P-1: ԦA=Ԧi +Ԧj -5k and B =Ԧi - 7Ԧj -6k. Detemine;?????? - A B B A A B B A B A B A 7
P-: The scew ee is subjected to two foces, Ԧ 1 and Ԧ. Detemine the magnitude and diection of the esultant foce.
P-: The vecto V lies in the plane defined b the intesecting lines L A and L B. Its magnitude is 400 units. Suppose that ou want to esolve V into vecto components paallel to L A and L B. Detemine the magnitudes of the vecto components. Also detemine the pojections P a and P b of V onto the lines L A and L B. L B V 60 80 L A
P-4: Epess foces as the vecto fom in tems of the unit vectos Ԧi and Ԧj. (a) (b) (c) 40 o i j 500cos 40i - 500sin 40 j 8.0i - 1.9 j 0 o i j -400cos0i 400sin 0 j -46.4i 00 j i j 1-5. 1 5 5-5. 1 5 1-4.8i - j 1
P-5: Detemine the magnitude of the esultant V=V 1 +V and the angle q which positive ais. V makes with the V = 7 units 4 0 V 1 = 9 units 5
P-6:
P-7: Detemine the - components of the tension T which is applied to point A of the ba OA. Neglect the effects of the small pulle at B. Assume that and q ae known. Also detemine the n-t components of the tension T fo T=100 N and q=5 o.
- coodinates AB - sinq cosq cosq - sinq - sin q q b T b cosq 1 cosq cos b cosq - sinq cosq - sinq - sinq 1- sinq sin b cosq - sinq cosq - sinq T T T cos b T -T sin b T 1 cosq cosq - sin q sin q -1 cosq - sin q + cos q n-t coodinates (fo q=5 o and T=100 N) 1- b actan 1 T T n t T cos T sin sin cos 5 1. 19 5 q b 100cos5 1. 19 66. 67 q b 100 sin5 1. 19 74. 54 N o N q b T
P-8: In the design of the obot to inset the small clindical pat into a close-fitting cicula hole, the obot am must eet a 90 N foce P on the pat paallel to the ais of the hole as shown. Detemine the components of the foce which the pat eets on the obot along aes (a) paallel and pependicula to the am AB, and (b) paallel and pependicula to the am BC.
the obot am must eet a 90 N foce P on the pat foce which the pat eets on the obot vetical P=90 N (a) paallel and pependicula to the am AB P=90 N AB hoiontal P AB 15 o vetical 45 o 0 o P //AB 60 o //AB P //AB = P AB =90 cos45=6.64 N (b) paallel and pependicula to the am BC. vetical //BC hoiontal P=90 N P //BC 0 o 45 o 15 o 45 o 45 o P //BC =90 cos0=77.94 N P BC =90 sin0=45 N P BC BC
P-9: The unstetched length of the sping is. When pin P is in an abita position q, detemine the - and -components of the foce which the sping eets on the pin. Evaluate ou answe fo =400 mm, k=1.4 kn/m and q =40.
P-10: Thee foces act on the backet. Detemine the magnitude and diection q of so that the esultant foce is diected along the positive u ais and has a magnitude of 50 N. =5 N 1 =80 N
if R=50 N q =? =? Resultant R R cos 5i - Rsin 5 j R 50cos 5i - 50sin 5 j R 45.15i - 1.1 j 1 80i cos q 5 i - sinq 5 j 5 5 i 1 R 1 5 1 i j 0i cos sin 80 q 5-54. 685 cos q 5 - sinq q 5 69. 1 5 j 48 j 0 1 45.15 48-1.1 1 =5 N 1 =80 N R q 5 69.1 q 5-54. 685-1.64 sin cos tan q 5 q 10.5 88. 14 N
P-11: Detemine the magnitude and diection angles of the esultant foce acting on the backet. Resultant R 1-450cos 45sin 0i 1 1-159.1i 450cos 45cos0 j 75.57 j 1 1 18.k 450sin 45k
Diection angles fo 45 q 60 q q Diection cosines cos q cos l m q n cos 1 q 1 90 cos 45 cos 60 cos q 1 cos q 0.5 cosq 0.5 q 90 cosq -0.5 q 10 600cos 45i 600cos60 j 600 cos10k 44.6i 00 j 00k -
-159.1i 75.57 j 1 18.k Resultant R 1 R 44.6i 00 j - -159.1 44.6i 75.57 00 j 18. - 00k R 65.16i 575.57 j 18.k 00k Magnitude of Resultant oce R R 65.16 575.57 18. 6. 97 Diection Cosines of Resultant oce cosq cosq R R cosq R R 65.16 0.418 6.97 Diection angles fo R q accos( 0.418) 65. cosq cosq R R 575.57 6.97 0.907 q accos(0.907) 4.9 N cosq 18. 6.97 0.09 q accos(0.09) 88.
P-1: The tunbuckle T is tightened until the tension in cable OA is 5 kn. Epess the foce Ԧ acting on point O as a vecto. Detemine the pojection of Ԧ onto the -ais and onto line OB. Note that OB and OC lies in the - plane.
O plane acting on point O 65 o C =5sin50=.8 kn =5cos50=.14 kn = sin65=.14sin65=.91 kn = cos65=.14cos65=1.58 kn 1.58i.91j.8k Pojection onto ais j. 91 kn OB OB OB Pojection onto line OB O Unit vecto of line OB n OB 0 o n OB cos0i cos60 j B n 1.58i.91j.8k 0.866i 0.5 j OB 1.580.866.910.5.6 kn
P-1: The fame shown is subjected to a hoiontal foce Ԧ = 00Ԧj. Detemine the magnitude of the components of this foce paallel and pependicula to membe AB. Ԧ = 00Ԧj 19
P-14: The cable BC caies a tension of 750 N. Wite this tension as a foce T acting on point B in tems of the unit vectos Ԧi, Ԧj and k. The elbow at A foms a ight angle. T T acting on point B T Tn BC T C / B C / B The coodinates of points B and C ae B (1.6; 0.8sin0; 0.8cos0) B (1.6; 0.4; 0.69), C (0; 0.7; 1.) Theposition vecto BC is B -1.6i 1.1 j 0.507k C / B BC 1.6 1.1 0.507-1.6i 1.1 j 0.507k The unit vecto of T BC is nc / B nbc nt -0.797i 0.548 j 0.5k Tension T acting on point B in vecto fom T Tn BC C / - 0.797i 0.548 j 0.5k -597.75i 411j 189.75k 750 m
P-15: The sping of constant k =.6 kn/m is attached to the disk at point A and to the end fitting at point B as shown. The sping is unstetched when q A and q B ae both eo. If the disk is otated 15 clockwise and the end fitting is otated 0 counteclockwise, detemine a vecto epession fo the foce which the sping eets at point A.
P-16: The ectangula plate is suppoted b hinges along its side BC and b the cable AE. If the cable tension is 00 N, detemine the pojection onto line BC of the foce eeted on the plate b the cable. Note that E is the midpoint of the hoiontal uppe edge of the stuctual suppot.
If T=00 N, detemine the pojection onto line BC of the foce eeted on the plate b the cable. Coodinates of points A, B, C and E with espect to coodinate sstem A 400, 0, 0 B (0, 0, 0) C 0, 100sin5, 100cos5 C (0, 507.14, -1087.57) E 0, 100sin5, 600cos5 E (0, 507.14, -54.78) - 400i 507.14 j - T TnT 00 844. T -14.1i 180.19 j -19.1k Pojection of T onto line BC T T n BC BC 54.78k Unit vecto of line BC -cos 5k sin 5 j 0.4 j - 0.906k n BC - 14.1i 180.19 j -19.1k 0.4 j - 0.906k T BC 51. 6 N n BC 5 o 5 o
P-17: Conside the staight lines OA and OB. a) Detemine the components of a unit vecto that is pependicula to both OA and OB. b) What is the minimum distance fom point A to the line OB? B (6, 6, -) m O q A (10, -, ) m 4
a) Detemine the components of a unit vecto that is pependicula to both OA and OB. A (10, -, ) m q B (6, 6, -) m O b) What is the minimum distance fom point A to the line OB? O B / O A / k j i k j i O A O B 10 6 6 / / - - m d d d O B O A O B O A O A 9.71 6 6 87.6 sin sin 9 / / / / / q q Vecto nomal to both OA and OB is k j i i j i k j k k j i k j i O B O A 7 48 1 18 18 6 1 0 60 6 6 10 / / - - - - Unit vecto nomal to both OA and OB is k j i n k j i n 0.84 0.549 0.17 7 48 1 7 48 1 87.6 - - 5
P-18: Conside the tiangle ABC. a) What is the suface aea of ABC? b) Detemine the unit vecto of the oute nomal of suface ABC. c) What is the angle between AC and AB? C (0, 0, 5) m B (0, 16, 0) m A (1, 0, 0) m 6
a)what is the suface aea of ABC? A C AC C (0, 0, 5) m B s AB -1i 16 j AC -1i 5k AB AC -1i 16 j -1i 5k AB AC 60 j 19k 80i ABC 108.4 m B (0, 16, 0) m 80 60 19 A (1, 0, 0) m n AB AB AB b) Detemine the unit vecto of the oute nomal of suface ABC. AC AC 80i 60 j 19k 16.48 0.69i 0.77 j 0.887k c) What is the angle between AC and AB? AB AC AC cos -1i 16 j -1i 5k 1 16 1 5 144 AB 01cos 56.68 cos 7
P-19: The doo is held opened b two chains. oces in AB and CD ae A =00 N and C =50 N, espectivel. a) Wite these foces in vecto fom. b) Detemine the esultant of these foces and the diection angles of the esultant foce. c) Detemine the pojection of foce A on line CD. 8
