Fall 2008 Exam 1 Soln

# Fall 2008 Exam 1 Soln - Problem I(25 Points Given two...

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Unformatted text preview: Problem I (25 Points) Given two vectors, A (1, -2, “2), and B (6, 3, 2), ﬁnd: a) A unit vector paraiiei to B (7) b) The angle between A and B (10) c) llprojABll (8) Solution: a)B= [6,3,2], 1IB||=./(62+32+22):JE§=7 2m 6, 3, 2 , , Unit vector paraileitoB= [ 7 ], or [%!+—:-J+%-k] 5 pts b) To ﬁnd the angle betwen A and B, we evaiuate AB AB : llA'!.llBl].cos (8), 6 is the angle between A and B 2 pt A.B= (1x6+(-2)x3 +(~2)x2)=w4 3pts llAll=x/(12+(m2)2+(-2)2 --=~f§=3 23: or we can com ute'é m[~]—i—-E'—-g~k] ’ p A 3 3j 3 ’ cosB=[§i+§~j+i2—k].[ling—g~k]m 2—3—1 :~.19 7 7 7 3 3 3 7 7 21 —4 —4 e :__..=__=w,19 2 t COS” 7x3 21 p5 , 6 ""—-“ 300.930 lpt,0 ifacute angle . AB A "41—2—2 [4,8,8] ww _ : ___ :0 7 2 5 5 t °“’“°“ nAu liAlt Jim] 9 p5 llprojAB” =% (“4): + 82 + 82 )m g, or 1.33 3 pts Problem 2 (25 Points) As Shown in the: ﬁgure at {he rig-gmn block A. is supported by two cables which are securely attached at ring P. One ()FLhBSC cables is 6 ft in length and has its other end fastened main-31y in tlw Wall 211 (I). This other cable: passes over a iiictionlﬁss pulley and is then connected to block 8. The weight of block A is W4 2 SOIbS, distance y = 3 ft. and the system is in static equilibrium. 3) Draw the free body diagrams; needed to solve this problem. (5 points) b) Determ'im: the weight. ofblock B necessary to maintain the equilibrium gecmieiry Shawn. (10 points) 9) Determine the tension in cable OF. (H) pointS) F 80‘ ' A F” D- 8 Fr g A 'l v‘wT‘E; i 13 2 ‘ I’I‘JWI’S J ('35 1' you! we 51-30»: THE' gsoavuiji'ﬁy m? "NH? Sysrgm 32165749 W- €9*\$f'vmr(y2) :30" ' in“: li*'Wcs‘D C v 11 u Kim) TAM ‘P = {@3ch W (,9 :: Flaw; map-n xii? :; (~ gaggyf+ SIM3OJ;}‘1F;(:VSZ3.BDI' + gawzzia‘ﬁ) 4‘03 = 35 lbi® 09’s" . 3 29?: ~ 0.9mm. +-0.‘?JSF2 ‘: 0 Hal» ;, W "37865 F}? L331“? u“@ #3: 0:500'711- 0403:er ~§o 0 [H mm gags‘rn‘I/ITJNG (we) mm {W} WELD; F. , '" z.- C“ 0.5‘U-05vﬁj + O'VIOEVFR “g0 “‘7' 0’97“? F2 “’0 I! C 19;? c.) bMEEETlTVIT'N£‘ Fix R.er wru (1} Wifij a: ,2“? p; : (Loggﬂs’simi) : soc-1y Page2of4 Problem 3 {25 Paints) Use the (Pause—Jordan elimination method to solve the ‘ibllewi-ng system. of Einear equations: )51 + 8x3 '2 1 x] +2x2 +3x3 x 2 2x, +5x2 +3x3 m 3 :1.) Write the augmented matrix (4 peints) b) Use elementary row Operations Ee obtain {he reduced r0w~eehefen form of the augmented matrix (E8 pein‘ts). 0) Write the seiutien fer-the three unknowns (3 points) Note: No credit will be given for Paris (b). and (C) unless ALL intermediate work is shown For Part {b} I Augmented Matrix Action taken hem Jreviuus matrix 151 rew remains unchanged 0 2 ~— 5 i Add negative of 1“ row to 2.“(1 row 0 5 "#13 I Multipty J. by 1St row and add to 3rd row _ ISl rew remains unchanged 1 0 I ~% Muitiply 2'“1 row by 1/2 h 1‘ lMumpy '39 by 2nd row and add to 3“1 row 1”" row remains unchanged 0 l 0 8 Muitipiy 2'“1 row by 1/2 0 0 1 3 Muftipy \$512 by 2"d mw and add to 3” row Solution: x; :23, x1 "2 8, x e e CD Page ‘E 0'!" 1 Problem 4 (25 points) Given: The tension in cable AB is 777 N and in cable BC is 990 N. Deﬁne the x—y—z coordinate system to be centered at point 0 as shown, with associated unit vectors (i, j, k) Detezmine: it. the force vector acting on point B by cable BC“ {9 points) 2.. the force vector acting on point B by I _ H cable AB” {9 points) ‘ 3 the vector resultant ofthe forces exerted at point B by the two cables (AB and CB) ('7 points) Give all answers using the x»y-z coordinate system and associated unit vectors, (i, j, k), being careful to supply all physical units when applicabic‘ )m m A 5: Ed” O’?¢”’g¢€&c is W " (WWW 62X 231. 3’.qu s. [22.14 (m) W a; mam» (mew (A221 0’” M u 9.?m A ,0.?3 ,. 3.4%721: (m) . ' A far-‘5 '“" “tr e :2? 342:4- (WW) \ @é‘q" 5’4 6A 5A 2/ ‘ im+ﬂf.¢)z¥{72) 5’“ m W‘WW IL - W4 "h -...t " » A. .2342 (“M _ m:— M at 4:0 fg‘y)’c7z. (/Zﬂaitﬁ-ﬂ04); WE" s (ﬂaw): vii/‘5’ "' j Page 4 GM ...
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