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Physics 218 Lecture 22. Dr. David Toback. Checklist for Today. Things due Monday Chapters 12 & 13 in WebCT Things that are due Tuesday Read Chapters 14-16 Things that were due yesterday Chapter 14 problems Read Lab hand out on webpage Things due next Monday Chapter 14 in WebCT

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## Physics 218 Lecture 22

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**Physics 218Lecture 22**Dr. David Toback Physics 218, Lecture XXII**Checklist for Today**• Things due Monday • Chapters 12 & 13 in WebCT • Things that are due Tuesday • Read Chapters 14-16 • Things that were due yesterday • Chapter 14 problems • Read Lab hand out on webpage • Things due next Monday • Chapter 14 in WebCT • Next Tuesday • Exam 3, Chapters 10-13 • Mini-practice exam and bonus points available Physics 218, Lecture XXII**The Schedule**This week (4/7) • Mon: Chapter 12 & 13 material due in WebCT • Tues: Reading: Chap 14-16 • Wed: Recitation on Chap 14, Lab • Today: Chap 15, Part 1 Next Week (4/14) • Monday: Chapter 14 due in WebCT • Tues: Exam 3 (Chaps 10-13) • Wed: Recitation on Chap 15, Lab • Thurs: Lecture on Chap 15, Part 2 Week after that (4/21) • Monday: Chapter 15 & 16 due in WebCT • Tues: Reading for Chapter 18 • Tues: Lecture on Chapter 18 • Wed: Recitation on Chapter 18, Lab • 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 XXII**Overview**• Chapters 12-16 are about Rotational Motion • While we’ll do Exam 3 on Chapters 10-13, we’ll do the lectures on 12-16 in six combined lectures • Give extra time after the lectures to Study for the exam • The book does the math, I’ll focus on the understanding and making the issues more intuitive Physics 218, Lecture XXII**Rotational Motion**Chapters 12 through 16 in six combined lectures • This is the 5th of the 6 lectures • Concentrate on the relationship betweenlinearandangularvariables Today: Finish up topics Next time: Hard problems Physics 218, Lecture XXII**Angular Quantities**• Position Angle q • Velocity Angular Velocityw • Acceleration Angular Accelerationa Moving forward: • Force Torquet • Mass • Momentum • Energy Physics 218, Lecture XXII**Analogue of Mass**The analogue of Mass is called Moment of Inertia Example: A ball of mass m moving in a circle of radius R around a point has a moment of inertia F=ma t=Ia Physics 218, Lecture XXII**Calculate Moment of Inertia**Calculate the moment of inertia for a ball of mass m relative to the center of the circle R Physics 218, Lecture XXII**Moment of Inertia**• To find the mass of an object, just add up all the little pieces of mass • To find the moment of inertia around a point, just add up all the little moments Physics 218, Lecture XXII**Torque and Moment of Inertia**• Force vs. Torque F=ma t = Ia • Mass vs.Moment of Inertia Physics 218, Lecture XXII**Pulley and Bucket**A heavy pulley, with radius R, and known moment of inertia I starts at rest. We attach it to a bucket with mass m. The friction torque is tfric. Find the angular acceleration a Physics 218, Lecture XXII**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 XXII**Momentum**Momentum vs. Angular Momentum: Newton’s Laws: Physics 218, Lecture XXII**Angular Momentum**First way to define the Angular Momentum L: Physics 218, Lecture XXII**Angular Momentum Definition**Another definition: Physics 218, Lecture XXII**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 XXII**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 XXII**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 XXII**Problem Solving**For Conservation of Angular Momentum problems: BEFORE and AFTER Physics 218, Lecture XXII**Conservation of Angular Momentum**Before Physics 218, Lecture XXII**Conservation of Angular Momentum**After Physics 218, Lecture XXII**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 XXII**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 XXII**Rotational Kinetic Energy**KEtrans = ½mv2 KErotate = ½Iw2 Conservation of Energy must take rotational kinetic energy into account Physics 218, Lecture XXII**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 XXII**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 XXII**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 XXII**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 XXII**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 XXII**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 XXII**Same Problem: Forces**Same problem but with Forces Physics 218, Lecture XXII**Next Time**Exam 3!!! • Covers Chapters 10-13 • Get caught up on your homework!!! • Mini-practice exam 3 is now available Thursday: - Finish up angular “Stuff” Physics 218, Lecture XXII**End of Lecture Notes**Physics 218, Lecture XXII**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 XXII**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 XXII**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 XXII**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 XXII**Angular Quantities**• Position Angle q • Velocity Angular Velocityw • Acceleration Angular Accelerationa • Force Torquet Today we’ll finish: • Mass • Momentum • Energy Physics 218, Lecture XXII**Calculating Moments of Inertia**Here r is the distance from the axisof each little piece of mass Physics 218, Lecture XXII**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 XXII**Calculate the Moment of Inertia**Better example here… Calculate its moment of inertia R Physics 218, Lecture XXII**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 XXII**Parallel-Axis Theorem**• Quick Trick for calculating Moments • I = Icm + Mh2 • Example Physics 218, Lecture XXII**Old stuff**Physics 218, Lecture XXII**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 XXII**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 XXII**Why does the Bicycle Wheel Turn to the Right?**Physics 218, Lecture XXII

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