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Physics 218 Lecture 23. Dr. David Toback. Checklist for Today. Things due Monday Chapter 14 in WebCT Things that were due yesterday Chapter 15 problems as Recitation Prep Things due next Monday Chapter 15 & 16 in WebCT Next week Chapter 18 reading. The Schedule. This Week (4/14)

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## Physics 218 Lecture 23

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**Physics 218Lecture 23**Dr. David Toback Physics 218, Lecture XXIII**Checklist for Today**• Things due Monday • Chapter 14 in WebCT • Things that were due yesterday • Chapter 15 problems as Recitation Prep • Things due next Monday • Chapter 15 & 16 in WebCT • Next week • Chapter 18 reading Physics 218, Lecture XXIII**The Schedule**This Week (4/14) • Monday: Chapter 14 due in WebCT • Tues: Exam 3 (Chaps 10-13) • Wed: Recitation on Chap 15, Lab • Thurs: Last lecture on Chaps 12-16 Next Week (4/21) • Monday: Chapter 15 & 16 due in WebCT • Tues: • Reading for Chapter 18 • Lecture on Chapter 18 • Wed: Recitation on Chapter 18 • Thurs: Last lecture, Chapter 18 Week after that (4/28) • No lectures or recitations Week after that (5/5) • Final: Monday May 5th, 1PM-3PM in this room Physics 218, Lecture XXIII**Overview**• Chapters 12-16 are about Rotational Motion • Concentrate on the relationship betweenlinearandangularvariables • Do them in six combined lectures • Last lecture today • Angular Momentum and Energy • The book does the math, I’ll focus on the understanding and making the issues more intuitive Physics 218, Lecture XXIII**Angular Quantities**• Position Angle q • Velocity Angular Velocityw • Acceleration Angular Accelerationa • Force Torquet • Mass Moment of InertiaI Today we’ll finish: • Momentum • Energy Physics 218, Lecture XXIII**Momentum**Momentum vs. Angular Momentum: Newton’s Laws: Physics 218, Lecture XXIII**Angular Momentum Definition**Another definition: Physics 218, Lecture XXIII**Angular Motion of a Particle**Determine the angular momentum, L, of a particle, with mass m and speed v, moving in circular motion with radius r Physics 218, Lecture XXIII**Conservation of Angular Momentum**By Newton’s laws, the angular momentum of a body can change, but the angular momentum for a system cannot change Conservation of Angular Momentum Same as for linear momentum Physics 218, Lecture XXIII**Ice Skater**• This one you’ve seen on TV • Try this at home in a chair that rotates • Get yourself spinning with your arms and legs stretched out, then pull them in Physics 218, Lecture XXIII**Problem Solving**For Conservation of Angular Momentum problems: BEFORE and AFTER Physics 218, Lecture XXIII**Conservation of Angular Momentum**Before Physics 218, Lecture XXIII**Conservation of Angular Momentum**After Physics 218, Lecture XXIII**Clutch Design**As a car engineer, you model a car clutch as two plates, each with radius R, and masses MA and MB (IPlate = ½MR2). Plate A spins with speed w1 and plate B is at rest. You close them so they spin together Find the final angular velocity of the system Physics 218, Lecture XXIII**Angular Quantities**• Position Angle q • Velocity Angular Velocityw • Acceleration Angular Accelerationa • Force Torquet • Mass Moment of InertiaI Today we’ll finish: • Momentum Angular MomentumL • Energy Physics 218, Lecture XXIII**Rotational Kinetic Energy**KEtrans = ½mv2 KErotate = ½Iw2 Conservation of Energy must take rotational kinetic energy into account Physics 218, Lecture XXIII**Rotation and Translation**• Objects can both Rotate and Translate • Need to add the two KEtotal = ½ mv2 + ½Iw2 • Rolling without slipping is a special case where you can relate the two • V = wr Physics 218, Lecture XXIII**Rolling Down an Incline**You take a solid ball of mass m and radius R and hold it at rest on a plane with height Z. You then let go and the ball rolls without slipping. What will be the speed of the ball at the bottom? What would be the speed if the ball didn’t roll and there were no friction? Note: Isphere = 2/5MR2 Z Physics 218, Lecture XXIII**A bullet strikes a cylinder**A bullet of speed V and mass m strikes a solid cylinder of mass M and inertia I=½MR2, at radius R and sticks. The cylinder is anchored at point 0 and is initially at rest. What is w of the system after the collision? Is energy Conserved? Physics 218, Lecture XXIII**Rotating Rod**A rod of mass uniform density, mass m and length l pivots at a hinge. It has moment of inertia I=ml/3 and starts at rest at a right angle. You let it go: What is w when it reaches the bottom? What is the velocity of the tip at the bottom? Physics 218, Lecture XXIII**Less Spherical Heavy Pulley**A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed w