Chapter 9 – Rotation and RollingII. Rotation with constant angular accelerationIII. Relation between linear and angular variables- Position, speed, accelerationI. Rotational variables- Angular position, displacement, velocity, accelerationIV. Kinetic energy of rotationV. Rotational inertiaVI. TorqueVII. Newton’s second law for rotationVIII. Work and rotational kinetic energyIX. Rolling motionI. Rotational variablesRigid body: body that can rotate with all its parts locked together and without shape changes.rsradiuslengtharc==θRotation axis: every point of a body moves in a circle whose center lieson the rotation axis. Every point moves through the same angle duringa particular time interval. Angular position: the angle of the reference line relative to the positive direction of the x-axis.Units: radians (rad)Reference line: fixed in the body, perpendicular to the rotation axis and rotating with the body.revradradrrrev159.03.571223601=====ππNote: we do not reset θ to zero with each complete rotation of the reference line about the rotation axis. 2 turns  θ =4πTranslation: body’s movement described by x(t).Rotation: body’s movement given by θ(t) = angular position of the body’s reference line as function of time.Angular displacement: body’s rotation about its axis changing the angular position from θ1to θ2.12θθθ−=∆Clockwise rotation  negativeCounterclockwise rotation  positiveAngular velocity:tttavg∆∆=−−=θθθω1212dtdttθθω=∆∆=→∆ 0limAverage:Instantaneous:Units: rad/s or rev/sThese equations hold not only for the rotating rigid body as a whole but also for every particle of that body because they are all locked together.Angular speed (ω): magnitude of the angular velocity.Angular acceleration:tttavg∆∆=−−=ωωωα1212dtdttωωα=∆∆=→∆ 0limAverage:Instantaneous:Angular quantities are “normally” vector quantities right hand rule.Object rotates around the direction of the vector  a vector defines an axis of rotation not the direction in which something is moving.Examples: angular velocity, angular accelerationAngular quantities are “normally” vector quantities right hand rule.Exception: angular displacements The order in which you add two angular displacements influences the final result  ∆θ is not a vector.II. Rotation with constant angular acceleration20000202200021)(21)(221atvtxxtvvxxxxavvattvxxatvv−=−+=−−+=+=−+=Linear equationsAngular equations20000202200021)(21)(221ttttttαωθθωωθθθθαωωαωθθαωω−=−+=−−+=+=−+=III. Relation between linear and angular variablesPosition:rs⋅=θθ always in radiansSpeed:rvdtdrdtds⋅=→=ωθω in rad/sSince all points within a rigid body have the same angular speed ω, points located at greater distance with respect to the rotational axis have greater linear (or tangential) speed, v. v is tangent to the circle in which a point movesIf ω=constant, v=constant  each point within the body undergoes uniform circular motion.Period of revolution:ωπωππ222===rrvrTAcceleration:rarrdtddtrddtdvt⋅=→⋅==⋅=ααωω)(Tangential component of linear accelerationRadial component oflinear acceleration:rrvar⋅==22ωResponsible for changes in the direction of the linear velocity vector vIV. Kinetic energy of rotationReminder: Angular velocity, ω is the same for all particles within the rotating body. Linear velocity, v of a particle within the rigid body depends on the particle’s distance to the rotation axis (r).Units: m/s2Responsible for changes in the magnitude of the linear velocity vector v.222223222121)(2121...212121ωω=⋅==+++=∑∑∑iiiiiiiiirmrmvmmvmvmvKRotational inertia = Moment of inertia, I:∑=iiirmI2Indicates how the mass of the rotating body is distributed about its axis of rotation.The moment of inertia is a constant for a particular rigid body and a particular rotation axis.Units: kg m2Example: long metal rod.Smaller rotational inertia in (a)  easier to rotate.221ωIK =Kinetic energy of a body in pure rotationKinetic energy of a body in pure translation221COMMvK =V. Rotational inertiaDiscrete rigid body  I =∑miri2Continuous rigid body  I = ∫r2 dmParallel axis theorem2MhIICOM+=Proof:Rotational inertia about a given axis = RotationalInertia about a parallel axis that extends trough body’s Center of Mass + Mh2h = perpendicular distance between the given axis and axis through COM.[]∫ ∫∫∫∫ ∫++−−+=−+−== dmbaydmbxdmadmyxdmbyaxdmrI )(22)()()(2222222R22222 MhIMhbMyaMxdmRICOMCOMCOM+=+−−=∫VI. TorqueTorque: Twist  “Turning action of force F ”.FrFrFrFrt ⊥==⋅=⋅⋅=)sin()sin(ϕϕτr┴: Moment arm of Fr : Moment arm of FtUnits: NmSign: Torque >0 if body rotates counterclockwise. Torque <0 if clockwise rotation.Superposition principle: When several torques act on a body, the net torque is the sum of the individual torquesVector quantityTangential component, Ft: does cause rotation  pulling a door perpendicular to its plane. Ft= F sinφRadial component, Fr : does not cause rotation  pulling a door parallel to door’s plane.VII. Newton’s second law for rotationατImaF=→=Proof:Particle can move only along the circular path only the tangential component of the force Ft(tangent to the circular path) can accelerate the particle along the path.ααατImrrrmrmarFmaFtttt==⋅=⋅=⋅==)()(2ατInet=VIII. Work and Rotational kinetic energyTranslation RotationWmvmvKKKifif=−=−=∆222121WIIKKKifif=−=−=∆222121ωω∫=fixxFdxW∫⋅=fidWθθθτWork-kinetic energy TheoremWork, rotation about fixed axisdFW⋅=)(ifWθθτ−=Work, constant torquevFdtdWP ⋅==ωτ⋅==dtdWPPower, rotation about fixed axisProof:22222222222121)(21)(21)(21)(212121ifififififIImrmrrmrmmvmvKKKWωωωωωω−=−=−=−=−=∆=∫⋅=→⋅=⋅⋅==fidWddrFdsFdWttθθθτθτθωτθτ⋅=⋅==dtddtdWPIX. Rolling- Rotation + Translation combined.COMvRRdtddtdsRs =⋅==→⋅=ωθθSmooth rolling motionExample: bicycle’s wheel.The motion of any round body rolling smoothly over a surface can be separated into purely rotational and purely translational motions.- Pure rotation.Rotation axis  through point where wheel contacts ground.Angular speed about P = Angular speed about O for stationary observer.COMtopvRRv 2)(2)2)((===ωω- Kinetic