Wed. March 9th1 PHSX213 class Class stuff –HW6W returned. –HW7. Worth 1.5 times normal assignment. Due Wed Mar. 16 th 6PM. –Practice Exam. –Projects … –MidTerm2 : Wed. March 16 th 8:00 – 9:30 PM. Same place as before (Budig 120). –No classes next Wed. and Fri. (as planned) More ROTATION

Wed. March 9th2 Rotational Inertia Demo

Wed. March 9th3 Work and Rotational KE We saw for linear motion that, W = ∫ F dx =  K = K f – K i = ½ m (v f 2 – v i 2 ) For rotational motion about a fixed axis, W = ∫  d  =  K = K f – K i = ½ I (  f 2 –  i 2 )

Wed. March 9th4 Example Pulley, uniform disk, mass M=0.5 kg, radius R=0.12m. Mass m 1 =0.4 kg, m 2 = 0.6 kg Cord doesn’t slip. Disk rotates freely (ignore friction). What is the magnitude of the acceleration of the blocks? What is the tension in each cord ?

Wed. March 9th5 Reading Quiz Which force gives the torque needed for objects to roll without slipping: A) normal force B) force due to gravity C) kinetic friction D) static friction do rolling example demo in parallel

Wed. March 9th6 Reading Quiz Which force gives the torque needed for objects to roll without slipping: A) normal force B) force due to gravity C) kinetic friction D) static friction

Wed. March 9th7 Check-Point 1 A wheel rolls without slipping along a horizontal surface. The center of the disk has a translational speed, v. The uppermost point on the wheel has a translational speed of : A) 0 B) v C) 2v D) need more information

Wed. March 9th8 Rolling A rolling object can be considered to be rotating about an axis through the CoM while the CoM moves.

Wed. March 9th9 Rolling The rotational motion obeys  = I CoM  while the translational motion obeys  F = m a CoM Note that point P in contact with the ground has zero velocity ie. stationary. So static friction is what matters at that point. –Not convinced it is stationary – think about clean (unsmudged) tire tracks in the snow

Wed. March 9th10 Check-Point 2 A solid disk and a ring roll down an incline. The ring is slower than the disk if A. m ring = m disk, where m is the inertial mass. B. r ring = r disk, where r is the radius. C. m ring = m disk and r ring = r disk. D. The ring is always slower regardless of the relative values of m and r.

Wed. March 9th11 Forces in Rolling Rolls smoothly down the ramp without sliding/slipping. How much torque about the CoM does each force produce ? How big is f S ? How big is the acceleration ? Note : I’ll define +x down the incline

Wed. March 9th12 Rolling One can derive that : f S = I a/R 2 = [  /(1+  )] mg sin  And that, a = g sin  /(1 + I/(MR 2 )) = g sin  /(1+  ) Where we have defined , via I =  M R 2  hoop = 1,  cylinder = 0.5,  solid sphere = 0.4

Wed. March 9th13 Discuss rolling in terms of KE From energy conservation considerations. Note the frictional force in this case doesn’t oppose the angular motion, and the work done by this non-conservative force is transformed into rotational kinetic energy. –W = ∫  d  And we find that K = ½ m v 2 + ½ I  2 = ½ m (1+  ) v 2 ( = mg h) Since the contact point doesn’t move, there is no translational displacement.

Wed. March 9th14 Another way to look at this Stationary observer sees rotation about an axis at point P with  = v/R. Using the parallel axis theorem, I P = I CoM + mR 2 So, K = ½ I P  2 = ½ I CoM  2 + ½ m v 2 just as before.

Wed. March 9th15 Rolling Body on Inclined Plane Demo

Wed. March 9th16 Loop-the-loop

Wed. March 9th17 Torque Definition for a Particle  = r  F NB Only makes sense to talk about the torque wrt or about a certain point Vector product means that the torque is directed perpendicular to the plane formed by (r, F). Whether it is up or down is from the right-hand rule convention

Wed. March 9th18 Angular Momentum Demo

Wed. March 9th19 Angular Momentum Definition for a Particle l = r  p = r  m v NB Only makes sense to talk about the angular momentum wrt or about a certain point

Wed. March 9th20 Angular Momentum Definition for a Rigid Body About a Fixed Axis L =  l i = I  In analogy to P = M v

Wed. March 9th21 Newton II for Rotation  F = dp/dt ( the general form of Newton II)   = dl/dt It is the net torque that causes changes in angular momentum.

Wed. March 9th22 Angular Momentum Conservation Just as for Newton II for linear motion where if the net force was zero, dp/dt =0 => p conserved. For rotational motion, if the net torque is zero, dl/dt =0, so l conserved.

Wed. March 9th23 Angular Momentum Conservation Problems

Wed. March 9th24 Relating linear and angular variables a tan =  r a R = v 2 /r =  2 r v =  r Remarks on units and dimensional analysis