in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this tfric. What is this better estimate of the moment of Inertia? R Physics 218, Lecture XXIII**Person on a Disk**A person with mass m stands on the edge of a disk with radius R and moment ½MR2. Neither is moving. The person then starts moving on the disk with speed V. Find the angular velocity of the disk Physics 218, Lecture XXIII**Same Problem: Forces**Same problem but with Forces Physics 218, Lecture XXIII**Challenge Exam**• Announcement of this semester’s 218 Challenge Exam: Monday April 28th at 6:00PM • Not required (just for fun) • Does not (will not!!!) count as part of your final grade • Test your skills against the best Aggies from all the Physics 218 sections (not just this lecture) on Physics 218 material • Students who perform well will be recognized as Mechanics Scholars, and honored at a banquet in their honor. Other prizes, including cash. • Handout information on my WebSite Physics 218, Lecture XXIII**Next Week**• Reading: Chapter 18 “Harmonic Motion” • Last topic for the course Notes for the final: • No more “bonus” points • Exam is worth 200 points • Problem 8 will consist of two items: • 5 points for getting 100% on all your homework & WebCT quizzes • 5 more points for getting 100 on the Mini-practice Exam final Physics 218, Lecture XXIII**End of Lecture Notes**Physics 218, Lecture XXIII**Spherical Heavy Pulley**A heavy pulley, with radius R, starts at rest. We pull on an attached rope with a constant force FT. It accelerates to an angular speed of w in time t. What is the moment of inertia of the pulley? R Physics 218, Lecture XXIII**Less Spherical Heavy Pulley**A heavy pulley, with radius R, starts at rest. We pull on an attached rope with constant force FT. It accelerates to final angular speed w in time t. A better estimate takes into account that there is friction in the system. This gives a torque (due to the axel) we’ll call this tfric. What is this better estimate of the moment of Inertia? R Physics 218, Lecture XXIII**Exam II**• Mean = 75 • Please check to make sure they added your points correctly AND entered them into WebCT correctly!!! • Average on first two exam = 76% • Straight scale so far… • Reading quizzes should be passed back in recitation Physics 218, Lecture XXIII**Next Time**• Chapter 11 • Reading Questions: Q11.X & Q11.X • XXX FIXME!!! • Math, Torque, Angular Momentum, Energy again, but more sophisticated • The material will not be on the 3rd exam, but will help with the exam. It will all be on the final • HW 10 Due Monday • Exam 3 is next Thursday, April 22nd Physics 218, Lecture XXIII**Angular Quantities**• Position Angle q • Velocity Angular Velocityw • Acceleration Angular Accelerationa • Force Torquet Today we’ll finish: • Mass • Momentum • Energy Physics 218, Lecture XXIII**Calculating Moments of Inertia**Here r is the distance from the axisof each little piece of mass Physics 218, Lecture XXIII**Calculate the Moment of Inertia**A pulley has mass M, uniform density, radius R, and rotates around its fixed axis Calculate its moment of inertia R Physics 218, Lecture XXIII**Calculate the Moment of Inertia**Better example here… Calculate its moment of inertia R Physics 218, Lecture XXIII**Hollow Cylinder**Consider a hollow cylinder with uniform density, inner radius R1, outer radius R2 and total Mass M. Find the moment of Inertia Physics 218, Lecture XXIII**Parallel-Axis Theorem**• Quick Trick for calculating Moments • I = Icm + Mh2 • Example Physics 218, Lecture XXIII**Old stuff**Physics 218, Lecture XXIII**Kepler’s 2nd Law**2nd Law: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out area in equal periods of time Physics 218, Lecture XXIII**Atwood’s Machine**A pulley with a fixed center (at point O), radius R0 and moment of inertia I, has a massless rope wrapped around it (no slipping). The rope has two masses, m1 and m2 attached to its ends. Assume m2>m1 • What is the acceleration of the system? • Do some checks. Physics 218, Lecture XXIII**Why does the Bicycle Wheel Turn to the Right?**Physics 218, Lecture XXIII**Angular Momentum**Again we use the Cross Product: Derivation of St = dL/dt Physics 218, Lecture XXIII**L for a system of many bodies**• Have to be careful with Angular Momentum • t = dl/dt for a single particle • St = S(dl/dt) for a system of many particles • All internal torques cancel because of Newton’s law (all internal forces are equal and opposite) • Reference Frame matters. Only true for: • The origin is an inertial Reference Frame • The center of mass Physics 218, Lecture XXIII**L for a Rigid Body**Find the angular momentum, L, for this body given that it is rotating around the Z axis with angular velocity w. Physics 218, Lecture XXIII

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