#1. Two concurrent forces act at a point: F₁ = 120 N directed along the positive x-axis and F₂ = 90 N at 45° from the positive x-axis. Determine the magnitude of the resultant force.
#2. A force of 350 N acts at an angle of 40° below the positive x-axis. Determine the magnitude of the y-component of the force.
#3. Three forces act at a point: F₁ = 50 N along the positive x-axis, F₂ = 70 N along the positive y-axis, and F₃ = 40 N along the negative x-axis. Determine the magnitude of the resultant force.
#4. A force of 600 N has direction cosines cos(α) = 0.6 with the x-axis and cos(β) = 0.8 with the y-axis. Determine the x-component of the force.
#5. Two forces, F₁ = 250 N and F₂ = 180 N, act at a point with an angle of 60° between them. Determine the magnitude of the resultant force.
#6. A force F = (5i + 12j) N acts at a point. Determine the unit vector in the direction of this force.
#7. Three concurrent forces act at a point: F₁ = 80 N at 30°, F₂ = 100 N at 150°, and F₃ = 60 N at 270° from the positive x-axis. Determine the x-component of the resultant force.
#8. A resultant force R has components Rx = 180 N and Ry = -240 N. Determine the magnitude of R.
#9. A force of 450 N acts in the direction of vector (3, 4). Determine the y-component of the force.
#10. The resultant of two forces acting at a point has a magnitude of 200 N at an angle of 53.13° with the positive x-axis. If one force is 120 N along the positive x-axis, determine the magnitude of the second force.
#11. Four concurrent forces act at a point: F₁ = 30 N at 0°, F₂ = 40 N at 90°, F₃ = 50 N at 180°, and F₄ = 60 N at 270°. Determine the angle that the resultant makes with the positive x-axis.
#12. A force F = (8i – 6j + 2k) N acts at a point. Determine the magnitude of the force.
#13. Two forces of equal magnitude P act at a point with an angle of 120° between them. If the resultant has a magnitude of 150 N, determine the value of P.
#14. A force of 500 N is resolved into two components along directions making angles of 30° and 60° with the force direction. Determine the component along the 30° direction.
#15. The equilibrant of a system of forces has a magnitude of 75 N directed at 135° from the positive x-axis. Determine the x-component of the resultant force.
#16. A force of 200 N acts perpendicular to a wrench handle at a distance of 0.35 m from the bolt center. Determine the moment applied to the bolt.
#17. Two parallel forces of 80 N each act in opposite directions, separated by a perpendicular distance of 0.5 m. Determine the magnitude of the couple moment.
#18. A force of 150 N acts at a point located at coordinates (3, 4) m, directed along the positive x-axis. Determine the moment about the origin.
#19. Three couples act on a rigid body with moments of 40 N·m clockwise, 60 N·m counterclockwise, and 25 N·m clockwise. Determine the resultant couple moment.
#20. A force of 400 N acts in the positive y-direction at point (2, 0) m. Determine the magnitude of the moment about the origin.
#21. A couple consists of two forces of magnitude F separated by distance d. If the couple moment is 120 N·m and d = 0.4 m, determine F.
#22. A force of 300 N acts along a line passing through points (1, 2) m and (4, 6) m. Determine the perpendicular distance from the origin to the line of action.
#23. Two couple moments act on a body: M₁ = 80 N·m about the x-axis and M₂ = 60 N·m about the y-axis. Determine the magnitude of the resultant couple moment.
#24. A vertical force of 500 N acts at point (3, 0, 2) m downward. Determine the moment about the z-axis.
#25. A force F = (0, 0, 100) N acts at point (2, 3, 0) m. Determine the magnitude of the moment about the origin.
#26. A simply supported beam has a span of 8 m. A concentrated load of 24 kN acts at 3 m from the left support. Determine the reaction at the right support.
#27. A cantilever beam has a length of 5 m and carries a uniformly distributed load of 8 kN/m. Determine the reaction moment at the fixed support.
#28. A simply supported beam of span 10 m carries a uniformly distributed load of 4 kN/m over its entire length. Determine the maximum bending moment in the beam.
#29. An overhanging beam has supports at A and B, 6 m apart. The beam extends 2 m beyond support B. A load of 15 kN acts at the free end. Determine the reaction at support A.
#30. A simply supported beam of span 6 m carries two equal loads of 10 kN each at 2 m and 4 m from the left support. Determine the reaction at the left support.
#31. A cantilever beam of length 4 m carries a concentrated load of 20 kN at its free end. Determine the shear force at the fixed support.
#32. A simply supported beam has a span of 12 m. A triangular load varies from zero at the left support to 6 kN/m at the right support. Determine the reaction at the left support.
#33. A beam is supported by a pin at A and a roller at B, 5 m apart. A moment of 30 kN·m is applied at the midpoint. Determine the vertical reaction at A.
#34. A simply supported beam of span 8 m carries a uniformly distributed load of 6 kN/m over the left half only. Determine the reaction at the right support.
