7. The turnbuckle T is tightened until the tension in cable OA is 5 kN. Express the force F acting on point O as a vector. Determine the projection of F onto the y-axis and onto line OB. Note that OB and OC lies in the x-y plane. 8. Determine the magnitude and direction angles of the resultant force acting on the bracket. Resultant R F F1 F2 F1 450 cos 45 sin 30i 450 cos 45 cos 30 j 450 sin 45k F1 xy F1 xy F1 159.1i 275.57 j 318.2k Direction angles for 𝑭 𝟐 x 45 y 60 z 90 Direction cosines cos 2 x cos 2 y cos 2 z 1 l 2 m2 n2 1 cos 2 45 cos 2 60 cos 2 z 1 cos 2 z 0.25 cos z 0.5 z 90 cos z 0.5 z 120 F2 600 cos 45i 600 cos 60 j 600 cos120k F2 424.26i 300 j 300k F1 159.1i 275.57 j 318.2k Resultant R F F1 F2 F2 424.26i 300 j 300k R 159.1 424.26 i 275.57 300 j 318.2 300 k R 265.16i 575.57 j 18.2k Magnitude of Resultant Force R R 265.16 2 575.57 2 18.2 2 633.97 N Direction Cosines of Resultant Force Ry Rx cos x cos y R R 265.16 cos x 0.418 633.97 Rz cos z R 575.57 cos y 0.907 633.97 cos z 18.2 0.029 633.97 Direction angles for 𝑹 x arccos(0.418) 65.3 y arccos(0.907) 24.9 z arccos(0.029) 88.3 9. Determine the parallel and normal components of force 𝐹 in vector form with respect to a line passing through points A and B. Cartesian components of 𝑭 5 Fxy 75 5 3 2 2 64.311 kN A Fx Fxy cos 50 41.34 kN Fz 75 5 3 2 2 38.59 kN F 41.34i 49.265 j 38.59k (6, 4, 8) m (3, 2, 5) m F Fy Fxy sin 50 49.265 kN 3 B z 5 x 50° y 3 F=75 kN F 41.34i 49.265 j 38.59k B z Unit vector of line AB 3i 2 j 3k n AB 0.639i 0.426 j 0.639k 2 2 2 3 2 3 n AB A (3, 2, 5) m F y Parallel component of 𝑭 to line AB 3 5 (its scalar value) : 50° x F// F n AB F// 41.34i 49.265 j 38.59k 0.639i 0.426 j 0.639k (6, 4, 8) m F=75 kN F// 41.34 0.639 49.2650.426 38.59 0.639 72.14 kN Parallel component of 𝑭 to line AB (in vector form): F// F// n AB 72.14 0.639i 0.426 j 0.639k 46.14i 30.76 j 46.14k Normal component of 𝑭 to line AB (in vector form): F F F// 4.8i 18.505 j 7.55k 10. An overhead crane is used to reposition the boxcar within a railroad car-repair shop. If the boxcar begins to move along the rails when the x-component of the cable tension reaches 3 kN, calculate the necessary tension T in the cable. Determine the angle xy between the cable and the vertical x-y plane. Tx=3 kN, calculate tension T, the angle xy between the cable and the vertical x-y plane. Unit vector of 𝑻 nT 5i 4 j k 52 4 2 12 nT 0.77i 0.617 j 0.154k 𝑻 in vector form T T 0.77i 0.617 j 0.154k x-component of 𝑻 0.77T 3 T 3.896 kN (magnitude of 𝑻) y and z-components of 𝑻 Ty 0.6173.896 2.4 kN T 3i 2.4 j 0.6k Tz 0.1543.896 0.6 kN Txy Tx2 Ty2 32 2.4 2 3.84 kN Txy 3.84 cos xy 0.896 xy 9.598 T 3.89 11. The spring of constant k = 2.6 kN/m is attached to the disk at point A and to the end fitting at point B as shown. The spring is unstretched when A and B are both zero. If the disk is rotated 15° clockwise and the end fitting is rotated 30 ° counterclockwise, determine a vector expression for the force which the spring exerts at point A. 12. The rectangular plate is supported by hinges along its side BC and by the cable AE. If the cable tension is 300 N, determine the projection onto line BC of the force exerted on the plate by the cable. Note that E is the midpoint of the horizontal upper edge of the structural support. If T=300 N, determine the projection onto line BC of the force exerted on the plate by the cable. 𝑪𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆𝒔 𝒐𝒇 𝒑𝒐𝒊𝒏𝒕𝒔 𝑨, 𝑩, 𝑪 𝒂𝒏𝒅 𝑬 𝒘𝒊𝒕𝒉 𝒓𝒆𝒔𝒑𝒆𝒄𝒕 𝒕𝒐 𝒄𝒐𝒐𝒓𝒅𝒊𝒏𝒂𝒕𝒆 𝒔𝒚𝒔𝒕𝒆𝒎 𝑨 400, 0, 0 B (0, 0, 0) 𝑪 0, 1200𝑠𝑖𝑛25, −1200𝑐𝑜𝑠25 C (0, 507.14, 1087.57) 𝑬 0, 1200𝑠𝑖𝑛25, −600𝑐𝑜𝑠25 E (0, 507.14, 543.78) 400i 507.14 j 543.78k T TnT 300 844.33 T 142.12i 180.19 j 193.21k y x z Unit vector of 𝒍𝒊𝒏𝒆 𝑩𝑪 nBC cos 25k sin 25 j 0.423 j 0.906k Projection of T onto line BC TBC y z T nBC 142.12i 180.19 j 193.21k 0.423 j 0.906k TBC 251.26 N nBC 25o 25o 13. The y and z scalar components of a force are 100 N and 200 N, respectively. If the direction cosine l=cosx of the line of action of the force is 0.5, write 𝐹 as a vector. Fy 100 N Fz 200 N l 2 m2 n2 1 l 0.5 m 2 n 2 1 0.52 m 2 n 2 0.75 Fy2 Fz2 223.61 N Fyz Fy F cos y Fm Fz F cos z Fn F 0.75 223.61 F 258.2 N Fx F cos x 129.1 N F 129.1 i 100 j 200k
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