Welcome'back'to'Physics'215 Today s'agenda: Angular(momentum Rolling(without( slipping Physics'215' Spring'2017 Lecture'1272 1
Midterm'3' Thursday'April'5th Covers'all'previous'midterms'plus'Ch 7,'8,'9' and'10.1710.3'and'10.6710.8'(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
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
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
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
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
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
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
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
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'1272 10
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'1272 11
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'1272 12
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'1272 13
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'1272 14
Demo'7 wheels Physics'215' Spring'2017 Lecture'1272 15
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'1272 16
SG'A'hammer'is'held'horizontally'and' then'released.''which'way'will'it'fall? Physics'215' Spring'2017 Lecture'1272 17
Falling'bodies'rotate'about' their'center'of'mass! Physics'215' Spring'2017 Lecture'1272 18
QuickCheck 12.11 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'1272 19
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'1272 20
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$view)diagram Physics'215' Spring'2017 Lecture'1272 21
SG'Three'identical'rectangular'blocks'are'at'rest' on'a'flat,'frictionless'table.''the'same'force'is' exerted'on'each'of'the'three'blocks'for'a'very'short' time'interval.''the'force'is'exerted'at'a'different' point'on'each'block,'as'shown. After'each'force'has'stopped'acting,'which'block s' center'of'mass'will'have'the'greatest'speed? 1. A. 2. B. 3. C. 4. A,'B,'and'C'have'the'same'C.O.M.'speed. Top$view)diagram Physics'215' Spring'2017 Lecture'1272 22
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'1272 23
Rolling'without'slipping!xcm =!xcm = vcm = vcm = Physics'215' Spring'2017 Lecture'1272 24
Rolling'without'slipping translation rotation v cm = a cm = Physics'215' Spring'2017 Lecture'1272 25
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'1272 26
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'1272 31
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'1272 32