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Physics 218 Lecture 18 Dr. David Toback Physics 218, Lecture XVIII 1 Announcements • Chapter 9 HW due Wed Nov • Chapter 10 HW due Monday th Nov 13 , as usual th 8 • First Announcement of Exam 3: st –November 21 –Tuesday before Thanksgiving! Physics 218, Lecture XVIII 2 Rotational Motion Chapters 9 and 10 in four lectures • Lecture three of the four lectures • Concentrate on the relationship between linear and angular variables • Today: Finish up topics • Thursday: Hard problems Physics 218, Lecture XVIII 3 Physics 218, Lecture XVIII 4 Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a • Force Torque t • Mass Moment of Inertia I Today we’ll finish: – Momentum – Energy Physics 218, Lecture XVIII 5 Momentum Momentum vs. Angular Momentum: p mv L Iw Newton’s Laws: dp dL F t dt dt Physics 218, Lecture XVIII 6 Angular Momentum L Iw First way to define the Angular Momentum L: dw d ( Iw ) d ( L ) dL t Ia I dt dt dt dt dL t Ia dt Physics 218, Lecture XVIII 7 Angular Momentum Definition Another definition: L rp Physics 218, Lecture XVIII 8 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 XVIII 9 Conservation of Angular Momentum dL t dt if t 0 L Const 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 XVIII 10 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 L Iw Physics 218, Lecture XVIII 11 Problem Solving For Conservation of Angular Momentum problems: BEFORE and AFTER Physics 218, Lecture XVIII 12 Conservation of Angular Momentum Before Physics 218, Lecture XVIII 13 Conservation of Angular Momentum After Physics 218, Lecture XVIII 14 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 XVIII 15 Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a • Force Torque t • Mass Moment of Inertia I Today we’ll finish: – Momentum Angular Momentum L – Energy Physics 218, Lecture XVIII 16 Rotational Kinetic Energy 2 ½mv KEtrans = 2 KErotate = ½Iw Conservation of Energy must take rotational kinetic energy into account Physics 218, Lecture XVIII 17 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 XVIII 18 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 2/5MR Z Physics 218, Lecture XVIII 19 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 XVIII 20 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 XVIII 21 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? Physics 218, Lecture XVIII R 22 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 XVIII 23 Same Problem: Forces Same problem but with Forces Physics 218, Lecture XVIII 24 Next Time • More problems on Chapters 9 & 10 –Please get caught up on homework!!! –Believe it or not exam 3 is just st around the bend, Nov 21 –Tuesday before Thanksgiving! • Chap 9 HW due Wednesday Nov 8th th • Chap 10 HW due Nov 13 Physics 218, Lecture XVIII 25 Physics 218, Lecture XVIII 26 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 XVIII 27 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 XVIII 28 Angular Quantities • Position Angle q • Velocity Angular Velocity w • Acceleration Angular Acceleration a • Force Torque t Today we’ll finish: – Mass – Momentum – Energy Physics 218, Lecture XVIII 29 Calculating Moments of Inertia I mr 2 I r dm 2 Here r is the distance from the axis of each little piece of mass Physics 218, Lecture XVIII 30 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 Physics 218, Lecture XVIII R 31 Calculate the Moment of Inertia Better example here… R Calculate its moment of inertia Physics 218, Lecture XVIII 32 Physics 218, Lecture XVIII 33 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 XVIII 34 Parallel-Axis Theorem • Quick Trick for calculating Moments • I = Icm + Mh2 • Example Physics 218, Lecture XVIII 35