PHYS 1443 – Section 003 Lecture #13AnnouncementsRotational KinematicsRotational EnergyExample 10.4Calculation of Moments of InertiaExample 10.6Similarity Between Linear and Rotational MotionsNewton’s Second Law & Uniform Circular MotionMotion in Accelerated FramesExample 6.9Work Done by a Constant ForceKinetic Energy and Work-Kinetic Energy TheoremExample 7.8Work and Kinetic EnergyPowerGravitational Potential EnergyExample 8.1Elastic Potential EnergyConservative and Non-conservative ForcesConservation of Mechanical EnergyExample 8.2Example 8.3Work Done by Non-conserve ForcesExample 8.6General Energy Conservation and Mass-Energy EquivalenceLinear Momentum and ForcesConservation of Linear Momentum in a Two Particle SystemImpulse and Linear MomentumExample 9.5Elastic and Perfectly Inelastic CollisionsExample 9.9Slide 33Center of MassCenter of Mass of a Rigid ObjectExample 9.12Example 9.13Fundamentals on RotationMonesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu1PHYS 1443 – Section 003Lecture #13Monday, Oct. 28, 2002Dr. Jaehoon Yu1. Rotational Kinetic Energy2. Calculation of Moment of Inertia3. Relationship Between Angular and Linear Quantities4. ReviewThere is no homework today!! Prepare well for the exam!!Monesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu2Announcements•2nd Term exam –This Wednesday, Oct. 30, in the class–Covers chapters 6 – 10–No need to bring blue book–Some fundamental formulae will be given–Bring your calculators but delete all the formulaeMonesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu3Rotational KinematicsThe first type of motion we have learned in linear kinematics was under a constant acceleration. We will learn about the rotational motion under constant acceleration, because these are the simplest motions in both cases.tifJust like the case in linear motion, one can obtainAngular Speed under constant angular acceleration:Angular displacement under constant angular acceleration:221ttiifOne can also obtain ifif 222Monesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu4Rotational EnergyWhat do you think the kinetic energy of a rigid object that is undergoing a circular motion is? Since a rigid body is a collection of masslets, the total kinetic energy of the rigid object isBy defining a new quantity called, Moment of Inertia, I, asKinetic energy of a masslet, mi, moving at a tangential speed, vi, isWhat are the dimension and unit of Moment of Inertia?rimiOxyviiKRKiiirmI22IKR21The above expression is simplified as2mkg 2MLWhat similarity do you see between rotational and linear kinetic energies?What do you think the moment of inertia is?Measure of resistance of an object to changes in its rotational motion.Mass and speed in linear kinetic energy are replaced by moment of inertia and angular speed.221iivm2221iirmiiK2iiirm221iiirm221Monesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu5Example 10.4In a system consists 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 .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 MllmmbbOIRK2iiirm222200 mmMlMl22Ml221I 22221Ml22Ml2iiirm2222mbmbMlMl 222 mbMl 221I 2222221mbMl 222mbMl Monesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu6Calculation 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 10.5: 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. dmR22MRdVdmMonesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu7Example 10.6Calculate 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?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 33LLM32MLMonesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu8Similarity Between Linear and Rotational MotionsAll physical quantities in linear and rotational motions show striking similarity.Similar QuantityLinear RotationalMass Mass Moment of InertiaLength of motion Distance Angle (Radian)SpeedAccelerationForce Force TorqueWork Work WorkPowerMomentumKinetic Energy Kinetic Rotational dmrI2dtdrv dtddtdva dtdmaF IfixxFdxWvFP P221mvK 221IKRLMfidWvmp IL Monesday, Oct. 28, 2002 PHYS 1443-003, Fall 2002Dr. Jaehoon Yu9Newton’s Second Law & Uniform Circular
View Full Document
Unlocking...