mikemolt/staticsmechanics · Datasets at Hugging Face

id stringlengths 4 6	question stringlengths 96 868	answer stringlengths 4 244	topic stringclasses 25 values
S_Q1	Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. Determine the scalar components of F in the x and y directions.	Fx = 300 lbf, Fy = 519.6 lbf	force_vectors
S_Q2	Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. A new coordinate system is created by rotating the x- and y-axes 65 degrees clockwise. Determine the scalar components of F in the x' and y' directions.	Fx' = -344.1 lbf, Fy' = 491.5 lbf	force_vectors
S_Q3	Find the magnitude and direction of force F which has scalar components: Fx = -2 N and Fy = -6 N	6.325 N, 251.6 degrees	force_vectors
S_Q4	Vector R has vector components A and B where A has a magnitude of 4 kN and points 35 degrees CW from the positive x-axis, and B has a magnitude of 6 kN pointing 85 degrees CCW from the positive x-axis. Find the magnitude and direction of force of the resultant R.	5.3 kN, 44.1 degrees CCW from the positive x-axis	force_vectors
S_Q5	Determine the resultant of three forces A, B and C, with scalar components of A (-6.26, 12.5 lbf), B (35.4, -35.4 lbf) and C (-36.8, 27.6 lbf).	9.06 lbf, 148.3 degrees	force_vectors
S_Q6	Three forces act on a particle. Force A is 600 N acting in the negative y direction. Force B acts 15 degrees CCW from the positive y-axis. Force C acts 60 degrees CCW from the positive x-axis. Determine the magnitudes of forces B and C.	B = 424.3 N, C = 219.6 N	force_vectors
S_Q7	A small massless ring is located at (0, 0m). A is located at (0, -4m), B is located at (-2, 6m) and C is located at (4, 3m). A force of 400 N is applied to the ring in the direction towards C. The ring is held in equilibrium by two inextensible cables, one connected to point A and one connected to point B. Determine the tension in cable A and cable B.	Cable A = 1200 N, Cable B = 1012 N	particle_equilibrium
S_Q8	A mass of 10 kg is held in equilibrium by two cables, cable A and cable B. The mass is located at (0, 0m). Point A is located at (-4.25, 3.5m) and point B is located at (4.5, 3.5m). Determine the tension in cable A and cable B.	Cable A = 79.36 N, Cable B = 77.61 N	particle_equilibrium
S_Q9	A 2 lb cylinder rests in a trough with smooth frictionless surfaces. The left side of the trough is angled at 35 degrees from the horizontal. The right side of the trough is angled at 60 degrees to the horizontal. Determine the reactions on the cylinder from the left side and the right side of the trough.	Left side = 1.74 lb, right side = 1.15 lb	particle_equilibrium
S_Q10	Cable ABC has a length of 8 ft and the cable supports a weight of 100 lbf riding on a frictionless pulley of negligible radius located at B. The cable is supported at A (0, 0 ft) and C (6.5, 2.5 ft). Determine the horizontal distance from A of the pulley and the tension in the cable, required for equilibrium.	Distance = 1.51 ft, tension = 85.8 lbf	particle_equilibrium
S_Q11	A 320 mm x 550 mm steel plate 35 mm thick is suspended from a point P located 350 mm above the center of the plate (in the z direction). Three cables support the plate at A, B and D. The four corners of the plate are E located at (0, 0mm), C located at (0, 550mm), B located at (320mm, 550mm) and A located at (320, 0mm). Point D is located 165 mm from E along the plate edge EC. The density of steel is 7970 kg per cubic metre. Determine the tension in cable A, cable B and cable D.	Cable A = 97.6 N, cable B = 227.8 N, cable D = 275.4 N	particle_equilibrium
S_Q12	A system of cables are suspended by connections at B, E and D. A weight of 75 lbf is located at A and a weight of 55 lbf is located at C. Point B is located at the left end of the system of cables, and cable B to A forms an angle of 10 degrees to the horizontal where A is below and to the right of B. Cable A to C forms an angle of 15 degrees to the horizontal and C is above and to the right of A. Two cables connect to C, a cable from C to E which forms an angle of 70 degrees to the horizontal where E is above and to the right of C, and cable C to D which is in line with the horizontal and D is to the right of C. Determine the tensions in cables AB, AC, CE and CD.	Cable AB = 171.4 lbf, cable AC = 174.8 lbf, cable CE = 106.7 lbf, cable CD = 132.3 lbf	particle_equilibrium
S_Q13	Two cylinders are placed in a 14 ft wide container. The cylinder that sits on the bottom of the container has a radius of 5 ft and weighs 3750 lbf. The second cylinder sits on top of the other cylinder and has a radius of 4.5 ft and weighs 3000 lbf. Assuming smooth contact surfaces determine the magnitude at the contact point between the two cylinders.	3406 lbf	particle_equilibrium
S_Q14	Point A is located at (-6, -2m), point B is located at (8, -4m) and point C is located at (0, 6m). A force of magnitude 40 kN acts along the line passing through points A and B. Determine to moment of the force about point C, with CCW moments positive.	350.7 kN.m	moments
S_Q15	A 16 in long control rod is connected to the ground at B and extends in a direction 100 degrees CCW from the positive x-axis to its other end at A. A force is applied at the end of the rod at A, in a direction of 45 degrees CW from the line of action of the rod. If the force creates a CW moment of 893.8 lbf.in about point B, calculate the magnitude of the force.	79 lbf	moments
