Lectures Today: Rolling and Angular Momentum in ch 1 Homework 6 due Next time: Complete angular momentum (chapter 1) and begin equilibrium (chapter 13) By Monday, will post at website Sample midterm II and solution Midterm II two weeks from today (11/5) Covers especially chapters 7-13 slide 1
Finding C of M Empirically N 1 x 1 x N f 1 f f µ N mg N + N mg 1 1 xn xn 1 1 x N f > > 1 > f 1 x N So slips at (and vice versa) l l Allow rod to slide across hands Works because the forces and torques on the rod must sum to zero r rdm m com 1 M 1 D -- like stick dm dl M L L L 1 1 com M lmdl L 0 0 L ldl slide
Review: Torque-Producing Rotational Kinetic Energy t t t Definition of Torque r F rf sin f in direction rf plane rf rf t Torque produces Angular Acceleration Body s rotational energy results Work done by torque Work Fs Frq t tq Power tw slide 3
review Note that this form is similar to Newton s nd Law: Fma Motion driver (force) produces linear acceleration smaller if inertia (mass) larger Torque drives angular acceleration I, property of the body, is a measure of rotational inertia I larger, a smaller I I mr i i t a with DEMO: further masses are from rotation axis Larger I Smaller a slide 4
Note solution method Relies on understanding relation between translation of m and rotation of M Weight on Wheel Sample Prob 11-7 Answer: m m a g g 1 m + m + M I R slide 5
Problem Problem of pulley and well How much potential energy lost when bucket falls 1 meter? What is v after 1 meter? What is kinetic energy then? For m1.53kg, M4.0kg m U 1 m mgy 150. Joules m a g 0433. g m + M v 1 gy 91. m/s 1 1 K ( m + Mv ) 150. Joules M slide 6
w Rolling, rolling, rolling Wheel com moves distance, s, while wheel rotates angle, q. v w ds dt dq dt s v com Rq Rw Tells us immediately the velocity of com slide 7
Critical element of rolling The point at contact with the ground is at rest! Otherwise, that point would slip Implies that static friction is relevant when done properly v com Rw slide 8
Can view as rotation about pt of contact vcom Rw points on rolling wheel.mov slide 9
Rolling Ball (or cylinder ) d Determined by geometry of body Do with forces (as text: eqn 1-10) Use const acc equation to give speed Also do by energy (flick 49s) h a com 1 g + balls rolling down plane.mov 1 I MR uniform can sinq I MR v down the ramp v com com ad gh I 1 + MR So v at bottom depends only on shape and mass distribution of rolling body slide 10
Yo-yo: Rolls down String See section 1-4 a 1 + g I com MR 0 + + First define consistent conventions Sense of +ve rotation Sense of +ve translation Torques about c of m Forces Solve slide 11
New concept: Angular Momentum of a Particle Mass p mv l r p so l rp sin φ angular momentum vector.mov Angular momentum arises in many cases single particle as shown rotating rigid body (later) τ Note that right hand side is also Ia dl dt slide 1
Prove Torque-Ang Mom Relation dl d ( r mv ) dt dt a v dl Ø dv dr ø m r v dt Œ + º dt dt œ ß dl r ma r F t dt Here F is the total net force acting on m t is the net torque slide 13
w z Body Rotating about a Fixed Axis L z I ω For body rotating z about L fixed Iω axis L l r ( m )( ω r ) z iz i i z i proof L z ω r m z i i L z I z ω z slide 14
L τ If ext net τ dl dt ext net Conservation of Angular Momentum 0 0, is constant L L f dl dt i Important Conservation Rule if isolated system (no ext. torque) Reflects symmetry of Universe physical laws wrt orientation in space Many examples of its application Text Lecture slide 15
Demos of Ang Mom Conservation Rotation with weights (demo) Spin with weights outstretched Establish angular velocity pull weights in (reduce moment of inertia) change angular velocity Puck on air table with force along axis cons ang mom puck astronaut.mov I f w f L w f f I I L i f i I i w w i i slide 16
Examples from text Diver tucks in and reduces I, w gets large Demos with rotating platform Ang. Mom fixed at start But I is large, w small slide 17
Keep in Mind Leave HW 6 in box today Next time: Complete angular momentum (chapter 1) and begin equilibrium (chapter 13) By Monday, will post at website Sample midterm II and solution Midterm II two weeks from today Covers especially chapters 7-13 slide 18