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Welcome back to Physics 211

Welcome back to Physics 211. Today’s agenda: Rotational dynamics Angular momentum Problems with moving axis of rotation. Reminder. elastic energy, linear momentum, collisions, center of mass, equilibrium of rigid bodies, torque, rotational dynamics, angular momentum.

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Welcome back to Physics 211

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  1. Welcome back to Physics 211 Today’s agenda: Rotational dynamics Angular momentum Problems with moving axis of rotation

  2. Reminder elastic energy, linear momentum, collisions, center of mass, equilibrium of rigid bodies, torque, rotational dynamics, angular momentum • Exam 3 in class Thursday, November 17 • Practice exam in discussion section on Wednesday • Solutions to 2003 practice exam problems on website • HW 9 due Wednesday, November 16 • Blue homework book HW - 65, 66, 72 • MPHW 6 due Friday, November 18 - assignment will be online on Friday, November 11

  3. Simple problem Cable wrapped around cylinder. Pull off with constant force F. Suppose unwind a distance d of cable 2R F * what is final angular speed of cylinder ?

  4. cylinder+cable problemenergy method • Use work-KE theorem • work W = ? W = • Moment of inertia of cylinder ? • from table: (1/2)mR2  =

  5. cylinder+cable problemconst acceleration method extended free body diagram N F * no torque due to N or FW * why direction of N ? * torque due to t= FR * hence a= FR/[(1/2)MR2] = 2F/(MR)  w FW radius R

  6. Angular Momentum 1. q r p O Point particle: |L| = |r||p|sin(q) = m|r||v| sin(q) vector form  L = r x p – direction of L given by right hand rule (into paper here) L = mvr if v is at 900 to r for single particle

  7. Angular Momentum 2. w o rigid body: * |L| = I w(fixed axis of rotation) * direction – along axis – into paper here

  8. Rotational Dynamics • t=Ia DL/Dt = t • Equivalent statements • If no net external torques t=0  • * L is constant in time • * Conservation of Angular Momentum • * Internal forces/torques do not contribute to external torque.

  9. Double bicycle wheel demo • Two wheels on same axle; can rotate independently

  10. Bicycle wheel demo • Spin wheel, then step onto platform • Apply force to tilt axle of wheel

  11. Force Acceleration Momentum Kinetic energy Torque Angular acceleration Angular momentum Kinetic energy Linear and rotational motion

  12. General motion of extended objects • Net force  acceleration of CM • Net torque about CM  angular acceleration (rotation) about CM • Resultant motion is superposition of these two motions • Total kinetic energy K = KCM+Krot

  13. A hammer is held horizontally and then released. Which way will it fall?

  14. demos • hammer falling

  15. Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After the force has stopped acting on each block, which block will spin the fastest? 1. A. 2. B. 3. C. 4. A and C. Top-view diagram

  16. Three identical rectangular blocks are at rest on a flat, frictionless table. The same force is exerted on each of the three blocks for a very short time interval. The force is exerted at a different point on each block, as shown. After each force has stopped acting, which block’s center of mass will have the greatest speed? 1. A. 2. B. 3. C. 4. A, B, and C have the same C.O.M. speed. Top-view diagram

  17. Rolling without slipping translation rotation vcm = acm =

  18. A ribbon is wound up on a spool. A person pulls the ribbon as shown. Will the spool move to the left, to the right, or will it not move at all? 1. The spool will move to the left. 2. The spool will move to the right. 3. The spool will not move at all.

  19. A ribbon is wound up on a spool. A person pulls the ribbon as shown. Will the spool move to the left, to the right, or will it not move at all? 1. The spool will move to the left. 2. The spool will move to the right. 3. The spool will not move at all.

  20. A ribbon is wound up on a spool. A person pulls the ribbon as shown. Will the spool move to the left, to the right, or will it not move at all? 1. The spool will move to the left. 2. The spool will move to the right. 3. The spool will not move at all.

  21. Pulling ribbon at a special angle

  22. Combining linear and rotational motion • Sometimes it is easy to interrelate the linear and rotational motion • e.g. spool problem • rolling with slipping

  23. Rolling without slipping N F = maCM =  = Ia= Now aCM = Ra if no slipping So, m aCM = and F = F q W

  24. demos • rolling discs with different I down inclined plane • which goes faster (CM motion) ?

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