Welcome'back'to'Physics'215

Transcription

1 Welcome'back'to'Physics'215 Today s'agenda: Angular(momentum Rolling(without( slipping Physics'215' Spring'2017 Lecture'1272 1

2 Midterm'3' Thursday'April'5th Covers'all'previous'midterms'plus'Ch 7,'8,'9' and' 'and' '(work,'kinetic' and'potential'energy,'power,'center'of'mass,' Rotation'and'Torque) No'recitations'April'6 th. Prof'Manning'Office'hours: Monday'April'2:'273:30pm Wednesday'April'4:'3:3074:30pm Or'by'appointment Physics'215' Spring'2017 Lecture'1272 2

3 A'toy'race'car'of'mass'20'g'is'traveling'on' a'frictionless'loop7the7loop'track.'the' diagram'shows'the'track'from'the'side.'the' radius'of'the'loop'track'is'r=0.1'm.'the' initial'speed'of'the'car'at'the'bottom'of'the' loop'is'2.5'm/s.'what'is'the'speed'of'the' car'at'the'top'of'the'loop?'what'is'the' normal'force'on'the'car'at'the'top? al) R Physics'215' Spring'2017 Lecture'1272 3

4 A'sled'slides'along'a'horizontal'surface'with'a'coefficient'of' kinetic'friction'of'0.2.''its'velocity'at'point'a'is'8.0'm/s'and'its' velocity'at'point'b'is'5.0'm/s.'how'long'does'it'take'the'sled' to'travel'from'point'a'to'point'b?''how'far'did'it'slide?'use' the'work7kinetic'energy'and'the'impulse7momentum' theorems. Physics'215' Spring'2017 Lecture'1272 4

5 A block of mass m=2.0kg is set into motion down a rough inclined plane with speed 4.0 m/s as shown in the picture. The coefficient of kinetic friction between the block and the plane is 0.4 (take g=9.8 m/s2). What is the speed of the block after it has moved 1.0 m down the incline? v=2.0m/s 30 0 Physics'215' Spring'2017 Lecture'1272 5

6 A'child'sits'on'a'seesaw'of'length'L'='2m'with'his'father'and'his'pet'dog.'The' mass'of'the'seesaw'is'20'kg,'the'mass'of'the'father'is'50'kg,'the'mass'of'the' dog'is'5'kg'and'the'mass'of'the'child'is'20'kg.'the'father'sits'on'the'right'edge' of'the'seesaw,'while'the'child'sits'on'the'left'edge.''the'dog'is'0.5'm'from'the' father.'if'the'seesaw'is'in'equilibrium,'what'is'the'distance'from'the'pivot'to'the' child?'confirm'that'the'net'torque'about'this'pivot'point'is'zero. Physics'215' Spring'2017 Lecture'1272 6

7 Suppose'M'replaced'by'M/2'? vertical equilibrium? rotational dynamics? net torque?!f'=!" = which way rotates? initial angular acceleration? Physics'215' Spring'2017 Lecture'1272 7

8 Example:'Spinning'a'cylinder 2R F Cable wrapped around cylinder. Pull off with constant force F. Suppose unwind a distance d of cable What is final angular speed of cylinder? Use'work7KE'theorem W'= Mom.'of'inertia'of'cyl.?'! = Physics'215' Spring'2017 Lecture'1272 8

9 cylinder+cable'problem'77 constant'acceleration'method N F extended free body diagram * no torque due to N or F W * why direction of N? F W radius R torque due to! = hence'" = $% = (1/2)"t 2 = d/r; t = [(MR/F)(d/R)] 1/2! # = Physics'215' Spring'2017 Lecture'1272 9

10 Angular'Momentum can'define'rotational'analog'of'linear momentum'called'angular'momentum in'absence'of'external'torque it'will'be' conserved'in'time True'even'in'situations'where'Newton s' laws'fail'. Physics'215' Spring'2017 Lecture'

11 Definition'of'Angular'Momentum * Back to slide on rotational dynamics: m i r i2!"/!t = # i " * Rewrite, using l i = m i r i2 " :!l i /!t = # i pivot m i r i F i * Summing over all particles in body:!l/!t = # ext L ='angular'momentum'='i" Physics'215' Spring'2017 Lecture'

12 Angular'Momentum'1. Point particle: L = r p sin(!)'='m r v 'sin(!) O r vector'form'! L ='r x'p direction'of'l'given'by'right'hand'rule (into'paper'here) L'='mvr'if'v'is'at'90 0' to'r'for'single'particle! p Physics'215' Spring'2017 Lecture'

13 Angular'Momentum'2.! o rigid body: * L '='I! (fixed'axis'of'rotation/fixed'axle) *'direction' along'axis' into'paper'here' Physics'215' Spring'2017 Lecture'