-0: Conside foce Ԧ = 1Ԧi + 4Ԧj + 4k N. Resolve Ԧ nto two components, one paallel to line AB and one ependicula to line AB. Position vecto B / A B / A 4i 8 j -8k B n Unit vecto 14-10i 6 - - j 6-14 Paallel component // // n // n 4i 8 j -8k k / A - B / A 4 8 8 1 1 i A(10, -, 14) 1i 4 j 4k i j - k 1 4 4 4 1 i Nomal component - j - k 1 4 8 i 4 8 j - 8 k j 8 k 1 (in vecto fom) n 1i 4 j 4k - i j - k 10.67i 1. j 6.67k // // B(14, 6, 6) - 4 9 N
P-1: Obseves on Eath at points A and B measue diection cosines of position vectos to the space shuttle at C as o Ԧ C/A : cos q = 0.60, cos q = 0.480 and cos q <90. o Ԧ C/B : cos q = 0.515, cos q = 0.606 and cos q <90. Detemine the,, coodinates of the space shuttle located at point C and also the shotest distance fom point C to diection AB. C A B (50, 640, 0) km 0
P-: A house with 98000 kg mass is built on a steep slope defined b points A, B, and C. To help assess the possibilit of slope failue (mud slide), it is necessa to a) Detemine the components of the weight in diections nomal and paallel (tangent) to the slope. b) Detemine the component vectos of the weight in diections nomal and paallel (tangent) to the slope. c) Detemine the smallest distance fom point O to the slope. (g=9.81 m/s ) 1
a-b) Detemine the components of the weight in diections nomal and paallel (tangent) to the slope. W: Weight of house W n : Nomal component of the weight to slope W t : Paallel component of the weight to slope C A N AB AC N 10800i 700 j 1600k n N N AC B AB W -98000 - AB -180 j 60k AC 10i -180 j - 180 j 60k 10i -180 j 10800i 700 j 1600k 10800 700 1600 500 9.81k 96180k (Vecto nomal to the slope) 7 i 7 j 6 7 W n 6 n W n - 96180k i j k -84040 N (its magnitude) 7 7 7 6 n W n n - 84040 i j k -5160i - 5440 j - 7060k 7 7 7 k W W t (Unit vecto nomal to the slope) (in vecto fom)
W W t t W -Wn 5160i 5440 j - 55060k - 96180k - - 5160i - 5440 j - 7060k (in vecto fom) W W W t t 5160 495186.4 N 5440 55060 (its magnitude) c) Detemine the smallest distance fom point O to the slope. n B W n W t C O The smallest distance can also be detemined as follows d d OB n OA n 60k i 7 7 6 j k 51.4 m 7 180 j i j 6 k 51.4 m 7 7 7 6 d OC n 4 7 7 7 C 10i i j k 51. m
P-: An ovehead cane is used to eposition the boca within a aiload ca-epai shop. If the boca begins to move along the ails when the -component of the cable tension eaches kn, calculate the necessa tension T in the cable. Detemine the angle q between the cable and the vetical - plane.
T = kn, calculate tension T, the angle q between the cable and the vetical - plane. Unit vecto of T 5i 4 j k nt 5 4 1 n 0.77i 0.617 j 0.154k T T in vecto fom T T 0.77i 0.617 j 0.154k -component of T 0.77T T. 896 and -components of T T T kn (magnitude of T).896.4 kn T 0.154.896 0. kn 0.617 6 i.4 j 0.6k T T T.4. 84 kn T.84 cosq 0.896 q T.89 9.598
P-4: In opening a doo which is equipped with a heav-dut etun mechanism, a peson eets a foce P of magnitude 40 N as shown. oce P and the nomal n to the face of the doo lie in a vetical plane. Epess P as a vecto and detemine the angles q, q and q which the line of action of P makes with the positive -, - and -aes. P =40sin0=0 N P =40cos0=4.64 N plane 0 o // P = P cos0=4.64cos0=.55 N // n P P = P sin0=4.64cos0=11.848 N P.55i 11.848 j 0k angles q, q and q which the line of action of P makes with the positive -, - and -aes q a cos.55 40 5.56 q 11.848 a cos 40 7.77 q 0 a cos 40 60
P-5: The and scala components of a foce ae 100 N and 00 N, espectivel. If the diection cosine l=cosq of the line of action of the foce is -0.5, wite Ԧ as a vecto. 100 N 00 N l -0.5 l m n 1 m n 1-0.5 m n 0.75. 61 N cosq m cosq n 0.75.61 58. N cosq -19. 1 N -19.1i 100 j 00k