S_Q16	An L-shaped bracket ABC is pinned at A (0,0). Point B is located at (-12, 0cm) and point C is located at (-12, 6cm). A cable extends from point B to a support at point D (-15, 11cm). A force of 45 N is applied to the bracket at B, in a direction 50 degrees CW from the positive x-axis. Determine the moment of the force about A and the tension in the cable CD required for equilibrium.	MA = 413.7 N.cm, Cable CD = 57.43 N	moments
S_Q17	A rigid bent bar ABCD is pinned at B. The points are located at: A (0,0), B (1, 0ft), C (1, 1ft), D (4, 1ft). A force of 25 lbf is applied at A in a direction of 60 degrees CW from the positive x-axis. A force of 25 lbf is applied at C in a direction of 150 degrees CW from the positive x-axis. A force of 25 lbf is applied at D. Determine the direction (CW from the positive x-axis) of the force at D such that the sum of the moments about B are zero.	128.35 degrees	moments
S_Q18	A solid rectangle ABCD has corners A (0, 0m), B (0, 4m), C (10, 4m) and D (10, 0m). Two force couples act on the rectangle. One couple consists of 100 N acting in the negative x-axis direction at A and 100 N acting in the positive x-axis direction at C. The other couple consists of 35 N acting at B in a direction 120 degrees CW from the positive x-axis and 35 N acting at D in a direction 60 degrees CCW from the positive x-axis. Determine the magnitude of the resulting moment, with positive in the CW direction.	26.9 N.m	moments
S_Q19	An 8 m long cantilever beam has two vertical downwards forces applied, a force of 120 kN applied at the end of the cantilever and a force of 75 kN located at 5.2 m from the end of the cantilever. Determine the single equivalent resultant force and its location from the left end of the beam.	195 kN located 6 m from the left end	moments
S_Q20	An L-bracket has its corner located at C (0, 0 in), its horizontal leg extends 5 in to end A in the positive x direction and its vertical leg extends 4 in to end B in the positive y direction. A vertical downwards force of 150 lbf is applied at C. A force of 160 lbf is applied at A in a direction 60 degrees CCW from the positive x-axis. Replace the forces at A and C with an equivalent force-couple system acting at point B (force angle CCW from the positive x-axis and moment in the CCW direction).	80.8 lbf acting at -8.14 degrees, 1010 lbf.in	moments
S_Q21	A force couple systems acting at the origin (0, 0m) consists of a force of 90 N in a direction 20 degrees CCW from the positive x-axis and a couple of 369 N.m CW. Replace the force couple with a single force which has the same effect. Specify the magnitude and direction (CCW from the positive x-axis) of the resultant force, and the x coordinate of the point on the axis through which the line of action of the force passes.	90 N at 20 degrees, passing through the x-axis at -12 m	moments
S_Q22	The system consists of a rigid linkage. All dimensions are in inches. Point B is a pin support located at the origin, B = (0, 0). From point B, a rigid member extends upward and to the left to point A. Point A is located 24 in to the left of B and 20 in above B, giving A = (−24, 20). A cable connects point A to point C. Point C is a pin support located directly above point B at a vertical distance of 24 in, so C = (0, 24). A CCW moment M is applied at point B. Given that the tension in cable AC is 10 lbf, determine the magnitude of moment M.	236.7 lbf.in	rigid_body_equilibrium
S_Q23	A 75 kg pole is being raised by pulling a cable. Point A is a pin support located on the ground. The member AB is inclined upward from A at an angle of 70 degrees CCW from the positive x-axis. At the end of the pole at point B a cable is attached in order to raise the pole, and the cable extends towards the left creating an angle between the cable and the member AB of 74 degrees. Determine the tension in the cable.	130.9 N	rigid_body_equilibrium
S_Q24	The left end of a 1 m cantilever beam OABC is fixed at point O, which is located at O = (0, 0). Point A lies on the beam 0.2 m to the right of O. Point B lies a further 0.3 m to the right of A. Point C is located 0.5 m to the right of B, at the right end of the beam. An upward vertical force acts at point A. At point C, a force is applied in the direction 140 degrees CW from the positive x-axis. A CW moment is applied at B. Determine the reactions at O when A = 100 N, B = 325 N.m and C = 110 N.	MO = 375.7 N.m CCW, Ox = 84.3 N, Oy = -29.3 N	rigid_body_equilibrium
S_Q25	A 20 ft long beam ABC supports a frictionless pulley with radius 6 in at point B, located 13 ft from point A at the left end. Point A is a pin support. Point C is the right end of the beam, where the beam is supported by a roller on an inclined surface that makes an angle of 25 degrees with the horizontal. Point D lies on a vertical wall directly above A, and a cable runs from D through the pulley at B below which a weight of 150 lbf is supported. The cable BD is inclined such that it makes an angle of 65degrees with the vertical wall at point D. Determine the reactions at A and C.	Reaction C = 62.1 lbf at 115 degrees, Ax = 162.2 lbf, Ay = 30.3 lbf	rigid_body_equilibrium
