Review: Angular Momentum. Physics 201, Lecture 20

Transcription

1 q Physics 01, Lecture 0 Today s Topics More on Angular Momentum and Conservation of Angular Momentum Demos and Exercises q Elasticity (Section 1.4. ) Deformation Elastic Modulus (Young s, Shear, Bulk) q Next Tuesday: Static Equilibirium (Section 1.1-3) 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=ΣL j. For an object rotating about a fixed object: L=Iω d L / dt = Στ recall: P=mv L f = L i 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 Exercise: Momentum Conservation Jumping On Merry-Go-Round q 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 ω i L f = (I mgr + I child )ω f =(½ m mgr R mgr + m c R mgr ) ω f à ω f = ½ m mgr R mgr / (½ m mgr R mgr + m c R mgr ) ω i = ½ m mgr / (½ m mgr + m c ) ω i Right Hand Rule 1

2 Angular Momentum And Rotational Kinetic Energy q Recall: KE rot = ½ I ω q That is: and L = I ω Rotational Kinetic Energy KE rot = 1 Iω = 1 (Iω) I = L I Demos and Quizzes (Next Few Slides) A figure skater dances on ice with various poses. Which pose has larger moments of inertia? For same angular momentum, the larger the moment of inertia, the smaller the KE rot This or This or Same? Which Pose Has More Angular Momentum? Which Pose Spins Faster? L i = L f I i ω i = I f ω f ie. SAME (very little torque by ice ) i.e. ω i < ω f

3 Which Pose Has Larger Kinetic Energy? Demo and Discussion Turning the bike wheel L i = L f I i > I f KE rot_i < KE rot_f KE rot = L I Helicopters/Drones (Why Two+ Rotors?) Gyroscope For Navigation 3

4 Gyroscope And Precession q A top with spinning angular velocity ω at an inclination θ would precession around the z axis at frequency : ω p = Mgh/(Iωcosθ) (derivation out of scope of the course) q This type of motion is called precession and ω p is the precessional frequency. mg gives a torque θ h Read After class conceptual only Physical Objects q Physical Objects they can do Particles: No size, no shape. (hence do not rotate.) circular motion Extended objects: CM+Size+Shape though 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. v When ω is very very large, ω p 0, i.e the axis is spontenuously fixed. q Today: Deformable objects under small stress (elastic limit) à good for navigation Deformation and Elasticity Young s Modulus For Tensile Stress 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): 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): Strain = Stress / (Elastic modulus) Ø There are three general types of stress/strain: Y tensile stress tensile strain = F / A ΔL /L slope tensile shear bulk 4

5 Shear Modulus q When an object is subject to a shear stress, a shear strain can occur. shear strain 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 Shear Modulus (S): S shear stress shear strain = F / A Δx /h q Bulk Modulus (B): volume stress B volume strain = ΔF / A ΔV /V = ΔP ΔV /V Pressure: P=F/A Typical Elastic Moduli Special Announcement My office hours for today have been moved to 11am-1pm (from -4pm as scheduled). 5