1 Physics 201, Lecture 20 Today’s Topics More on Angular Momentum and Conservation of Angular Momentum • Demos and Exercises Elasticity (Section 12.4. ) Deformation Elastic Modulus (Young’s, Shear, Bulk) Next Tuesday: Static Equilibirium (Section 12.1-3) Hope you’ve previewed Chapter 11. Review: Angular Momentum A particle’s angular momentum relative to a chosen origin is defined as L ≡ rxP L is a vector. Angular momentum is always defined w.r.t an origin*. For a system with multiple particles, L=ΣL j . For an object rotating about a fixed object: L=Iω recall: P=mv τ Σ = dt d / L L f = L i if no torque Review: Angular Momentum of A Rotating Object For a rigid object about a fixed axis, its angular momentum is defined as: L= Iω For the same ω, the larger the I, the larger the L L is a vector, it has a direction. The direction of angular momentum can be determined by the “Right Hand Rule” Right Hand Rule Exercise: Momentum Conservation Jumping On Merry-Go-Round A freely spinning Merry-Go-Round of mass m mgr and radius R mgr has an initial angular speed ω i . After a child of mass m c jumps on it at the edge as shown, what is the new ω ? Solution: free spinning = no torque L f =L i L i = I mgr ω i = ½ m mgr R mgr 2 ω i L f = (I mgr + I child )ω f =(½ m mgr R mgr 2 + m c R mgr 2 ) ω f ω f = ½ m mgr R mgr 2 / (½ m mgr R mgr 2 + m c R mgr 2 ) ω i = ½ m mgr / (½ m mgr + m c ) ω i
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Exercise: Momentum Conservation - Department of Physics · 1 Physics 201, Lecture 20 origin Today’s Topics ! More on Angular Momentum and Conservation of Angular Momentum • Demos
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Physics 201, Lecture 20
Today’s Topics
q More on Angular Momentum and Conservation of Angular
q Next Tuesday: Static Equilibirium (Section 12.1-3)
q Hope you’ve previewed Chapter 11.
Review: Angular Momentum q A particle’s angular momentum relative to a chosen origin is defined as L ≡ rxP
§ L is a vector. § Angular momentum is always defined w.r.t an origin*. § For a system with multiple particles, L=ΣLj. § For an object rotating about a fixed object: L=Iω
recall: P=mv
τ
Σ=dtd /L Lf = Li if no torque
Review: Angular Momentum of A Rotating Object
q For a rigid object about a fixed axis, its angular momentum is defined as: L= Iω § For the same ω, the larger the I, the larger the L § L is a vector, it has a direction. The direction of angular
momentum can be determined by the “Right Hand Rule”
Right Hand Rule
Exercise: Momentum Conservation Jumping On Merry-Go-Round
q A freely spinning Merry-Go-Round of mass mmgr and radius Rmgr has an initial angular speed ωi . After a child of mass mc jumps on it at the edge as shown, what is the new ω ?
Solution: free spinning = no torque Lf=Li Li = Imgrωi = ½ mmgrRmgr
2 ωi Lf = (Imgr + Ichild )ωf =(½ mmgrRmgr
2 + mcRmgr2 ) ωf
à ωf = ½ mmgrRmgr
2 / (½ mmgrRmgr2 + mcRmgr
2 ) ωi = ½ mmgr/ (½ mmgr+ mc
) ωi
2
Angular Momentum And Rotational Kinetic Energy
q Recall: KErot = ½ I ω2 and L = I ω q That is:
For same angular momentum, the larger the moment of inertia, the smaller the KErot
Rotational Kinetic Energy
KErot =12
Iω 2 =
12
(Iω)2
I=
L2
2I
Demos and Quizzes (Next Few Slides) A figure skater dances on ice with various poses. Which pose has larger moments of inertia?
This or This or Same?
Which Pose Has More Angular Momentum?
Li = Lf ie. SAME (very little torque by ice )
Which Pose Spins Faster ?
Iiωi = Ifωf i.e. ωi < ωf
3
Which Pose Has Larger Kinetic Energy ?
Li = Lf Ii > If KErot_i < KErot_f
KErot =L2
2I
Demo and Discussion Turning the bike wheel
Helicopters/Drones (Why Two+ Rotors?) Gyroscope For Navigation
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Gyroscope And Precession q A top with spinning angular velocity ω at an inclination θ would precession around the z axis at frequency : ωp= Mgh/(Iωcosθ)
(derivation out of scope of the course)
q This type of motion is called precession and ωp is the precessional frequency.
v When ω is very very large, ωp 0, i.e the axis is spontenuously fixed.
à good for navigation
θ h
mg gives a torque
Read
Afte
r cla
ss
conc
eptu
al o
nly Physical Objects
q Physical Objects § Particles: No size, no shape. (hence do not rotate.) § Extended objects: CM+Size+Shape