S_Q26	A rigid V-shaped two-member linkage is pinned at A. From point A, one rigid member extends upward and to the left at an angle of 20 degrees above the horizontal, terminating at point B. A vertical downward force acts at point B. From point A, a second rigid member extends upward and to the right at an angle of 30 degrees above the horizontal, terminating at point C. At point C, a force is applied downward and slightly to the right, perpendicular to member AC. Member AB is 22 in long and AC is 22 in long. Determine the maximum force that can be applied at B if the magnitude of the reaction at A is not to exceed 950 lbf, and the force C that corresponds to this equilibrium state.	B = 507 lbf, C = 476.5 lbf	rigid_body_equilibrium
S_Q27	A car is parked on a 15 degree slope, with rear wheels locked by the parking brake. The vehicle dimensions (relative to when it is sitting horizontal) are: the wheelbase (distance between the front and rear wheels) is 2579 mm, the mass is 1291 kg, and the location of the center of gravity is 722 mm behind and 664 mm above the front wheel. Find the components of the forces acting on each of the front and rear wheels, parallel and perpendicular to the roadbed.	Front parallel = 0 N, front perpendicular = 3982 N, rear parallel = 1639 N, rear perpendicular = 2134 N	rigid_body_equilibrium
S_Q28	A triangular plate ABC is secured by a pin at A and a roller at C. The interior angles of the triangle at the base are both 65 degrees. At the top point B a force is applied in the direction 60 degrees CCW from edge AB. Determine the reactions at the pin and the roller when the plate is subjected to a 500 lbf load at B.	C = 512.3 lbf, Ax = 286.8 lbf, Ay = 102.7 lbf	rigid_body_equilibrium
S_Q29	A truss is pinned at the left side at A, and sits on a roller at the right side at D. The bottom horizontal chord AFED consists of members AF, FE and ED which are all 2 m in length. The top horizontal chord is BC, where B is located 2.6 m above F and C is located 2.6 m above E. The truss has other members AB, BF, BE, EC and CD. Loads in the positive x direction are located at B of 3 kN and C of 5 kN. Vertical downloads loads of 1 kN are applied at F and also at E. Determine the reaction at D.	D = 4.5 kN upwards	trusses
S_Q30	A horizontal beam ABC is loaded with a force of 150 kN at B in the direction 30 degrees CW from the positive x-axis. Distance AB is 0.4 m and distance BC is 1.6 m. Left end A is pinned and right end C rests on a roller. Determine the magnitudes of the resultant reactions at A and C.	A = 143.1 kN, C = 15 kN	trusses
S_Q31	A cantilever truss is connected to a wall with pinned joints at E and vertically above E at D. The horizontal bottom chord of the truss extends a distance of X m from E to C, then a further 2X m from C to A. The diagonal top chord extends right and downwards from D to B, then B to A. The angle between EC and internal member DC is 60 degrees. The angle between CA and internal member CB is 60 degrees. The angle between CA and BA is 30 degrees. Vertical downwards forces are 400 kN applied at A, 950 kN applied at B and 750 kN applied at C. Use the method of joints to find the forces in all members. Indicate tension as positive and compression as negative.	AB = 800 kN, AC = -693 kN, BC = -823 kN, BD = 1275 kN, CD = 1689 kN, CE = -1949 kN	trusses
S_Q32	A triangle truss has horizontal bottom chord members AC and CD, and is pinned at A at the left and sits on a roller at D at the right. A vertical internal member extends vertically upwards from C to B. The angle between members AB and AC is 30 degrees. The angle between members CD and BD is 30 degrees. A vertical downwards force of 625 kN is applied at C. A force of 150 kN is applied at B in a direction 70 degrees CW from the positive x-axis. Determine the forces in all members. Indicate tension as positive and compression as negative.	BD = -796 kN, CD = 689 kN, AC = 689 kN, BC = 625 kN, AB = -736 kN	trusses
S_Q33	A symmetric triangular truss is pinned at the left side at A. The horizontal bottom chord consists of members extending from A to C, from C to E, from E to G and from G to H. Members AC, CE, EG and GH are all 6 ft long. The truss sits on a roller at H. Joint B is located 5 ft vertically above C. Joint D is located 10 ft vertically above E. Joint F is located 5 ft vertically above G. The other members of the truss are AB, BD, DF, FH, CB, EB, ED, EF and GF. Vertical downwards loads are applied as 500 lbf at A, 500 lbf at H, 400 lbf at B, 400 lbf at D and 400 lbf at F. Using the method of joints, determine the forces in all members. Indicate tension as positive and compression as negative.	AB = -937 lbf, AC = 720 lbf, BC = 0, BD = -625 lbf, BE = -312 lbf, CE = 720 lbf, DE = 400 lbf, DF = -625 lbf, EF = -312 lbf, EG = 720 lbf, FG = 0, FH = -937.2 lbf, GH = 720 lbf	trusses
S_Q34	A cantilever truss connects to a wall with pins at F and G, and F is 6.5 ft vertically above G. The top chord extends horizontally from F to D, from D to B and from B to A. Members FD, DB and BA are each 5 ft long. The diagonal bottom chord extends from G right and upwards towards A, including members GE where joint E is vertically below D, EC where joint C is vertically below B and CA. Other internal members are FE, DE, DC and BC. A force of 325 lbf is applied at A in a direction 100 degrees CW from the positive x-axis. Determine the forces in members DB, DC and EC. Indicate tension as positive and compression as negative.	DB = 682 lbf, DC = 0, EC = -805 lbf	trusses
