exam4_soln_ch11_12_14 - 1 A very thin sheet has a mass...

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Unformatted text preview: 1. A very thin sheet has a mass density a = 720333;, where C is a positive constant. It lies in the m—y plane and its sides are both of length a. (a) Determine the inertia tensor for the plane. (b) What are the principal moments of inertia? It? 1 Sf 55'5“ka ‘ Yixcl' a In - SOZCLJ)(3L+%) AKAJ - ”HQ 37%;: _ ‘ica‘ IL '—‘ —- - I .5: ’_ , { ._. I 382033) K7 Axel) 71c 31 - {E L, [:3 : SQ'KJQDX? 1‘:- it} 1. $910.13) i)? fibeokj ‘- f) [-33 2 SGLCkJ\ Ck‘fj-B Akdj :. 72 Q (—53%: Jr 352; 2:3) : {90.5“ I 2 ”\Ca‘ ..ch.‘ o v 6 CAL; $0 a,‘ b O o 1350-3 ‘1—1 0 _ _ 1 ca‘ 6 By. :9 0" IX“ ”I” “‘0 g (H D LIL—1y a? -I%14—'L1"3 ° 0 lirl Qe—‘QCI’PIXI—I) =7 I: 1, 19,11 2. A projectile is shot from a latitude A as shown below. Its initial velocity is 110 due south, and its angle relative to the local horizontal is 45° as shown below. Because of the Earth’s rotation, the projectile appears to deviate from its original path. Let the Earth’s angular velocity be u). (a) As a. function of time (t) determine the component of the projectile’s velocity that results from the Coriolis effect. Hint: Project all vectors onto the :17 and y axes as shown. (is) Determine how far the particle appears to deviate from its original path as a function of t. A F; = —2m\23“\! L3: "‘ LDC-05X? tog-AXE} V 3’ Va Cote; r} Vbs\inOJA ~86§ ‘ V0 45° “ :1 ‘1‘ F: 3'- Zm VDCMB vD sme-fit a ‘UCDJX wSinX (3 ~ A E: lmi VOC-Jeusink + UC05\(V931A6*5'tj-jz A . )4 a: % - 26/0 casewsink {- gco>\&65‘h3—3f) '3 V?— fjvocosaws‘mXé + 2% Sin Qucoaké _. Cu coaxsfii‘ ML!- CzD Slfltt 411". h m.» Conrail: b\ V: €531 Q’QL Ohm ’“tk TMLLLc 73%;, 6 73‘) f 7' [Vgu (C056 Slnx 4- SiA9C03}3€L .. (.1605). .6319? /—”}’-§”’ 7 if 3. A body has an inertia tensor given by: 2Mb2 0 0 0 5Mb2 3Mb2 0 {5sz 511/1322 where b is a positive constant. The body rotates about the vertical axis with an angular frequency w. The angle between the vertical and the symmetry axis of the body is as, where the origin is located at the intersection of the principal axes. (a) Find the principal moments of inertia for this system. (b) Determine the angular momentum about the origin in the body frame. (0) What is the magnitude and direction of the torque in the inertial frame. Zi'vul)1L -’f_ <3 C) o grub: 3my¢l —_ o o 3"“34. S'n Lt-l figmbt—llli SIMS-13'“ ~ and?) lo .. Um51"1)(28‘h§b" - lumbt-‘IL— qmg‘]: a x (my 'I)( 8mL - LXML‘il .. 1. Tu “: 11.1.: Zmz’t 1:13" firm; A 3N2" *UC§\r~°L L 2 ZmbL b 6M5." +mu3°L fix: - 2mbluslni La?“ ’4'“- imbtu‘JM'L m... Car-w E 'A " (“'- Ug‘mr’t; '4' MCQJ-é‘X 'Zmlzu$l\qoi’\\c -}— fimllwcoso'la) 2 r' E gmll UL S‘IOJCD3‘! ’i‘ ZMBI “153‘“06565‘13 ': ...
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