final_soln - Name Student ID Physics 321 Final Professor E...

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Unformatted text preview: Name Student ID Physics 321 Final May 11, 2005 Professor E. Cheu 0 Clearly Show all of your work. 0 Work: that is scratched out or erased will not be counted. 0 All answers must include units where appropriate. 0 Clearly indicate your answer by putting a has around it. 1. A thin, uniform ring of mass M has a radius R. A small mass m0 is located a distance 2 above the center of the ring as shown. (a) Find the gravitational potential due to the ring at a point 2 above the center of the ring. (b) Determine the force on the small mass as a function of z. (c) If the mass is released from rest at 2:, what is its velocity when it reaches the center of the ring? ‘3 f: 92 = - G S :97 “I r! the top so that the rod swings M is attached to a pivot at d towards the 2. A thin rod of length L and mass M / 3 starts crawling from the top of the 1:0 in a vertical plane. A bug of mass bottom. (a) Determine the equations of motion for the system. Explicitly calculate the required moments of inertia. (1)) Determine the period of small oscillations as crawled from the pivot point. You can assume that it can be ignored. a function of the distance the bug has that the bug’s velocity is small enough ©1311! 3. A very small bullet of mass mb is shot at a cylinder of mass “:71C and radius R. The bullet comes to rest at the center of the cylinder. (a) Calculate the moment of inertia of the cylinder. about its axis of symmetry. (b) Determine the velocity of the system after the bullet comes to rest if the cylinder rolls t the bullet passes through without slipping on the horizontal surface. You can assume the the cylinder without changing the mass distribution and that the time between when the bullet hits the cylinder and when it comes to a stop is negligible. (0) Now assume that the cylinder slides without rolling, what is the ratio of the velocity for rolling relative to sliding? ml: |'_'D——> a) I) -: SQCX‘W‘S') ald/ :- $92-7- rau‘JSdt- ‘-= {7 $21k 2 ML __ - z‘ i a 1v s2: ‘0) car-s. arc {firm alw- mbmlm‘l'vn’)‘. m in: we‘ve ar- V9; 4. The components of the inertia tensor, I , of a rigid body in coordinate system 0 are given by: 3—D 0 0 I: 0 B D 0 D B (a) Find the principal moments of inertia. (b) Find the orientation of the principal axes in the 0 system. (A g-n “1 o o 0 8-1. D t (5 b 39 9-1 a) a; wash—ow] (3’D’D(I‘ ~21?» +e‘eb‘3 = 0 MM“ £ n: 23M”) 3 1 T’ 1 L ’— = st 0 ’Hu. sun Q H6 was, 84'th is o - 9mm» 0 o w 0 g-(GBPD) '1) “a :0 O D g-LCS+D\ “3 z; ~2Du‘50 ~90” Dag-1° ,+DuL¥-Dwz°° 79 “1—:qu ...
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