S_Q35	A triangular Howe truss has overall dimensions of 36 ft wide and 9 ft high. The truss is pinned at A at the left and sits on a roller at G at the right. The horizontal bottom chord consists of members from left to right of AH, HJ, JK, KL, LM, MG and each of these members is 6 ft long. Joint B is located vertically above H. Joint C is located vertically above J. Joint D is located 9 ft vertically above K. Joint E is located vertically above L. Joint F is located vertically above M. Top chords extend right and upwards from A towards D, and left and upwards from G to D, and include members AB, BC, CD, DE, EF and FG. Internal members are HB, JB, JC, KC, KD, KE, LE, LF and MF. Vertical downwards forces applied are 250 lbf at C, 100 lbf at H and 350 lbf at J. Determine the forces in members DE, KE and KL. Indicate tension as positive and compression as negative.	DE = -484 lbf, KE = 0, KL = 433 lbf	trusses
S_Q36	A rectangular Pratt truss has overall dimensions of 16 m wide and 4.5 m high. The top and bottom chords are both horizontal. The truss is pinned at A at the left and sits on a roller at G at the right. The bottom chord consists of members from A to H, from H to J, from J to K and from K to G. Members AH, HJ, JK and KG are each 4 m long. Joint B is located 4.5 m vertically above A. The top chord consists of members B to C, C to D, D to E and E to F. Members BC, CD, DE and EF are each 4 m long. Vertical members are AB, HC, JD, KE and GF and are each 4.5 m long. Diagonal members are HB, JC, JE and KF. Vertical downwards loads are 300 N at H and 150 N at K. A horizontal load of 425 N is applied at F in the positive x-axis direction. Determine the forces in members DE, JE and JK. Indicate tension as positive and compression as negative.	DE = 12.5 N, JE = 210 N, JK = 273 N	trusses
S_Q37	A trussed frame is connected to a wall at A and B. B (0, 0m) is a roller joint. A (0, 6m) is a pin joint. Joint F is located at (6, 6m) and joint C is located at (10, 10m). Truss members include AB, members from A right and upwards towards C and members from B right and upwards towards C. Joint D is at the midpoint between A and C, and creates two equal length members AD and DC. Joint E is at the midpoint between B and F and creates two equal length members BE and EF. Member FC connects F to C. Internal members are AE, AF and FD. A cable connects to the wall 4 m vertically above A and runs through a frictionless pulley at C, and there is a mass suspended vertically from the cable below C. Determine the maximum mass that may be supported by the cable without exceeding 60 kN in tension or 85 kN in compression in any member of the frame.	6120 kg	trusses
S_Q38	An A-frame consists of three members. Member AC is pinned at A (0, 0m) and connected to member BC at C (2, 4m) with a pin joint. Member BC is supported by a roller at B (4, 0m). Member DF is 1.5m long and is connected to AC at D (1.5, 3m) and BC at E (2.5, 3m) with pin joints. Member DF cantilevers out from E a distance of 0.5m to free end F. A vertical downwards load of 110 N is applied at F. Determine the planar components of the forces acting on member AC at joints A, D and C.	Ax = 0 N, Ay = 27.5 N, Dx = 82.5 N, Dy = 55 N, Cx = -82.5 N, Cy = -82.5 N	frames
S_Q39	A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (8, 2m). The two members are pin connected at B (3, 4m). A vertical downwards force of 250 N is applied at B. Determine the scalar components of the reactions at pins A and C.	Ax = 144.2 N, Ay = 192.3 N, Cx = -144.2 N, Cy = 57.7 N	frames
S_Q40	A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 3ft). The two members are pin connected at B (5, 3ft). A horizontal force of 175 lbf in the positive x direction is applied at the midpoint of AB. Determine the scalar components of the reactions at pins A and C.	Ax = -118.8 lbf, Ay = -18.8 lbf, Cx = -56.3 lbf, Cy = 18.8 lbf	frames
S_Q41	A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 0ft). The two members are pin connected at B (4, 5ft). A horizontal force of 475 lbf in the negative x direction is applied at the midpoint of AB. A CCW moment of 800 lb.ft is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.	Ax = 436.3 lbf, Ay = 248.4 lbf, Cx = 38.8 lbf, Cy = -248.4 lbf	frames
S_Q42	A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (10, -1m). The two members are pin connected at B (5, 3m). A horizontal force of 75 N in the negative x direction is applied at the midpoint of AB. A CW moment of 100 N.m is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.	Ax = 44.6 N, Ay = 4.29 N, Cx = 30.4 N, Cy = -4.29 N	frames
S_Q43	Given the function (y = -1.5.x^2 + 2.x + 2.5), determine the area under the curve from the y-axis to the point where the curve crosses the positive x-axis. The x- and y- coordinates are in inches.	5.03 in^2	centroids
S_Q44	Use integration with horizontal strips to determine the area of the spandrel bounded by the function (y = h - k.x^0.5) and the lines y = h and x = b, and also find the coordinates of its centroid. Note: b and h are constants. Give your answers for the area and the centroid in terms of b and h.	Area = (b.h / 1.5), centroid x coordinate = (1.5.b / 2.5), centroid y coordinate = (2.5.h / 4)	centroids
S_Q45	Determine the coordinates of the centroid of the shaded area bounded by the y-axis, a parabola passing through the origin (0, 0m) and point A, and a straight line passing through points A and B. Point A is located at (9, 3m) and B is located at (0, 9m).	(3.2, 3.7m)	centroids
S_Q46	A 16 ft long beam is supported by a pin at A located 1 ft from the left end, and a roller at B located 2 ft from the right end. A downwards vertical uniformly distributed load of 20 lbf/ft is applied between 7 ft from the left end and 4 ft from the right end. The beam weight is negligible. Determine the reactions at A and B.	Ax = 0 lbf, Ay = 34.6 lbf, By = 65.4 lbf	beam_reactions
S_Q47	A 15 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (15, 0m). Point C is located at (1, 0m) and D is located at (12, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 45 kN/m at C to 60 kN/m at D. The beam self weight is 15 kN/m. Determine the reactions at A and B.	Ax = 0 kN, Ay = 429.7 kN, B = 372.8 kN	beam_reactions
S_Q48	A 20 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (20, 0m). Point C is located at (6, 0m), D is located at (12, 0m) and E is located at (19, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 0 kN/m at C to 60 kN/m at D, then remains constant at 60 kN/m to E. The beam weight is negligible. Determine the reactions at A and B.	Ax = 0 kN, Ay = 184.5 kN, By = 415.5 kN	beam_reactions
S_Q49	An 18 m long beam supports a vertical downwards load that varies uniformly from 50 N/m at the left end to 150 N/m at the right end. The beam is pinned at A (0, 0m) and sits on a roller at B (10, 0m). Determine the internal shear and bending moment at a section passing through point C (6, 0m).	V = -490 N, M = -1640 N.m	beam_internal_forces
S_Q50	A beam BC is 4 ft long and is pinned at the right end C (4, 0ft). At the left end B (0, 0ft) the beam is pin connected to member AB. A is located to the left and upwards from B, creating an angle of 25 degrees with the x-axis. Beam BC supports a vertical downwards uniformly distributed load of 15 lbf/ft. Determine the internal forces at point D located at the midlength of BC.	V = 0 lbf, N = 64.3 lbf, M = 30 lbf.ft	beam_internal_forces
S_Q51	A 12 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 500 kN at 8 m from A, and 650 kN at 10 m from A. Find the shear and bending moment equations as function of x for the three ranges: 0m < x < 8m, 8m < x < 10m, 10m < x < 12m.	(0m < x < 8m: V = 275 kN, M = 275.x kN.m), (8m < x < 10m: V = -225 kN, M = 275.x - 500.(x - 8)), (10m < x < 12m: V = -875 kN, M = 275.x - 500.(x - 8) - 650.(x - 10))	beam_internal_forces
S_Q52	A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 165 kN at 2 m from A, and 15 kN at 9 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).	V (m, kN): (0, 0), (0, 157), (2, 157), (2, -8.3), (9, -8.3), (9, -23.3), (20, -23.3), (20, 0) M (m, kN.m): (0, 0), (2, 314), (9, 256), (0, 0)	beam_internal_forces
S_Q53	A 6 ft long cantilever beam is fixed at the left end A. Vertical downwards forces are 140 lbf at 3 m from A, and 240 lbf at 6 m from A. A CCW moment of 2340 lbf.ft is applied at 4 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).	V (ft, lbf): (0, 0), (0, 380), (3, 380), (3, 240), (6, 240), (6, 0) M (ft, lbf.ft): (0, 0), (0, 480), (3, 1620), (4, 1860), (4, -480), (6, 0)	beam_internal_forces
S_Q54	A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards uniformly distributed loads are: 200 kN/m from A to 9 m from A, and 180 kN/m from 15 m from A to 20 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).	V (m, kN): (0, 0), (0, 1508), (9, -293), (15, -293), (20, -1192), (20, 0) M (m, kN.m): (0, 0), (7.5, 5681), (9, 5468), (15, 3713), (20, 0)	beam_internal_forces
S_Q55	A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards loads are 400 kN at 2 m from A and a uniformly distributed load of 300 kN/m from 12 m from A to 17 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).	V (m, kN): (0, 0), (0, 773), (2, 773), (2, 373), (12, 373), (17, -1128), (20, -1128), (20, 0) M (m, kN.m): (0, 0), (2, 1545), (12, 5270), (13.2, 5501), (17, 3383), (20, 0)	beam_internal_forces
S_Q56	A 18 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards linearly varying distributed loads are 0 kN/m at A varying linearly to 25 kN/m at 9 m from A, then varying linearly to zero at B. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).	V (m, kN): (0, 0), (0, 113), (9, 0), (18, -113), (18, 0) M (m, kN.m): (0, 0), (9, 675), (18, 0)	beam_internal_forces
S_Q57	A 20 m long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards symmetric distributed parabolic loading w(x), given by the equation w(x) = -2.x^2 + 40.x kN/m. Find the maximum values of shear and moment and their locations in the beam.	V = 1333 kN at x = 0 m and x = 20 m, M = 8333 kN.m at x = 10 m	beam_internal_forces
S_Q58	A 20 ft long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards distributed parabolic loading w(x), given by the equation w(x) = - 0.007.x^3 - 0.03.x^2 + 3.x lbf/ft. Find the bending moment function in terms of x.	M = (x^4)/400 + (x^5)/2857 - (x^3)/2 + 124.x	beam_internal_forces
S_Q59	A 220 kN horizontal force P is applied to a box with weight 200kN resting on a surface with an incline of 20 degrees from the horizontal. The line of action of P passes through the center of gravity of the box. The box is 4 m wide and 3 m tall, and the static and kinetic coefficients of friction between the box and the surface are 0.7 and 0.57, respectively. Determine the magnitude of the friction force acting on the box.	138.3 kN	friction
