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Chapter 10 Dynamics of Rotational Motion PowerPoint® Lectures for University Physics, Twelfth Edition – Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 5_31_2012 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Goals for Chapter 10 • To see how torques cause rotational dynamics (just as linear forces cause linear accelerations) • To calculate work done by a torque • To study angular momentum and its conservation • To relate rotational dynamics and angular momentum Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Torque • A force (F) applied at a distance ( r) and at an angle (θ) will generate a torque (τ ). ! ! ! " #r$F ! ! r ! Magnitude of torque =| " |= rFsin# . ! by the Direction of torque is given ! "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 " Moment force and Lever arm Compute cross product using î,ĵ,k̂: In this example: ! r=rsin! î+rsin! ĵ ! F = Fĵ ! ! r ! F = rsin! î+rcos! ĵ ! Fĵ ( ) = rsin! î ! Fĵ+rcos! ĵ! Fĵ =rFsin! k̂ + 0 Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Direction of torque vector • Mathematically, the direction of torque vector is given by the right-hand rule (RHR) by convention. • Physical effect of torque is tend to rotate the object counterclockwise or clockwise. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Calculate an applied torque • Consider Example 10.1. • Refer to Figure 10.5. ! Which angle should you use in | " |= rFsin# ? Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! Τ = Iα is just like F = ma • A 9.0N force is applied to the wheel for 2s and then released. • What is the angular acceleration as a function of time? • What is the angular velocity as a function of time? (Given the wheel was initially at rest) Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! Another look at the unwinding cable Given the mass (m) and the wheel (M), find the acceleration of m. (Assume no airresistance 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 The yo-yo - rolling without slipping • Calculate the yoyo 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 | # | (no slipping condition) Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Energy method (translation + rotation) 1 2 1 K = mvCM + ICM " 2 2 2 ! Rolling without slipping condition " vCM = R# 2 ' $ vCM 1 2 1 " K = mvCM + ICM & 2 ) 2 2 %R ( Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Application of Conservation of Energy • Find speed of yo-yo after falling a distance h. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley The race of objects with different moments of inertia • Use energy method to determine which object will reach the bottom of the incline first (i.e. which object reach the bottom with the largest speed?) 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 Why conservation of mechanical energy works when there is friction? • What is the role of friction? Answer: The role of static friction is to do negative work on the center of motion while doing positive work on the rotation motion. Net result, the work done by static friction is zero. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Angular momentum ! 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" ) ! = =# dt dt ! ! If I = constant $ I% = # as before. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! Compare linear and rotational dynamics Linear ! ! F = ma (if m = constant) ! ! p = mv ! dp! F= dt For a system of objecys : ! If net Fexternal = 0 ! then p total is conserved. Rotational dynamics of a rigid object ! ! " = I# (if I = constant) ! ! L = I$ (valid for rotation about a symmetry axis) ! ! dL "= dt ! If net " external = 0, ! then, L total is conserved. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Conservation of Angular momentum • The frictionless platform ensures the external torque along the z-direction is zero=>Lz is conserved I1"1 = I2" 2 ; I # " $ Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley ! This is how a car s clutch works-conservation of angular momentum ! ! ! IA" A + IB" B = (IA + IB )" ! Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley Conservation of angular momentum in daily devices: Gyroscope - A fast spinning top whose angular momentum vector points at a fixed direction in space => a navigation device. 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 Gyroscopic precession • If external torque =/ =0, there could be precession motion. Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley