Goals for Chapter 10 • To see how torques cause rotational dynamics (just Chapter 10 as linear forces cause linear accelerations) • To calculate work done by a torque Dynamics of • To study angular momentum and its conservation Rotational Motion • To relate rotational dynamics and angular momentum PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 6_8_2011 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Torque Moment force and Lever arm • A force (F) applied at a distance ( r) and at an angle (θ) will Cross product computed using ˆi,ˆj,k:ˆ generate a torque (τ ). In this example: ! ! ! r=rsin!ˆi+rsin!ˆj ! ! r ! " # r $ F F Fjˆ " (moment force) = ! ! ! r " F = rsin!ˆi+rcos!ˆj " Fjˆ Magnitude of torque =| " |= rFsin#. ( ) = rsin!ˆi " Fˆj+rcos!ˆj " Fˆj Direction of torque is given! by the =rFsin!kˆ + 0 ! ! " right - hand rule"- see next slide. Which force on the figure produces the largest torque ! about point O and which one produces the smallest torque? Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 1 Direction of torque vector Calculate an applied torque • Mathematically, the • Consider Example 10.1. direction of torque vector is given by the • Refer to Figure 10.5. right-hand rule (RHR) by convention. • Physical effect of torque is tend to rotate the object counterclockwise or clockwise. ! Which angle should you use in | " |= rFsin# ? Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as! Pearson Addison-Wesley Τ = Iα is just like F = ma Another look at the unwinding cable • A 9.0N force is applied to the wheel for 2s and then released. Given the mass (m) and the wheel (M), find the • What is the angular acceleration as a function of time? acceleration of m. • What is the angular velocity as a function of time? (Given the (Assume no air- wheel was initially at rest) resistance or friction at the axle of the wheel, assume the no-slipping) ! ! Concept 1: net F = ma ! ! Concept 2 : net " = I# ! ! Concept 3: | a |= R |# | ( no slipping condition) ! Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 2 The yo-yo - rolling without slipping Energy method (translation + rotation) • Calculate the yo- yo’s acceleration and then the final v after it has fallen a distance h. ! ! Concept 1: net F = Ma ! ! Concept 2 : net " = I# ! ! Concept 3: | a |= R |# | 1 2 1 2 K = mvCM + ICM " (no slipping condition) 2 2 Rolling without slipping condition " vCM = R# $ 2 ' ! 1 2 1 vCM ! " K = mvCM + ICM & ) 2 2 % R2 ( Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! Application of Conservation of Energy The race of objects with different moments of inertia • Find speed of • Use energy method to determine which object will reach the bottom of the incline first (i.e. which object yo-yo after reach the bottom with the largest speed?) falling a distance h. The object with the smallest moment of inertia will spend less energy rotating and hence has more energy for translation. The sphere has the smallest moment of inertia => will have the largest speed. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 3 Why conservation of mechanical energy works when there is friction? Angular momentum ! • What is the role of friction? Angular momentum (L) for a moving point mass : ! ! ! ! ! L " r # p = r # (mv ). ! dL ! ! ! $ = r # F = % dt Angular momentum for a rotating rigid object ! about a symmetry axis: ! ! L = I" ! ! dL d(I" ) ! = = # Answer: The role of static friction is to do negative work on dt dt the center of motion while doing positive work on the rotation ! ! If I = constant $ I% = # as before. motion. Net result, the work done by static friction is zero. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! Compare linear and rotational dynamics Conservation of Angular momentum • The frictionless platform ensures the external torque along Linear Rotational dynamics of a rigid object the z-direction is zero=>L is conserved ! ! ! ! z F = ma (if m = constant) " = I# (if I = constant) ! ! ! ! p = mv L = I$ (valid for rotation about a symmetry axis) ! ! ! dp ! dL F = " = dt dt For a system of objecys: ! ! If net Fexternal = 0 If net " external = 0, ! ! then p is conserved. then, L is conserved. total total I1"1 = I2"2; I#" $ ! Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! 4 This is how a car’s clutch works-conservation of Conservation of angular momentum in daily devices: angular momentum Gyroscope - A fast spinning top whose angular momentum vector points at a fixed direction in space => ! ! ! a navigation device. I A" A + IB" B = (IA + IB )" Fast rotating bicycle wheels keep the bicycle stable. What is the purpose of the small propeller at the back of ! a helicopter? Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Gyroscopic precession • If external torque =/ =0, there could be precession motion. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley 5 .