S_Q60	A block weighs 500 N and is resting on an incline of 30 degrees from the horizontal. The coefficient of friction between the block and the inclined surface is 0.4 . A force P is applied to the box. The line of action of P passes through the center of gravity of the box and is inclined 65 degrees from the horizontal. Determine the maximum value of P before the block starts to move up the incline (i.e. impending upwards motion), and the minimum value of P required to prevent the block from moving down the incline (i.e. impending downward motion).	Maximum = 404 N, minimum = 130 N	friction
S_Q61	A block with mass 10 kg rests on a horizontal surface. A block with mass 35 kg rests on top of the 10 kg block, and is restrained from moving horizontally. The coefficient of friction between all surfaces is 0.2. A horizontal force P is applied to the 10 kg block with a line of action through its center of gravity. Determine the smallest force P required to start the 10 kg block moving.	157 N	friction
S_Q62	A 2D object is defined by 5 vertices in the x-y space with straight lines between them. The vertices are: (-1, 0cm), (-7, 0cm), (0, 7cm), (1, 7cm), (1, 6cm). Calculate the moment of inertia and radius of gyration with respect to the x-axis.	Ix = 278.4 cm^4, kx = 3.3 cm	moments_of_inertia
S_Q63	A square has vertices in the x-y space of (0, 6in), (-4, 6in), (-4, 10in) and (0, 10in). Calculate the moments of inertia of the object about both the x- and y-axes.	Ix = 1045 in^4, Iy = 85.3 in^4	moments_of_inertia
S_Q64	A 2D object is defined by 6 vertices in the x-y space with straight lines between them. The vertices are: (0,0ft), (0, 9ft), (1, 9ft), (1, 3ft), (4, 3ft), (4, 0ft). Calculate the moments of inertia of the object about both the x- and y-axes.	Ix = 270 ft^4, Iy = 66 ft^4	moments_of_inertia
S_Q65	A circle has a diameter of 8 in and is positioned in the x-y space such that its centre is located at (5, 4in). Determine the moment of inertia and radius of gyration about the y-axis.	Iy = 1458 in^4, ky = 5.4 in	moments_of_inertia
S_Q66	A rectangle has vertices in the x-y space of (0, 0cm), (0, 8cm), (10, 8cm), (10, 0cm). Inside this rectangle is a rectangular void with vertices of (1, 6cm), (1, 7cm), (9, 7cm), (9, 6cm). Determine the moment of inertia of the composite shape about the horizontal neutral axis of the composite shape.	370.4 cm^4	moments_of_inertia
S_Q67	A 2D object is defined by 8 vertices in the x-y space with straight lines between them. The vertices are: (0, 2ft), (7, 2ft), (7, 8ft), (8, 8ft), (8, -8ft), (7, -8ft), (7, -2ft), (0, -2ft). Calculate the moment of inertia of the object about the vertical neutral axis of the object.	278.6 ft^4	moments_of_inertia
S_Q68	Two steel channel sections are placed back to back to each other with a distance of 120 mm separating them. They are placed upright such that their large dimension (depth) is in line with the vertical y-axis. A built up beam is constructed by welding two plates to these channels, symmetrically placed, one on the top and one on the bottom. The plates are both 300 mm wide and 35 mm thick. The channel sections have properties: Area = 4740 mm^2, depth = 305 mm, flange width = 77.5 mm, Ixx = 59900000 mm^4, Iyy = 1850000 mm^4, and the horizontal distance to the section centroid is 17.1 mm. Determine the moment of inertia and radius of gyration of the built up beam with respect to the x- and y-axes.	Ix = 728800000 mm^4, kx = 154.6 mm, Iy = 217600000 mm^4, ky = 84.5 mm	moments_of_inertia
M_Q1	A pedestal has a triangular cross section. If it is subjected to a compressive force P of 500 lb, specify the x and y coordinates for the location of point P(x, y), where the load must be applied on the cross section, so that the average normal stress is uniform. Compute the stress. Vertices of the triangular cross section: Vertex 1: (0 in., 0 in.), Vertex 2: (0 in., 9 in.), Vertex 3: (12 in., 3 in.)	x = 4in, y = 4in, stress = 9.26psi	stress
M_Q2	A metal rod specimen failed in a tension test at an angle of 52 degrees (to the longitudinal axis) when the axial load was 19.80 kip. If the diameter of the specimen is 0.5 in., determine the average normal and average shear stress acting on the area of the inclined failure plane. Also, what is the average normal stress acting on the cross section when failure occurs?	Average normal stress on cross section: 100.9 ksi, Average normal stress on inclined failure plane: 62.7 ksi, Average shear stress on inclined failure plane: 48.9 ksi	stress
M_Q3	Rods AB and BC have diameters of 4 mm and 6 mm, respectively. The geometry: Point A: Pin support on left, Point B: Ring connection where both rods meet, Point C: Pin support on right, Rod AB: Horizontal, connects A to B, Rod BC: Inclined at theta = 60 degrees above horizontal, connects B to C. If a vertical downwards load of 8 kN is applied to the ring at B, determine the average normal stress in each rod if theta = 60 degrees.	AB = 368MPa, BC = 327MPa	stress
M_Q4	The bars of the truss each have a cross-sectional area of 1.25 in squared. Geometry: Joint A: (0 ft, 0 ft), Joint B: (4 ft, 3 ft), Joint C: (8 ft, 3 ft), Joint E: (4 ft, 0 ft), Joint D: (8 ft, 0 ft), Joint C: Pinned to wall, Joint D: Pinned to wall. Members: AB, AE, ED, EB, BC, BD. Force P = 8 kip acting downward at joint A, Force 0.75P = 6 kip acting downward at joint E. Determine the average normal stress in each member due to the loading P = 8 kip. State whether the stress is tensile (T) or compressive (C).	AB=10.7ksi T, AE=8.53ksi C, ED=8.53ksi C, EB=4.8ksi T, BC=23.5ksi T, BD=18.7ksi C	stress