#35. A cantilever beam has a length of 6 m. A load of 12 kN acts at 4 m from the fixed end. Determine the bending moment at the fixed support.
#36. A simply supported beam of span 10 m carries a concentrated load of 40 kN at the center. Determine the bending moment at a section 3 m from the left support.
#37. An overhanging beam is supported at points 8 m apart, with a 3 m overhang on the right. A uniformly distributed load of 5 kN/m acts over the entire beam length. Determine the reaction at the left support.
#38. A simply supported beam of span 6 m carries a trapezoidal load varying from 4 kN/m at the left to 8 kN/m at the right. Determine the total load on the beam.
#39. A horizontal beam is pinned at A and has a roller support at B, 4 m to the right. A horizontal force of 10 kN is applied at B. Determine the horizontal reaction at A.
#40. A simply supported beam of span 8 m carries a uniformly distributed load of 10 kN/m. Determine the shear force at a section 2 m from the left support.
#41. A simple triangular truss has members forming a right triangle with legs of 3 m (vertical) and 4 m (horizontal). A vertical load of 12 kN is applied at the top joint. Determine the force in the horizontal bottom member.
#42. In a truss joint where two collinear members meet with a third non-collinear member, and no external load acts at the joint, what is the force in the non-collinear member?
#43. A Pratt truss has a span of 16 m divided into 4 equal panels with a height of 4 m. A vertical load of 40 kN acts at the second bottom chord joint from the left. Using method of sections, determine the force in the top chord at the first panel.
#44. A plane truss has 7 joints and 11 members. The truss is supported by 3 reaction components. Determine whether the truss is statically determinate.
#45. In a symmetric truss loaded symmetrically at the center, what is the force in the central vertical member if the applied load is 30 kN?
#46. A Warren truss has diagonal members at 60° to the horizontal. If a diagonal member in tension carries 20 kN, determine its horizontal component.
#47. A truss has 8 joints, 13 members, and 3 reaction components. What is the degree of static indeterminacy?
#48. A simply supported truss has a span of 12 m and height of 3 m. A single 24 kN load acts at the center bottom chord. Determine the force in the diagonal member connected to the support.
#49. At a joint in a truss, only two non-collinear members meet and no external load is applied. What can be concluded about both members?
#50. A Howe truss has vertical members between the top and bottom chords. In a typical loading condition with loads on the bottom chord, the vertical members are generally in:
#51. A cantilever truss extends 6 m from a wall with a height of 2 m. A vertical load of 18 kN acts at the free end on the top chord. Determine the force in the horizontal top chord member nearest the wall.
#52. For a truss to be stable and statically determinate, the relationship between members (m), joints (j), and reactions (r) must satisfy:
#53. A truss has a span of 10 m with panels of 2.5 m each. The truss height is 2 m. A load of 20 kN acts at each of the intermediate bottom chord joints (3 joints). Determine the total reaction at one support.
#54. In analyzing a truss by the method of joints, one should start at a joint with:
#55. A K-truss configuration at a joint shows three members connected at angles of 0°, 135°, and 225° from the horizontal. If the horizontal member carries 50 kN in tension, and no external load acts at the joint, determine the force in each inclined member.
#56. A rectangular plate 8 m × 4 m has a rectangular hole 2 m × 2 m centered at coordinates (6, 2) m from the lower-left corner. Determine the x-coordinate of the centroid of the remaining area.
#57. A triangular plate has vertices at (0, 0), (9, 0), and (6, 6) m. Determine the x-coordinate of the centroid.
#58. A semi-circular plate has a diameter of 120 mm. Determine the distance from the diameter to the centroid.
#59. An L-section consists of a vertical leg 15 mm × 100 mm and a horizontal leg 60 mm × 15 mm at the bottom. Determine the y-coordinate of the centroid from the bottom.
#60. A quarter-circle has a radius of 60 mm. Determine the distance from the center to the centroid of the area.
#61. A T-section has a flange 80 mm × 15 mm on top of a web 15 mm × 70 mm. Determine the distance from the bottom of the web to the centroid.
#62. A rectangular area 100 mm × 50 mm has a semi-circular area of radius 20 mm removed from the center of the long edge. Determine the y-coordinate of the centroid from the bottom edge.
#63. A wire is bent into an L-shape with a horizontal segment of 80 mm and a vertical segment of 60 mm. Determine the x-coordinate of the centroid of the wire from the corner.
#64. A circular sector has a radius of 100 mm and a central angle of 90°. Determine the distance from the center to the centroid of the sector.
#65. A composite area consists of a rectangle 120 mm × 80 mm with a triangle (base 40 mm, height 30 mm) added to one end. Determine the total area.
#66. A parabolic spandrel is bounded by y = x²/4, the x-axis, and x = 6 m. Determine the x-coordinate of the centroid.
#67. An inverted T-section has a flange 100 mm × 20 mm at the bottom and a web 20 mm × 80 mm on top. Determine the distance from the bottom to the centroid.
#68. A right triangle has a base of 60 mm (along x-axis) and height of 45 mm (along y-axis). Determine the y-coordinate of the centroid from the base.
