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Physics 201, Lecture 20

Today’s Topics

q  More on Angular Momentum and Conservation of Angular

Momentum •  Demos and Exercises

q  Elasticity (Section 12.4. )

§  Deformation §  Elastic Modulus (Young’s, Shear, Bulk)

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

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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

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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

•  Rigid objects: Translation + Rotation, non deformable •  Deformable objects:

– Regular solids: »  Shape/size change under stress »  Eventually break down when stress gets large

–  Liquids: » Do not have fixed shape »  Size (volume) can change under stress.

q  Today: Deformable objects under small stress (elastic limit)

they can do circular motion though

Deformation and Elasticity q  Regular deformable objects under stress

§  Small stress deformation in “linear” (elastic) fashion §  Larger stress deformation in non-linear fashion §  Even larger stress break down

q  Small deformation (strain under small stress):

Strain = Stress / (Elastic modulus) Ø  There are three general types of stress/strain:

tensile shear bulk

Young’s Modulus For Tensile Stress q  When an object is stressed in the direction of its length, its

length will change with strength of the stress §  definitions:

•  Tensile stress = F/A •  Tensile strain = ΔL/L •  Young’s Modulus (Y):

€

Y ≡tensile stresstensile strain

=F /AΔL /L slope

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Shear Modulus q  When an object is subject to a shear stress, a shear strain can

occur.

q  Shear Modulus (S):

shear strain

€

S ≡shear stressshear strain

=F /AΔx /h

Bulk Modulus q  When any object is subject to a uniform stress in all direction

(called pressure, or volume stress), its volume can change

q  Bulk Modulus (B):

€

B ≡volume stressvolume strain

=ΔF /AΔV /V

=ΔP

ΔV /VPressure: P=F/A

Typical Elastic Moduli Special Announcement My office hours for today have been moved to 11am-1pm (from 2-4pm as scheduled).