circular MotionLecture 7, Example Gravity, Normal Forces etc.Slide 18Loop-the-loop 1Loop-the-loop 1Loop-the-loop 3Example ProblemLecture 8, Example Circular Motion Forces with Friction (recall mac = m v2 / r Ff ≤ ms n )Lecture 8, ExampleBanked CurvesSlide 26Navigating a hillLoop-the-loop 2Physics 207: Lecture 8, Pg 1Physics 207, Physics 207, Lecture 8, Oct. 1, by Lecture 8, Oct. 1, by Prof. Pupa GilbertProf. Pupa GilbertAgenda:Agenda:•Chapter 7 (Circular Motion, Dynamics III ) Uniform and non-uniform circular motion•Next time: Problem Solving and Review for MidTerm IAssignment: (NOTE special time!)Assignment: (NOTE special time!)MP Problem Set 4 due Oct. 3,Wednesday, 4 PMMP Problem Set 4 due Oct. 3,Wednesday, 4 PMMidTerm Thurs., Oct. 4, Chapters 1-6 & 7 (lite), 90 minutes, MidTerm Thurs., Oct. 4, Chapters 1-6 & 7 (lite), 90 minutes, 7:15-8:45 PM7:15-8:45 PMRooms: Rooms: B102 & B130 in B102 & B130 in Van VleckVan Vleck.. Honors class this Friday 8.50 AM, in Honors class this Friday 8.50 AM, in 53105310 Chamberlin HallChamberlin HallNotice new location!!! Notice new location!!! this Friday onlythis Friday onlyPhysics 207: Lecture 8, Pg 2Rotation requires a new lexiconRotation requires a new lexiconArc traversed s = rTangential velocity vtPeriod and frequencyAngular position Angular velocityPeriod (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 timedemorvtsPhysics 207: Lecture 8, Pg 3Relating rotation motion to linear velocityRelating rotation motion to linear velocityAssume a particle moves at constant tangential speed vt around a circle of radius r. Distance = velocity x timeOnce around… 2r = vt T or, rearranging (2/T) r = vt r = vtDefinition: UCM is uniform circular motion (constant)rvtsPhysics 207: Lecture 8, Pg 4Angular displacement and velocity Angular 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/dtif is constant, integrating = d/dt, we obtain: = + tCounter-clockwise is positive, clockwise is negativeand we note that vradial (or vr) and vz (are both zero if UCM)bb3rvtsPhysics 207: Lecture 8, Pg 5Lecture 7, Lecture 7, Exercise 1Exercise 1A Ladybug sits at the outer edge of a merry-go-round, and a June bug sits halfway between the outer one and the axis of rotation. The merry-go-round makes a complete revolution once each second. What is the June bug’s angular velocity?JLA.A.half the Ladybug’s.half the Ladybug’s.B.B.the same as the Ladybug’s.the same as the Ladybug’s.C.C.twice the Ladybug’s.twice the Ladybug’s.D.D.impossible to determine.impossible to determine.Physics 207: Lecture 8, Pg 6Lecture 7, Lecture 7, Exercise 1Exercise 1A Ladybug sits at the outer edge of a merry-go-round, and a June bug sits halfway between the outer one and the axis of rotation. The merry-go-round makes a complete revolution once each second. What is the June bug’s angular velocity?JLA.A.half the Ladybug’s.half the Ladybug’s.B.B.the same as the Ladybug’s.the same as the Ladybug’s.C.C.twice the Ladybug’s.twice the Ladybug’s.D.D.impossible to determine.impossible to determine.demoPhysics 207: Lecture 8, Pg 7Then angular velocity is no longer constant so d/dt ≠ 0Define tangential acceleration as at=dvt/dtCan we relate at to d/dt?= o + o t + t2 = o + t Many analogies to linear motion but it isn’t one-to-oneMost importantly: even if the angular velocity is constant, there is always a radial acceleration.And if And if is increasing… is increasing…12at rat rbb4Physics 207: Lecture 8, Pg 8Newton’s Laws and Circular MotionNewton’s Laws and Circular Motion(Chapter 7)(Chapter 7)Centripetal Acceleration ac = vt2/rCircular motion involves continuous radial accelerationand this means a central force.Fc = mac = mvt2/r = m2rvtracUniform circular motion involves only changes in the direction of the velocity vector, thus acceleration is perpendicular to the trajectory at any point, acceleration is only in the radial direction. Quantitatively (see text)No central force…no UCM!Physics 207: Lecture 8, Pg 9Examples of central forces are:Examples of central forces are:T on the string holding the ballGravity (moon, solar system, satellites, space crafts….)Positive charge of the nucleus, for the e-Centripetal friction acting on the ladybugBall on a loop-the-loop track ( n on the track)Physics 207: Lecture 8, Pg 10Circular MotionCircular MotionUCM enables high accelerations (g’s) in a small spaceComment: In automobile accidents involving rotation severe injury or death can occur even at modest speeds. [In physics speed doesn’t kill….acceleration (i.e., force) does.]demo: cut the stringPhysics 207: Lecture 8, Pg 11Mass-based separation with a Mass-based separation with a centrifugecentrifugeHow many g’s?Before Afterac=v2 / r and f = 104 rpm is typical with r = 0.1 m and v = r = 2 f rbb5Physics 207: Lecture 8, Pg 12Mass-based separation with a Mass-based separation with a centrifugecentrifugeHow many g’s?Before Afterac=v2 / r and f = 104 rpm is typical with r = 0.1 m and v = r = 2 f rv = (2 x 104 / 60) x 0.1 m/s =100 m/sac = 1 x 104 / 0.1 m/s2 = 10,000 gbb5Physics 207: Lecture 8, Pg 13g-forces with respect to humansg-forces with respect to humans1 g Standing1.2 g Normal elevator acceleration (up).1.5-2g Walking down stairs.2-3 g Hopping down stairs.1.5 g Commercial airliner during takeoff run.2 g Commercial airliner at rotation3.5 g Maximum acceleration in amusement park rides (design guidelines).4 g Indy cars in the second turn at
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