hw1_solutions

hw1_solutions - Problem 3.29 Two tow trucks lift a...

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Unformatted text preview: Problem 3.29 Two tow trucks lift a motorcycle out of a ravine following an accident. It the IOO-kg motorcycle is in equilibrium in the position shown. what are the tensions in cables AB and AC"? SOIlItIOll: We need to find unit vectors 6.49 and eAc. Then write TAB = TA 36.43 and TAC = TA 1:9.4C'. Finally. Write and solve the equations of equilibrium. For the ring at A. From the known lorations of points A. B. and C'. TAB [AC 9.43 = C‘AC = “148' IFACI ma = -3i + 3-55 "1 II‘ABI = 4131 m ['30 ll'Acl =G.02m e“; = —0.651i+ 0.759j QAC = + T” = —o.051ngi + [17591.35 T.“ = 0.661T.4ci + ommw j w = —mgj = —(100)(9.81)j N Forequilibrium. TAB + TAC +“tr = 0 In component form. we have 2 F, = 4mm“; + 0.6mmC = 0 Z a, = +0.70%” + 0.7471“~ — 981 = o Solving. we get mg=(100)(9.8l)N TAB = 658 N. TAC = 645 N Problem 3.33 The unstretched length of the spring AB is 660 mm. and the spring constant k = 1000 N/m. What is the mass of the suspended object? Solution: Use the litrnr spring force-extension relation to find the magnitude of the tension in spring AB. Isolate juncture A. The forces are the weight and the tensions in the cables. The angles m are tana = ($) = 0.5333. 0 = 3026‘. 350 m) = 0.875. a = 41.2°. tan:3=( The angle between the 1' axis and the spring is a. The tension is T33 = ITAgIU coso + j sina J. The angle between the 1' axis and AC is (180 — :3). The tension is TAC- = |T_4c|(icos(180 _ a) +js‘n(180 _ 3)) TM: = HACK—“063 +58in 3)- The weight is: VV = 0i — The equilibrium conditions: 2 F = ‘V + TAB + TAC = 0. Substitute and collat like terms The tension is |TAB| = IcAL = (1000)(0.03462) = 34.6 N. ZF, = (|T_“;|coso — |T_4c-| cos :3.)i = 0. Z P” = (ITACI sine + ITABISin 3 _ “VI” = 0. Fora = 30.20: and .‘3 = 4112"“. the weight is 0.918 a ' o .3 ‘V = 34.6 = 43.62 N; sow: Incl = (2:3) ITAtaland [WI = (%$4) ITABI- ' ' (0.702% ’ The tension ITAEI is found from the linear spring forte-deflection relation. The spring extension is The mass is m = ( AL = «(350)? + (000)2 — 060 = 694.02 - 660 = 34.62 mm Problem 3.50 The system is in equilibrium. Whatare the coordinates of A? Solutlon: Determine from geometry the coordinates I. y. Isolate the cable juncture A, Since the frictionless pulleys do not change the magnitude of cable tension. and since each cable is loaded with the same weight. arbitrarily set this weight to unity. |VV| = 1. The angle between the cable AB and the positiye 1‘ axis is a: the tension in A Bis ITABI = icoscr + j sine. The angle between AC and the positive 1' axis is (180° — :3): the tension is TAG = |T,4c|(—icos.'3 +jsin 3). The weight is |VV| = 0i — jl. The equilibrium conditions are ZF =TAB+TAC+VV=U Substitute and collect like terms. 2 F, = (case — cos Sill = 0. 21"” =(sino+sin.‘3—l)j=0. From the first equation eoso = cos :3. 0n the realistic assumption that both angles are in the same quadrant. then a = :3. From the second equation sin a = oro = 30’. With the angles known. geometry can be used to determine the coordinates .r. y. The origin of the .l‘. y coordinate system is at the pulley B. so that the coordinate a- of the point A is positive. Define the positive distance 5 as shown. sothat ( ) =tano. h Similarly. (b + ) =tana. —I H In. It. Reduce to obtain 1' = b—hcota — acotct. Substitute into the first equation to obtain 1‘: (b—hcota). Multiply this equation by tano and use 5 = .rtana to obtain t s =( ago) (b— hcoto‘). The sign of the coordinate y is determined 15 follows: Since the co- ordinate :- is positive. the condition (b — hcoto) > 0 is required; with this inequality satisfied (as it must be. or the problem is invalid}. sis also positive. as required. Butthe angle a is in the first quadrant. so that the point A is below the pulley 3. Thus 5/ = —s and the coordinates of the point A are: l l -.r = (b— hcotov). y= —§(btana — h). a = 30° Problem 3.64 Prior to its launch. a balloon carrying a set of experiments to high altitude is held in place by groups of student volunteers holding the tethers at B. C. and D. The mass of the balloon. experiments package. and the gas it contains is 90 kg. and the buoyancy force on the balloon is 1000 N. The supervising professor con- sen'atively estimates that each student can exert at least a 40-N tension on the tether for the necessary length of time. Based on this estimate. what minimum numbers of students are needed at B. C. and D? Solution: 21-“, =1000—(90)(9.81)—T =0 T = 117.1 N A(0,8,0) 8(16. 0. is) 000.0, —12) D(—16,0.d) We need to write unit vectors e33, 9A0. and 6A0. aw = 0.667i — 0.3335 + 0.6671: GAG = 0.570i — 0.456j — 0.68:1]: BAD = —0.873i — 0.436j + 0.218]: We now write the forces in terms of magnitudes and unit vectors FAB = 0.607FAgi—0.333FABJ+0.667FABR PAC = 0.570FAci — 0.456}:ch — 0.684FAck FAD = —0.873FADi — + 0.218FAck T = 117.1] (N) The equations of equilibrium are 2 F, = 0.657519 + 0.5mm: — 0.8mm; = 0 X: F” = 4.3331233 _ 0.4501?“ _ 0.4361?“ +117.1= 11 Zr. = 0.6671?” — 0.93.11?“ + 0.2131?“ = 0 Solving. we 3d FAB = 64.8 N ~ 2students FAC = 99.8 N ~ 3 students FAD =114.6N ~ 3 students C (I0.0.-12)m (46.0.4) 111 x B “6.0. I6) m C(l0.0.Dl2)In B (16.0.l6) m HIDN 190) g B(l6.0.l6)rn N Problem 3.76 The cable AB keeps the 8-kg collar .4 in place on the smooth bar CD. The y axis points upward. What is the tension in the cable? Solution: The coordinates of points C and D are C (0.4. 0.3. 0). and D (0.2. 0. 0.25). The unit vector from C toward D is given by 9CD = ecp,i + ecpyj + acmk = —0.456i — now + 0.570k. The location of point A is given by r}. = 1C + dCAeCD, with similar equations for 515 and :A. From the figure. dCA = 0.2 m. From this. we find the coordinmes of A are A (0.309. 0.162. 0.1l4). From the figure. the coordinates of B are B (0. 0.5. 0.15). The unit vector from A toward Bis then given by (I 9A3 = 9,43,] + eAgvj + eAB:k = —0.674i + 0.735j + 0.079k. The tension force in the cable can now be written as TAa = —0.674TAai + 0.735TAaj + 0.079TABk. From the free body diagram. the equilibrium equations are: FN: +TABeAB: = 0- FNy + 73486.43” - m9 = 0. and FN: +TAB°A8: = 0 We haw: three equmion in four unknowns. We get another equation from the condition that the bar CD is smooth. This means that the normal force has no component parallel to CD. Mathematically. this can be stated as FN - ec-D = 0. Expanding this. we get FNreCDx + FNyeC'Dy + FNeeC-‘D: = 0- We now have four equations in our four unknowns. Substituting in the numbers and solving. we get :3 to u L1 7.7 N. FM, = 38.9 N. FNV = 36.1 N. and FN; = —~‘1.53 N. Problem 3.87 Cable AB is attached to the top of the vertical 3-m post. and its tension is 50 kN. What are the tensions in cables A0. AC. and AD? Solution: Get the unit vectors pnmllel to the cables using the coordinates of the end points. Expres the tensions in terms of these unit vectors. and solve the equilibrium conditions The coor- dinates of points A. B. C. D. O are found from the problem sketch: The coordinates ofthe points are A(6, ‘2, 0). 3(12, 3, 0). C(0. 8, 5). D(0. 4, —5). 0(0. 0. 0). The vector locations of these pains one: I" =6i+2j+0k, r5=12i+3j+0k, rc=0i+85+5k, ro=0i+4i—5k, ro=0i+0j+0k. The unit vector parallel tothe tension acting between the points A. B in the direction ofB isby definition a” = &_ Ira - I'Al Perform this for each of the unit vectots a“; = +0.9864i + 0.164“ + 0k eAc, = —0.6092i + 0.60% + 0.5077k 9,”; = —0.7442i + 0.2481j — 0.62021: er = -0.94871- 0.3162] + 0k The tensions in the cdtles are atplessed in terms of the unit vectots. TAB = ITABIGAB = 508,43. TAC = ITACIGACs TAD = ITADIBADe TAO = ITAoler- The equilibrium conditions are ZF=0= TAB+TAc+TAD+TAo =0- Substitutc and collect lilac terms. 2 F. = (0.9864(50) — 0.6092ITACI — 0.7422|T,.D| —0.9<187]TA0|)i = 0 Z P, = (0.1644(50) + 0.6092|TAc| + 0.2481|TAD| —0.3162|TA0|)j = 0 2 F: = (+0.5077|TAC| — 0.6202|T,.D|)k = o. This set of simultaneous equnions in the unknown forces may be solsed using any of several standard algorithms. The results are: iTAOi = 43-3 kN. ITACI = 6.8 IcN. ITAD| = 5.5 m. ...
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This note was uploaded on 04/26/2009 for the course MAE 101 taught by Professor Orient during the Spring '08 term at UCLA.

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hw1_solutions - Problem 3.29 Two tow trucks lift a...

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