#69. A circular arc (wire) has a radius of 80 mm and subtends an angle of 180°. Determine the distance from the center to the centroid of the arc.
#70. A channel section has a web 10 mm × 100 mm and two flanges each 50 mm × 10 mm. Determine the x-coordinate of the centroid from the back of the web.
#71. A rectangular cross-section has a width of 120 mm and a height of 180 mm. Determine the moment of inertia about the horizontal centroidal axis.
#72. A solid circular cross-section has a diameter of 100 mm. Determine the moment of inertia about a centroidal axis.
#73. A hollow circular section has an outer diameter of 120 mm and an inner diameter of 80 mm. Determine the moment of inertia about the centroidal axis.
#74. Using the parallel axis theorem, determine the moment of inertia of a rectangle (60 mm × 40 mm) about an axis 50 mm below and parallel to the centroidal horizontal axis.
#75. A triangular cross-section has a base of 80 mm and a height of 100 mm. Determine the moment of inertia about the base.
#76. A rectangular section 80 mm wide × 120 mm high has a rectangular hole 40 mm × 60 mm centered within it. Determine the moment of inertia about the horizontal centroidal axis.
#77. A solid circular shaft has a diameter of 50 mm. Determine the polar moment of inertia.
#78. Determine the radius of gyration about the horizontal centroidal axis for a rectangle 50 mm wide × 80 mm high.
#79. An I-beam has flanges 100 mm × 15 mm and a web 15 mm × 120 mm. The total depth is 150 mm. Determine the moment of inertia about the horizontal centroidal axis.
#80. A semi-circular area has a diameter of 80 mm. Determine the moment of inertia about the diameter.
#81. A rectangular cross-section has dimensions 40 mm × 100 mm. Determine the section modulus about the horizontal centroidal axis.
#82. A T-section has a flange 120 mm × 20 mm and a web 20 mm × 100 mm. The centroid is 78.5 mm from the bottom. Determine the moment of inertia about the horizontal centroidal axis.
#83. A circular section of diameter 60 mm has its moment of inertia calculated about an axis tangent to the circle. Determine this moment of inertia.
#84. An equal-leg angle section has dimensions 80 mm × 80 mm × 10 mm. Determine the moment of inertia about a centroidal axis parallel to one leg.
#85. A hollow rectangular section has outer dimensions 100 mm × 60 mm and inner dimensions 80 mm × 40 mm. Determine the moment of inertia about the horizontal centroidal axis.
#86. A 40-kg block rests on a horizontal surface with coefficient of static friction 0.35. Determine the minimum horizontal force required to start motion. Use g = 9.81 m/s².
#87. A 60-kg block rests on an inclined plane at 25° to the horizontal. The coefficient of static friction is 0.4. Determine the friction force acting on the block. Use g = 9.81 m/s².
#88. A flat belt passes over a pulley with an angle of wrap of 150°. The coefficient of friction is 0.25. If the tight side tension is 800 N, determine the slack side tension.
#89. A wedge with a 10° angle is used to lift a load. If the coefficient of friction on all surfaces is 0.15, determine the friction angle.
#90. A 50-kg crate on a horizontal floor requires a horizontal force of 200 N to start sliding. Determine the coefficient of static friction. Use g = 9.81 m/s².
#91. A block weighing 100 N rests on an inclined plane. The block just begins to slide when the plane angle reaches 30°. Determine the coefficient of static friction.
#92. A square-threaded screw has a mean radius of 25 mm and a pitch of 5 mm. The coefficient of friction is 0.12. Determine the lead angle.
#93. A ladder 5 m long rests against a smooth vertical wall at an angle of 70° with the horizontal floor. If the ladder weighs 180 N, determine the normal force from the wall.
#94. A journal bearing supports a shaft of diameter 60 mm under a radial load of 3 kN. If the coefficient of friction is 0.1, determine the friction torque.
#95. A rectangular block 0.4 m wide and 0.8 m tall, weighing 200 N, rests on a horizontal surface. A horizontal force P is applied at the top. Will the block tip or slide first if μs = 0.3?
#96. A band brake has a coefficient of friction of 0.3 and an angle of wrap of 240°. If the tension on the tight side is 1200 N, determine the braking torque for a drum of radius 200 mm.
#97. A 25-kg block on a 30° incline is connected by a rope over a pulley to a hanging weight W. If μs = 0.2 between block and incline, determine the minimum W to prevent the block from sliding down. Use g = 9.81 m/s².
#98. A cone clutch has a mean radius of 100 mm and a cone angle of 15°. If the axial force is 2 kN and μ = 0.25, determine the friction torque.
#99. A disk clutch has inner radius 50 mm and outer radius 100 mm. For uniform pressure distribution, determine the location of the resultant friction force from the center.
#100. A wheel of radius 300 mm supports a load of 1500 N. The coefficient of rolling resistance is 0.4 mm. Determine the force required to maintain constant velocity on a horizontal surface.