M_Q5	A beam and hanger assembly is used to support a distributed loading of w = 0.8 kip/ft. Beam: Pinned to the wall at C (0, 0ft) and extends horizontal for 6ft. Hanger rod AB: Pinned to the wall at B (0, 3ft) and connected to the beam with a bolt at A (4, 0ft). Uniformly distributed load w = 0.8 kip/ft acting downward over the 6 ft beam length. Determine the average shear stress in the 0.40 in. diameter bolt at A and the average tensile stress in rod AB, which has a diameter of 0.5 in. If the yield shear stress for the bolt is 25 ksi, and the yield tensile stress for the rod is 38 ksi, determine the factor of safety with respect to yielding in each case.	Bolt: shear stress=23.9ksi, FOS=1.05. Rod: stress=30.6ksi, FOS=1.24	stress
M_Q6	A rod BC is made of steel having an allowable tensile stress of 155 MPa and supports the left end of beam AB. Joint A: (4.5, 0 m) right end, pinned support, Joint B: (0, 0 m) left end, connection point on beam for rod BC, Joint C: Above point B, upper connection of rod to support. Distributed downwards load increases from zero at B to w = 15 kN/m at (3, 0m) then decreases to zero at A. Determine its smallest diameter so that it can support the load.	11.1 mm	stress
M_Q7	A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip.	0.0472 in/in	strain
M_Q8	A bar CA has length L m and is supported at its right end A by a wire AB. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at B which is L m vertically above C. When the wire AB is unstretched it makes an angle of theta = 45 degrees below the horizontal at B. If a load is applied to the bar AC which causes the angle to change to theta = 47 degrees, determine the normal strain in the wire.	0.0343 m/m	strain
M_Q9	Two wires are connected together at A with a ring. Wire CA: Connects C to A, length = 300 mm, oriented at 30 degrees above horizontal (measured from horizontal at A). Wire BA: Connects B to A, length = 300 mm, oriented at 30 degrees below horizontal (measured from horizontal at A). B and C are connections to a wall and C is vertically above B. A horizontal force P is applied at A. If the force P causes point A to be displaced horizontally 2 mm away from the wall, determine the normal strain developed in each wire.	Strain in both wires = 0.0058 mm/mm	strain
M_Q10	A right-angled triangular plate ABC has vertices: A (0, 0mm), B (0, 800mm), C (800, 0mm). Edge BC is fixed, and its apex A is given a displacement of 5 mm in the direction perpendicular to edge BC. Determine the shear strain at A.	0.0088 rad	strain
M_Q11	A rectangular plastic block is glued at its top and bottom to rigid plates. Point A: Bottom-left corner (0, 0). Point D: Bottom-right corner. Point B: Top-left corner (0, 2in). Point C: Top-right corner. A horizontal force P is applied to the top plate in the positive x direction, and causes the material to deform so that its sides are described by the equation y = 3.56x^(1/4). Determine the shear strain in the material at its corners A and B.	A = 0, B = 0.199 rad	strain
M_Q12	The stress - strain diagram for an elastic fibre has coordinates: (0 in/in, 0 psi), (2 in/in, 11 psi), (2.25 in/in, 55 psi). Determine the modulus of elasticity and estimate the modulus of toughness and modulus of resilience.	modulus of elasticity = 5.5psi, modulus of toughness = 19.25psi, modulus of resilience = 11 psi	mechanical_properties
M_Q13	The stress-strain (s-e) diagram for a bone is described by the equation e = 0.45(10^-6)s + 0.36(10^-12)s^3, where s is in kPa. Determine the modulus of toughness and the amount of elongation of a 200 mm long region just before it fractures if failure occurs at e = 0.12 mm/mm.	613MPa, 24mm	mechanical_properties
M_Q14	A specimen is originally 1 ft long, has a diameter of 0.5 in., and is subjected to a force of 500 lb. When the force is increased to 1800 lb, the specimen elongates 0.9 in. Determine the modulus of elasticity for the material if it remains elastic.	88.3ksi	mechanical_properties
M_Q15	A bar BC has length 10 ft and is supported at its right end B by a wire AB with diameter 0.2 in made of A-36 steel. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at A which is vertically above C. Wire AB makes an angle of 30 degrees above the horizontal at B. A downwards uniformly distributed load is applied to the full length of the rod BC. Determine the magnitude of the distributed load if the end B is displaced by 0.75 in downwards.	0.246kip/ft	mechanical_properties
M_Q16	An aluminum block has a rectangular cross section and is subjected to an axial compressive force of 8 kip. If the 1.5 in. side changed its length to 1.500132 in., determine Poisson's ratio and the new length of the 2 in. side. The modulus of elasticity is 10(10^3) ksi.	Poissons ratio = 0.33, side length = 2.00018 in	mechanical_properties
M_Q17	An 8 mm diameter brass rod has a modulus of elasticity of 100 GPa. If it is 3 m long and subjected to an axial load of 2 kN, determine its elongation. What is its elongation under the same load if its diameter is 6 mm?	For 8mm diameter = 1.19mm, for 6mm diameter = 2.12m	mechanical_properties
M_Q18	An assembly consists of a steel rod CB of length 3 m and an aluminum rod BA of length 2 m, each having a diameter of 12 mm. The rod is fixed at C and free at A and is subjected to axial loadings at A and at B: 6 kN acting at B in the negative x-direction, 18 kN acting at end A in the positive x-direction. Determine the displacement at the free end A. Modulus of elasticity (steel) is 200 GPa and Modulus of elasticity (aluminum) is 70 GPa.	6.14mm	axial_load
