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Classical Mechanics Lecture 15 Today’s Concepts: a) Parallel Axis Theorem b) Torque & Angular Acceleration Mechanics Lecture 15, Slide 1 Schedule One unit per lecture! I will rely on you watching and understanding pre-lecture videos!!!! Lectures will only contain summary, homework problems, clicker questions, Example exam problems…. Midterm 3 Wed Dec 11 Mechanics Lecture 14, Slide 2 Main Points Mechanics Lecture 15, Slide 3 Main Points Mechanics Lecture 15, Slide 4 Main Points Mechanics Lecture 15, Slide 5 Parallel Axis Theorem Mechanics Lecture 15, Slide 6 Parallel Axis Theorem Smallest when D = 0 Mechanics Lecture 15, Slide 7 Clicker Question A. B. C. D. A solid ball of mass M and radius is connected to a rod of mass m and length L as shown. What is the moment of inertia of this system about an axis perpendicular to the other end of the rod? 2 1 2 2 2 A) I = MR ML mL 5 3 2 1 2 2 B) I = MR mL ML2 5 3 R M 2 1 2 2 C) I = MR mL 5 3 m L 58% 1 D) I = ML2 mL2 3 25% 17% 0% axis Mechanics Lecture 15, Slide 8 CheckPoint A ball of mass 3M at x = 0 is connected to a ball of mass M at x = L by a massless rod. Consider the three rotation axes A, B, and C as shown, all parallel to the y axis. For which rotation axis is the moment of inertia of the object smallest? (It may help you to figure out where the center of mass of the object is.) A C B y x 3M M L 100% got this right !!! 0 L/4 L/2 Mechanics Lecture 15, Slide 9 Right Hand Rule for finding Directions Why do the angular velocity and acceleration point perpendicular to the plane of rotation? Mechanics Lecture 15, Slide 11 Clicker Question A. B. C. D. A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular velocity vector point when the ball is rolling up the ramp? 64% A) Into the page 14% 14% 7% B) Out of the page C) Up D) Down Mechanics Lecture 15, Slide 12 A. Enter Question Text B. C. D. A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling up the ramp? 54% 46% A) Into the page 0% 0% B) Out of the page Mechanics Lecture 15, Slide 13 A. Enter Question Text B. C. D. A ball rolls across the floor, and then starts up a ramp as shown below. In what direction does the angular acceleration vector point when the ball is rolling back down the ramp? 75% A) into the page 25% B) out of the page 0% 0% Mechanics Lecture 15, Slide 14 Torque t = rF sin(q ) Mechanics Lecture 15, Slide 15 Mechanics Lecture 15, Slide 16 CheckPoint In Case 1, a force F is pushing perpendicular on an object a distance L/2 from the rotation axis. In Case 2 the same force is pushing at an angle of 30 degrees a distance L from the axis. In which case is the torque due to the force about the rotation axis biggest? A) Case 1 B) Case 2 C) Same axis axis L/2 100% got this right 90o Case 1 F L 30o F Case 2 Mechanics Lecture 15, Slide 17 CheckPoint In which case is the torque due to the force about the rotation axis biggest? axis A) Case 1 B) Case 2 C) Same 90o L/2 F A) Perpendicular force means more torque. B) F*L = torque. L is bigger in Case 2 and the force is the same. C) Fsin30 is F/2 and its radius is L so it is FL/2 which is the same as the other one as it is FL/2. Case 1 axis L 30o F Case 2 Mechanics Lecture 15, Slide 18 Torque and Acceleration Rotational “2nd law” Mechanics Lecture 15, Slide 19 Similarity to 1D motion Mechanics Lecture 15, Slide 20 Summary : Torque and Rotational “2nd law” Mechanics Lecture 15, Slide 21 Clicker Question A. B. Strings are wrapped around the circumference of two solid disks and pulled with identical forces. Disk 1 has a bigger radius, but both have the same moment of inertia. C. Which disk has the biggest angular acceleration? A) Disk 1 w2 w1 43% 36% 21% B) Disk 2 C) same F F Mechanics Lecture 15, Slide 22 Clicker/Checkpoint A. B. C. Two hoops can rotate freely about fixed axles through their centers. The hoops have the same mass, but one has twice the radius of the other. Forces F1 and F2 are applied as shown. How are the magnitudes of the two forces related if the angular acceleration of the two hoops is the same? 83% A) F2 = F1 B) F2 = 2F1 17% F2 0% F1 C) F2 = 4F1 Case 1 Case 2 Mechanics Lecture 15, Slide 23 CheckPoint How are the magnitudes of the two forces related if the angular acceleration of the two hoops is the same? A) F2 = F1 F2 F1 B) F2 = 2F1 C) F2 = 4F1 M, R Case 1 M, 2R Case 2 B) twice the radius means 4 times the moment of inertia, thus 4 times the torque required. But twice the radius=twice the torque for same force. 4t = 2F x 2R Mechanics Lecture 15, Slide 24 t = RF sin q q q = 900 q = 00 q = 90 36 = 54 Mechanics Lecture 15, Slide 25 t = RF sin q Direction is perpendicular to both R and F, given by the right hand rule tx = 0 ty = 0 t z = t F t F t F 1 2 3 Mechanics Lecture 15, Slide 26 1 MR 2 2 (i) I DISK = (ii) t = I 1 2 (iii) K = Iw 2 Use (i) & (ii) Use (iii) Mechanics Lecture 15, Slide 27 Moment of Inertia I total = I rod I sphere 1 2 2 I total = mrod L2rod msphere Rsphere msphere ( Lrod Rsphere ) 2 3 5 1 2 2 I total = msphere (4 Rsphere ) 2 msphere Rsphere msphere (5Rsphere ) 2 15 5 16 2 I total = ( 25)msphere ( Rsphere ) 2 15 5 7 I total = (26 )msphere ( Rsphere ) 2 15 Mechanics Lecture 15, Slide 28 Moment of Inertia Lrod 2 t = rF sin q = F sin 90 t = I = Lrod F 2 t I = (26 7 )msphere ( Rsphere ) 2 15 Mechanics Lecture 15, Slide 29 Moment of Inertia RCM = mrod RCM ,rod msphere RCM , sphere mrod msphere = 4.5Rsphere I total = I rod I sphere 5 1 1 2 2 I total = mrod L2rod mrod ( Rsphere ) 2 msphere Rsphere msphere ( Rsphere ) 2 2 2 12 5 1 5 1 1 2 2 I total = msphere (4 Rsphere ) 2 msphere ( Rsphere ) 2 msphere Rsphere msphere ( Rsphere ) 2 5 2 2 60 5 16 5 2 1 I total = msphere ( Rsphere ) 2 60 4 5 4 Mechanics Lecture 15, Slide 30 Moment of Inertia 5 2 t = rF sin q = Rsphere F sin 0 t = I = t I = 0 16 5 2 1 2 msphere ( Rsphere ) 60 4 5 4 Mechanics Lecture 15, Slide 31 Moment of Inertia I total = I rod I sphere 1 2 2 I total = mrod L2rod mrod (4Rsphere ) 2 msphere Rsphere msphere ( Rsphere ) 2 12 5 Mechanics Lecture 15, Slide 32