Chapter 12. Rotation of a Rigid Body - Physics & Astronomy

1 Chapter 12. Rotation of a Rigid BodyChapter 12. Rotation of a Rigid BodyNot all motion can be described as that of a particle. Rotation requiresthe idea of an extended object. This diver is moving toward the water along a Copyright 2008 Pearson Education, Inc., publishing as Pearson the water along a parabolic trajectory, and she s rotating rapidly around her center of Goal:To understand the Physics of rotating : Rotational Motion Rotation About the Center of Mass Rotational Energy Calculating Moment of Inertia Torque Chapter 12. Rotation of a Rigid BodyChapter 12. Rotation of a Rigid BodyCopyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. Torque Rotational Dynamics Rotation About a Fixed Axis Static Equilibrium Rolling Motion The Vector Description of Rotational Motion Angular Momentum of a Rigid Body Chapter 12. Reading QuizzesChapter 12. Reading QuizzesCopyright 2008 Pearson Education, Inc., publishing as Pearson 12.

2 Reading QuizzesChapter 12. Reading QuizzesA new way of multiplying two vectors is introduced in this Chapter . What is it called:A. Dot ProductCopyright 2008 Pearson Education, Inc., publishing as Pearson ProductC. Tensor ProductD. Cross ProductE. Angular ProductA new way of multiplying two vectors is introduced in this Chapter . What is it called:A. Dot ProductCopyright 2008 Pearson Education, Inc., publishing as Pearson ProductC. Tensor ProductD. Cross ProductE. Angular ProductMoment of inertiais A. the rotational equivalent of point at which all forces appear to 2008 Pearson Education, Inc., publishing as Pearson point at which all forces appear to the time at which inertia an alternative term for moment of inertiais A. the rotational equivalent of point at which all forces appear to 2008 Pearson Education, Inc., publishing as Pearson point at which all forces appear to the time at which inertia an alternative term for moment Rigid body is in equilibrium if Copyright 2008 Pearson Education, Inc.

3 , publishing as Pearson neither A nor either A or both A and Rigid body is in equilibrium if Copyright 2008 Pearson Education, Inc., publishing as Pearson neither A nor either A or both A and 12. Basic Content and ExamplesChapter 12. Basic Content and ExamplesCopyright 2008 Pearson Education, Inc., publishing as Pearson 12. Basic Content and ExamplesChapter 12. Basic Content and ExamplesRotational MotionThe figure shows a wheel rotating on an axle. Its angular velocity isThe units of are rad/s. If Copyright 2008 Pearson Education, Inc., publishing as Pearson units of are rad/s. If the wheel is speeding up or slowing down, its angular acceleration isThe unitsof are rad/s2. Rotational MotionCopyright 2008 Pearson Education, Inc., publishing as Pearson A rotating crankshaftQUESTION:Copyright 2008 Pearson Education, Inc., publishing as Pearson A rotating crankshaftCopyright 2008 Pearson Education, Inc., publishing as Pearson A rotating crankshaftCopyright 2008 Pearson Education, Inc.

4 , publishing as Pearson A rotating crankshaftCopyright 2008 Pearson Education, Inc., publishing as Pearson A rotating crankshaftCopyright 2008 Pearson Education, Inc., publishing as Pearson About the Center of MassAn unconstrained object ( , one not on an axle or a pivot ) on which there is no net force rotates about a point called the center of Copyright 2008 Pearson Education, Inc., publishing as Pearson called the center of mass. The center of mass remains motionless while every other point in the object undergoes circular motion around About the Center of MassThe center of mass is the mass-weighted center of the 2008 Pearson Education, Inc., publishing as Pearson EnergyA rotating Rigid body has kinetic energy because all atoms in the object are in motion. The kinetic energy due to Rotation is called rotational kinetic 2008 Pearson Education, Inc., publishing as Pearson the quantity Iis called the object s moment of units of moment of inertia are kg m2.

