Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.March 18, 2008.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.March 18, 2008.• Course Information• Topics to be discussed today:• Variables used to describe rotational motion• The equations of motion for rotational motionFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterCourse Announcements.• Homework set # 6 is now available on the WEB and will bedue on Saturday morning, March 22, at 8.30 am.• All the material to be covered on Exam # 2 has now beendiscussed. Today we will start on material that will becovered on Exam # 3.• Exam # 2 will take place on Tuesday March 25 at 8 am inHubbell. It will cover the material discussed in Chapters 7,8, and 9.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterPhysics 121.Homework Set # 6.Equations of motionwith constant α.Rotational kinetic energy. Note: thisis not a rigid body!Calculating themoment of inertiaTorque and accelerationEquations of motionwith constant α.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• In our discussion of rotationalmotion we will first focus on therotation of rigid objects around afixed axis.• The variables that are used todescribe this type of motion aresimilar to those we use todescribe linear motion:• Angular position• Angular velocity• Angular accelerationFrank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• The angular position, measuredin radians, is the angle of rotationof the object with respect to areference position.• The angular position of point P atthis point in time is equal to θ. Inorder to uniquely define thisposition, we have assume that anthe angular position is measuredwith respect to the x axis.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• Note:• The angular position is alwaysspecified in radians!!!!• One radian is the angulardisplacement corresponding to alinear displacement l = R.• Make sure you keep track of thesign of the angular position!!!!!• An increase in the angularposition corresponds to a counter-clockwise rotation; a decreasecorresponds to a clockwiserotation.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterComplex motion in Cartesian coordinates issimple motion in rotational coordinates.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• If we look at an object carryingout a rotation around a fixed axis,we will see that the angularposition becomes a function oftime.• To describe the rotational motion,we introduce the concepts ofangular velocity and angularacceleration.• Remember: for linear motion wefound it useful to introduce theconcepts of linear velocity andlinear acceleration.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• For both velocity andacceleration we can talk aboutinstantaneous and average.• Angular velocity:• Definition: ω = dθ/dt• Symbol: ω• Units: rad/s• Angular acceleration:• Definition: α = dω/dt = d2θ/dt2• Symbol: α• Units: rad/s2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: constant acceleration.• If the object experiences aconstant angular acceleration,then we can describe itsrotational motion with thefollowing equations of motion:• Note how similar these equationsare to the equation of motion forlinear motion!!!!!!!t( )=!0+"t#t( )=#0+!0t +12"t2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: constant acceleration.Example problem.• A wheel starting from rest, rotates with a constant angularacceleration of 2.0 rad/s2. During a certain 3.0 s interval itturns through 90 rad. (a) How long had the wheel beenturning before the start of the 3.0 s interval ? (b). What wasthe angular velocity of the wheel at the start of the 3.0 sinterval ?• Define time t = 0 s as the time that the wheel is at rest. Theangular velocity and the angle of rotation at a later time tare given by!t( )="t!t( )=12"t2Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: constant acceleration.Example problem.• The change in the angular position Δθ during a time periodΔt can now be calculated:• Since the problem specifies Δθ and Δt we can now calculatethe time t and the angular velocity at that time:!"t( )="t + !t( )#"t( )=12$t + !t( )2#12$t2=12$!t( )2+$t!tt =!"#12$!t( )2$!t%=$t =!"#12$!t( )2!t=!"!t#12$!t( )Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• The linear velocity of a part ofthe rigid body is related to theangular velocity of the object.• Consider point P:• It this point makes one completerevolution, it travels a distance2πR.• When the angular positionchanges by dθ, point P moves adistance dl = 2πR(dθ/2π) = Rdθ.• The linear velocity of point P isequal to v = dl/dt = Rdθ/dt = Rω.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: variables.• Things to consider when lookingat the rotation of rigid objectsaround a fixed axis:• Each part of the rigid object hasthe same angular velocity.• Only those parts that are locatedat the same distance from therotation axis have the same linearvelocity.• The linear velocity of parts of therigid object increases withincreasing distance from therotation axis.Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRelation between rotational and linearvariables.• Although in rotational motion weprefer to use rotational variables,we can also express the motion interms of linear variables:s = r!v =dsdt=ddtr!( )= rd!dt= r"at=dvdt=ddtr"( )= rd"dt= r#Frank L. H. Wolfs Department of Physics and Astronomy, University of RochesterRotational motion: acceleration.• Note: the acceleration at = rα isonly one of the two