PHYS 1443 – Section 001 Lecture #14AnnouncementsMoment of InertiaExample for Moment of InertiaCalculation of Moments of InertiaExample for Rigid Body Moment of InertiaParallel Axis TheoremExample for Parallel Axis TheoremTorque & Angular AccelerationExample for Torque and Angular AccelerationRotational Kinetic EnergyTotal Kinetic Energy of a Rolling BodyKinetic Energy of a Rolling SphereExample for Rolling Kinetic EnergyWork, Power, and Energy in RotationAngular Momentum of a ParticleAngular Momentum and TorqueAngular Momentum of a System of ParticlesExample for Angular MomentumMonday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu1PHYS 1443 – Section 001Lecture #14Monday, June 26, 2006Dr. Jaehoon Yu•Moment of Inertia•Parallel Axis Theorem •Torque and Angular Acceleration•Rotational Kinetic Energy•Work, Power and Energy in Rotation •Angular Momentum & Its Conservation•Similarity of Linear and Angular QuantitiesMonday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu2Announcements•Quiz Results–Average: 60.2–Top score: 100•Last quiz this Wednesday–Early in the class–Covers Ch. 10 – what we cover tomorrow•Final exam–Date and time: 8 – 10am, Friday, June 30–Location: SH103–Covers: Ch 9 – what we cover by Wednesday–No class this ThursdayMonday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu3Moment of Inertia Rotational Inertia:What are the dimension and unit of Moment of Inertia?I �2mkg 2MLMeasure of resistance of an object to changes in its rotational motion. Equivalent to mass in linear motion.Determining Moment of Inertia is extremely important for computing equilibrium of a rigid body, such as a building.I �For a group of particlesFor a rigid body2i iim r�2r dm�Dependent on the axis of rotation!!!Monday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu4Example for Moment of InertiaIn a system of four small spheres as shown in the figure, assuming the radii are negligible and the rods connecting the particles are massless, compute the moment of inertia and the rotational kinetic energy when the system rotates about the y-axis at angular speed .ISince the rotation is about y axis, the moment of inertia about y axis, Iy, isRKThus, the rotational kinetic energy is Find the moment of inertia and rotational kinetic energy when the system rotates on the x-y plane about the z-axis that goes through the origin O.xyThis is because the rotation is done about y axis, and the radii of the spheres are negligible.Why are some 0s?M MllmmbbOIRK2iiirm2Ml=22Ml221I 22221Ml22Ml2iiirm2Ml= 222 mbMl 221I 2222221mbMl 222mbMl 2Ml+20m+ �20m+ �2Ml+2mb+2mb+Monday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu5Calculation of Moments of InertiaMoments of inertia for large objects can be computed, if we assume the object consists of small volume elements with mass, mi.It is sometimes easier to compute moments of inertia in terms of volume of the elements rather than their massUsing the volume density, , replace dm in the above equation with dV.The moment of inertia for the large rigid object isHow can we do this?iiimmrIi20lim dmr2dVdmThe moments of inertia becomes dVrI2Example: Find the moment of inertia of a uniform hoop of mass M and radius R about an axis perpendicular to the plane of the hoop and passing through its center.xyROdmThe moment of inertia is dmrI2What do you notice from this result?The moment of inertia for this object is the same as that of a point of mass M at the distance R. dmR22MRdVdmMonday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu6Example for Rigid Body Moment of InertiaCalculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis perpendicular to the rod and passing through its center of mass.The line density of the rod is What is the moment of inertia when the rotational axis is at one end of the rod.xyLxdxLMso the masslet is dmThe moment of inertia is I dmrI2Will this be the same as the above? Why or why not?Nope! Since the moment of inertia is resistance to motion, it makes perfect sense for it to be harder to move when it is rotating about the axis at one end.dxdxLM dmr2dxLMxLL2/2/22/2/331LLxLM33223LLLM433LLM122MLdxLMxL02LxLM0331 033 LLM 33LLM32MLMonday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu7xy(x,y)xCM(xCM,yCM)yCMCMParallel Axis TheoremMoments of inertia for highly symmetric object is easy to compute if the rotational axis is the same as the axis of symmetry. However if the axis of rotation does not coincide with axis of symmetry, the calculation can still be done in simple manner using parallel-axis theorem.2MDIICMyxrMoment of inertia is defined dmrI2Since x and y arex’y’'xxxCMOne can substitute x and y in Eq. 1 to obtain dmyyxxICMCM22''Since the x’ and y’ are the distances from CM, by definition0'dmxDTherefore, the parallel-axis theoremCMIMD 2What does this theorem tell you?Moment of inertia of any object about any arbitrary axis are the same as the sum of moment of inertia for a rotation about the CM and that of the CM about the rotation axis.( )2 2 (1)x y dm= +�'yyyCM dmyxdmyydmxxdmyxCMCMCMCM2222'''2'20' dmy dmyxdmyxICMCM2222''Monday, June 26, 2006 PHYS 1443-001, Summer 2006Dr. Jaehoon Yu8Example for Parallel Axis TheoremCalculate the moment of inertia of a uniform rigid rod of length L and mass M about an axis that goes through one end of the rod, using parallel-axis theorem.The line density of the rod is Using the parallel axis theoremLMso the masslet is dxLMdxdm The moment of inertia about the CM CMIMDIICM2The result is the same as using the definition of moment of inertia.Parallel-axis theorem is useful to compute moment of inertia of a rotation of a rigid object with complicated shape about an arbitrary axisxyLxdxCMMLML22212
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