5 An object s moment of inertia depends on the axis of 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodQUESTION:Copyright 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodCopyright 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodCopyright 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodCopyright 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodCopyright 2008 Pearson Education, Inc., publishing as Pearson The speed of a rotating rodCopyright 2008 Pearson Education, Inc., publishing as Pearson the common experience of pushing open a door. Shown is a top view of a door hinged on the left. Four pushing forces are shown, all of equal strength. Which of these will be most effective at opening the door?Copyright 2008 Pearson Education, Inc., publishing as Pearson ability of a force to cause a Rotation depends on three factors:1.

6 The magnitude Fof the the distance rfrom the point of application to the the angle at which the force is 2008 Pearson Education, Inc., publishing as Pearson s define a new quantity called torque (Greek tau) asEXAMPLE Applying a torqueQUESTION:Copyright 2008 Pearson Education, Inc., publishing as Pearson Applying a torqueCopyright 2008 Pearson Education, Inc., publishing as Pearson Applying a torqueCopyright 2008 Pearson Education, Inc., publishing as Pearson Applying a torqueCopyright 2008 Pearson Education, Inc., publishing as Pearson between Linear and Rotational DynamicsCopyright 2008 Pearson Education, Inc., publishing as Pearson the absence of a net torque ( net= 0), the object either does not rotate ( = 0) or rotates with constantangular velocity ( = constant).Problem-Solving Strategy: Rotational Dynamics ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Strategy: Rotational Dynamics ProblemsCopyright 2008 Pearson Education, Inc.

7 , publishing as Pearson Strategy: Rotational Dynamics ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Strategy: Rotational Dynamics ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Starting an airplane engineQUESTION:Copyright 2008 Pearson Education, Inc., publishing as Pearson Starting an airplane engineCopyright 2008 Pearson Education, Inc., publishing as Pearson Starting an airplane engineCopyright 2008 Pearson Education, Inc., publishing as Pearson Starting an airplane engineCopyright 2008 Pearson Education, Inc., publishing as Pearson Starting an airplane engineCopyright 2008 Pearson Education, Inc., publishing as Pearson Equilibrium The condition for a Rigid body to be in static equilibriumis that there is no net forceand no net torque. An important branch of engineering called staticsanalyzes buildings, dams, bridges, and other structures in total static equilibrium.

8 Copyright 2008 Pearson Education, Inc., publishing as Pearson total static equilibrium. No matter which pivot point you choose, an object that is not rotating is not rotating about that point. For a Rigid body in total equilibrium, there is no net torque about any point. This is the basis of a problem-solving Strategy: Static Equilibrium ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Strategy: Static Equilibrium ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Strategy: Static Equilibrium ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Strategy: Static Equilibrium ProblemsCopyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?QUESTION:Copyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc.

9 , publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc., publishing as Pearson Will the ladder slip?Copyright 2008 Pearson Education, Inc., publishing as Pearson and StabilityCopyright 2008 Pearson Education, Inc., publishing as Pearson Without SlippingFor an object that is rolling without slipping, there is a rolling constraintthat links translation and Rotation :Copyright 2008 Pearson Education, Inc., publishing as Pearson Without SlippingWe know from the rolling constraint that R is the center-of-mass velocity vcm. Thus the kinetic energy of a rolling object isCopyright 2008 Pearson Education, Inc., publishing as Pearson other words, the rolling motion of a Rigid body can be described as a translation of the center of mass (with kinetic energy Kcm) plus a Rotation about the center of mass (with kinetic energy Krot).

10 The Angular Velocity VectorCopyright 2008 Pearson Education, Inc., publishing as Pearson Addison-Wesley. The magnitude of the angular velocity vector is . The angular velocity vector points along the axis of Rotation in the direction given by the right-hand rule as illustrated Momentum of a ParticleCopyright 2008 Pearson Education, Inc., publishing as Pearson particle is moving along a trajectory as shown. At this instant of time, the particle s momentum vector, tangent to the trajectory, makes an angle with the position Momentum of a ParticleCopyright 2008 Pearson Education, Inc., publishing as Pearson define the particle s angular momentum vector relative to the origin to beAnalogies between Linear and Angular Momentum and EnergyCopyright 2008 Pearson Education, Inc., publishing as Pearson 2008 Pearson Education, Inc., publishing as Pearson 12. Summary SlidesChapter 12. Summary SlidesCopyright 2008 Pearson Education, Inc., publishing as Pearson 12.