14 Rotational'Dynamics! = "# $L/$t'='!! These'are'equivalent'statements! If'no'net'external'torque:'! = 0 " *'L is'constant'in'time *'Conservation+of+Angular+Momentum *'Internal'forces/torques'do'not'contribute''' to'external'torque. Physics'215' Spring'2017 Lecture'

15 Demo'7 wheels Physics'215' Spring'2017 Lecture'

16 Linear'and'rotational'motion Force Acceleration F net = Momentum p = m v Kinetic'energy K = 1 2 mv 2 F = m a Torque Angular'acceleration τ net = τ = I α Angular'momentum** L = I ω Kinetic'energy K = 1 2Iω 2 **"about"a"fixed"axle"or"axis" of"symmetry Physics'215' Spring'2017 Lecture'

17 SG'A'hammer'is'held'horizontally'and' then'released.''which'way'will'it'fall? Physics'215' Spring'2017 Lecture'

18 Falling'bodies'rotate'about' their'center'of'mass! Physics'215' Spring'2017 Lecture'

19 QuickCheck SG'Two'buckets'spin'around'in'a'horizontal' circle'on'frictionless'bearings.'suddenly,'it' starts'to'rain.'as'a'result, A. The'buckets'speed'up'because'the'potential'energy'of'the' rain'is'transformed'into'kinetic'energy. B. The'buckets'continue'to'rotate'at'constant'angular'velocity' because'the'rain'is'falling'vertically'while'the'buckets'move' in'a'horizontal'plane. C. The'buckets'slow'down'because'the'angular'momentum'of' the'bucket'+'rain'system'is'conserved. D. The'buckets'continue'to'rotate'at'constant'angular'velocity' because'the'total'mechanical'energy'of'the'bucket'+'rain' system'is'conserved. E. None'of'the'above. Physics'215' Spring'2017 Lecture'

20 General'motion'of'extended'objects Net'force'! acceleration'of'cm Net'torque'about'CM'! angular'acceleration' (rotation)'about'cm Resultant'motion'is'superposition'of'these' two'motions Total'kinetic'energy'K'='K CM' +'K rot Physics'215' Spring'2017 Lecture'

21 SG'Three'identical'rectangular'blocks'are'at'rest' on'a'flat,'frictionless'table.''the'same'force'is' exerted'on'each'of'the'three'blocks'for'a'very' short'time'interval.''the'force'is'exerted'at'a' different'point'on'each'block,'as'shown. After'the'force'has'stopped'acting'on'each'block,' which'block'will'spin'the'fastest? 1. A. 2. B. 3. C. 4. A'and'C. Top$view)diagram Physics'215' Spring'2017 Lecture'

22 SG'Three'identical'rectangular'blocks'are'at'rest' on'a'flat,'frictionless'table.''the'same'force'is' exerted'on'each'of'the'three'blocks'for'a'very'short' time'interval.''the'force'is'exerted'at'a'different' point'on'each'block,'as'shown. After'each'force'has'stopped'acting,'which'block s' center'of'mass'will'have'the'greatest'speed? 1. A. 2. B. 3. C. 4. A,'B,'and'C'have'the'same'C.O.M.'speed. Top$view)diagram Physics'215' Spring'2017 Lecture'

23 EXAMPLE'12.21'Two'Interacting'Disks A 20-cm diameter, 2.0 kg solid disk is rotating at 200 rpm. A 20-cm-diameter, 1.0 kg circular loop is dropped straight down onto the rotating disk. Friction causes the loop to accelerate until it is riding on the disk. What is the final angular velocity of the combined system? Physics'215' Spring'2017 Lecture'

24 Rolling'without'slipping!xcm =!xcm = vcm = vcm = Physics'215' Spring'2017 Lecture'

25 Rolling'without'slipping translation rotation v cm = a cm = Physics'215' Spring'2017 Lecture'

26 Rolling'without'slipping F N "F = ma CM W! "# = I$ Now'a CM' ='R$ if'no'slipping So, ma CM and'f'= Physics'215' Spring'2017 Lecture'

27 Kinetic'energy'of'rolling The'total'kinetic'energy'of'a'rolling' object'is'the'sum'of'its'rotational'and' translational'kinetic'energies: Physics'215' Spring'2017 Lecture'

28 SG.'Two'cylinders'with'the'same'radius'and'same'total' mass'roll'down'a'ramp.'in'cylinder'a,'a'set'of'8'point' masses'are'equally'spaced'in'a'circle'with'radius'r1' around'the'cylinder s'axis'of'rotation,'while'in'cylinder'b,' the'8'point'masses'are'a'distance'r2'>'r1'from'the' center.''which'cylinder'reaches'the'bottom'of'the'ramp' first? A. Cylinder'A B. Cylinder'B C. They'both'reach'the'bottom'at'the'same'time D. Not'enough'information'to'tell Physics'215' Spring'2017 Lecture'