Transcription
Classical MechanicsLecture 20Today’s Concepts:A) Angular MomentumB) PrecessionMechanics Lecture 20, Slide 1
Torques and GyroscopesMechanical Universe: Episode 19
Student on StoolThere are no external torques acCng on the student-stoolsystem, so angular momentum will be conserved.IniCally: Li Ii ωiFinally: Lf If ωfωfωiIiIfLiLfMechanics Lecture 20, Slide 3
Clicker QuestionA student sits on a freely turning stool and rotates withconstant angular velocity ω1. She pulls her arms in and herangular velocity increases to ω2.In doing this her kineCc energy:A) IncreasesB) Decreases C) Stays the sameωfωiIiIfLiLfMechanics Lecture 20, Slide 4
(using L Iω)L is conserved:Ii IfK increases!Kf KiωfωiIiIfLiLfMechanics Lecture 20, Slide 5
Since the student has to force her arms to move towardher body, she must be doing posiCve work!The work/kineCc energy theorem states that this will increase thekineCc energy of the system!ωfωiIiIfLiLfMechanics Lecture 20, Slide 6
I would like to see the student siRng in the chair with a top in a real life demo.Mechanics Lecture 20, Slide 7
Clicker Question (like CheckPoint)A puck slides in a circular path on a horizontal fricConless table. It isheld at a constant radius by a string threaded through a fricConlesshole at the center of the table. If you pull on the string such that theradius decreases by a factor of 2, by what factor does the angularvelocity of the puck increase?A) 2B) 4C) 8Mechanics Lecture 20, Slide 8
Clicker QuestionSince the string is pulled through a hole at the center of rotation,there is no torque: Angular momentumis conserved. RMechanics Lecture 20, Slide 9
Food for thought (not on any test)RWe just usedButto findSo how do we get an α without a τ ?usually 0,but not nowMechanics Lecture 20, Slide 10
Food for thought (not on any test)Now suppose τEXT 0:So in this case we can have an α without an external torque!Mechanics Lecture 20, Slide 11
PrecessionThe magnitude of the torque about the pivot is τ Mgd.The direc on of this torque at the instant shown is out of thepage (using the right hand rule). extd L dtThe change in angular momentum at the instant shown must also be out of thepage!Mechanics Lecture 20, Slide 13
PrecessionAerial ViewdφτEXTpivotΩMechanics Lecture 20, Slide 14
PrecessionIn this example:DirecCon:The Cp of L movesin the direcCon of τ.Mechanics Lecture 20, Slide 15
CheckPointA disk is spinning with angular velocity ω on a pivoted horizontalaxle as shown. Gravity acts down. In which direcCon does precessioncause the disk to move?A) Out of the page B) Into the pageC) UpD) DownTorque is outof the pageMechanics Lecture 20, Slide 16
CheckPointIn which direcCon doesApoint?BMechanics Lecture 20, Slide 17
CheckPointIn which direcCon does precession cause the disk to move?A) Into the page B) Out of the pageC) UpD) DownTorque is outof the pageA) The torque provided by gravity is going out of the page.Therefore, L must change out of the page. However, since Lis in the opposite direc on of the axel when drawn fromthe pivot, precession must make the disk go into the page.B) Gravity exerts a net torque out of the page, so the diskprecesses in a direc on out of the page.D) Torque is down so precession is e/instructor/charts/stdhw choice histogram.aspx?qid 593054Mechanics Lecture 20, Slide 18
Clicker QuestionA disk is spinning with angular velocity ω on a pivotedhorizontal axle as shown. If the mass of the disk were doubled but itsradius and angular velocity were kept the same:A) The angular momentum of the disk doublesB) The torque about the pivot doublesC) Both A and BL IωMechanics Lecture 20, Slide 19
CheckPointA disk is spinning with angular velocity ω on a pivoted horizontal axle as shown.Gravity acts down and the disk has a precession frequency Ω. If the mass of thedisk were doubled but its radius and angular velocity were kept the same, theprecession frequency would:A) Increase B) DecreaseC) Stay the sameMechanics Lecture 20, Slide 20
CheckPointIf the mass of the disk were doubled but its radius and angular velocity werekept the same, the precession frequency wouldA) IncreaseB) DecreaseC) Stay the sameA) torque due to weight increases in thenumerator.B) When mass is doubled, angular momentumdoubles, so precession frequency decreases.C) Increasing the mass increases both theangular momentum and the torque equally, sothe precession frequency stays the e/instructor/charts/stdhw choice histogram.aspx?qid 593056Mechanics Lecture 20, Slide 21
Clicker QuestionA disk is spinning with angular velocity ω on a pivoted horizontal axleas shown. If the radius of the disk were doubled but its mass andangular velocity were kept the same:A) The angular momentum of the disk doublesB) The torque about the pivot doublesC) Both A and BL IωMechanics Lecture 20, Slide 22
CheckPointA disk is spinning with angular velocity ω on a pivoted horizontal axleas shown. Gravity acts down and the disk has a precession frequencyΩ. If the radius of the disk were doubled but its mass and angularvelocity were kept the same, the precession frequency wouldA) Increase B) DecreaseC) Stay the sameMechanics Lecture 20, Slide 23
CheckPointIf the radius of the disk were doubled but its mass and angular velocity werekept the same, the precession frequency wouldA) IncreaseB) DecreaseC) Stay the sameA) The moment of iner a would increase, so theangular momentum would increase, so omega wouldalso increaseB) If you increase radius, you increase angular momentum,and thus decrease the precession frequency.C) the frequency would remain the same because theradius has no impact on the frequencyMechanics Lecture 20, Slide 24
All of this stuff doesn't really seem prac cal. If you can't prove me wrong then weshould all just get A's for this sec on of the course.http://www.youtube.com/watch?v cquvA IpEsA (see 2:30)Prac cal Applica on: Keeps you from falling off your bike when you rideusing no hands!Riding straight(τ 0)Wheel steers straightLean LeY(τ out of page)Wheel steers leYLean Right(τ into page)Wheel steers rightMechanics Lecture 20, Slide 25
dMgMechanics Lecture 20, Slide 26
L(using right hand rule)Mechanics Lecture 20, Slide 27
Mechanical Universe: Episode 19. Student on Stool There are no external torques acCng on the student-stool system, so angular momentum will be conserved. IniCally: L i I i ω i Finally: L f I f ω f ω f I f L f ω i I i L i Mechanics Lecture 20, Slide 3. Clicker Question A student sits on a freely turning stool and rotates with constant angular velocity ω 1. She pulls her arms in and her .