M_Q19	A truss is made of three A-36 steel members AB, BC and CA, each having a cross-sectional area of 400 mm squared. Joint A: (0 m, 0 m), Joint C: (1.4 m, 0 m), Joint B: (0.8 m, 0.8 m). Joint A is pinned and joint C is a roller. Forces: Force P acting downward at joint B, force 5 kN acting horizontally to the right at joint B. Determine the magnitude P required to displace the roller to the right 0.2 mm.	21.7kN	axial_load
M_Q20	A linkage is made of three pin-connected A-36 steel members DA, CA and AB, each having a cross-sectional area of 0.730 in squared. Joint D: (0 ft, 0 ft), Joint C: (6 ft, 0 ft), Joint A: (3 ft, -4 ft), Joint B: (3 ft, -10 ft). Joints C and D are pinned. A vertical force of P = 50 kip is applied to the end B of member AB. Determine the vertical displacement of point B.	0.281in	axial_load
M_Q21	A 500 mm long steel pipe is filled with concrete and subjected to a compressive force of 80 kN. The steel pipe outer diameter is 80 mm and inner diameter is 70 mm. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (concrete) is 24 GPa. Determine the stress in the concrete and the steel due to this loading.	steel stress = 48.8MPa, concrete stress = 5.85MPa	axial_load
M_Q22	A 10 mm diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. The assembly is subjected to a compressive load P. The yield stress for the steel is 640 MPa and the bronze is 520 MPa. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (bronze) is 100 GPa. Determine the magnitude of the largest elastic load P that can be applied to the assembly.	126kN	axial_load
M_Q23	A tapered member is fixed connected at its ends A and B and is subjected to a downwards load P. The member is 2 in thick and 60 in long. The height tapers linearly from 6 in at A to 3 in at B. Determine the distance from A of the load position and its greatest magnitude, if the allowable normal stress for the material is 4 ksi.	28.9in, 60.4kip	axial_load
M_Q24	A post is made from 6061-T6 aluminum and has a diameter of 50 mm and length 0.5 m. It is fixed supported at A and B and at its center C there is a coiled spring with stiffness 200 MN/m attached to a rigid collar. If the spring is originally uncompressed, determine the compression of the spring when a load of P = 50 kN is applied to the collar at C.	0.039mm	axial_load
M_Q25	A thermo gate consists of two 6061-T6 aluminum plates that have a width of 15 mm and are fixed supported at their ends. One plate is 600 mm length, the other is 400 mm length, and both plates have thickness 10 mm. If the gap between them is 1.5 mm when the temperature is 25 degrees C, determine the temperature required to just close the gap. The coefficient of thermal expansion is 24(10^-6)/degrees C.	87.5 degrees C	axial_load
M_Q26	A member is to be made from a steel plate that is 0.25 in thick. If a 1 in hole is drilled through its center, determine the approximate width of the plate so that it can support an axial force of 3350 lb. The allowable stress is 22 ksi.	2.49in	axial_load
M_Q27	A 0.25 in diameter steel rivet having a temperature of 1500 degrees F is secured between two plates such that at this temperature it is 2 in long and exerts a clamping force of 250 lb between the plates. Determine the approximate clamping force between the plates when the rivet cools to 70 degrees F. Assume that the heads of the rivet and the plates are rigid and use a coefficient of thermal expansion of 8(10^-6)/degrees F, and modulus of elasticity of 29(10^3) ksi.	16.5kip	axial_load
M_Q28	A copper pipe has outer diameter 40 mm and inner diameter 37 mm. It is fixed at a wall at A and the axis is the longitudinal axis, positive away from the wall. Three torques are applied, sequentially moving away from the wall: -30 Nm, 20 Nm then -80 Nm. Determine the maximum shear stress in the pipe.	26.7MPa	torsion
M_Q29	A steel tube having an outer diameter of 2.5 in. is used to transmit 350 hp when turning at 27 rev/min. Determine the inner diameter of the tube if the allowable shear stress is 10 ksi.	2.48in	torsion
M_Q30	Two steel 20 mm diameter steel shafts are connected using a brass coupling. If the yield point for the steel is 100 MPa and for the brass is 250 MPa, determine the required outer diameter of the coupling so that the steel and brass begin to yield at the same time when the assembly is subjected to a torque T. Assume that the coupling has an inner diameter of 20 mm.	21.9mm	torsion
M_Q31	The drive shaft of an automobile is made of a steel having an allowable shear stress of 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft's wall.	0.174in	torsion
M_Q32	A 0.75 in. diameter shaft for an electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is 6 in long and supported by ball bearings at its ends.	1.51lbft, 219psi	torsion

S_Q1

Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. Determine the scalar components of F in the x and y directions.

Fx = 300 lbf, Fy = 519.6 lbf

force_vectors

S_Q2

Force F has a magnitude of 600 lbf acting 60 degrees CCW from the positive x-axis. A new coordinate system is created by rotating the x- and y-axes 65 degrees clockwise. Determine the scalar components of F in the x' and y' directions.

Fx' = -344.1 lbf, Fy' = 491.5 lbf

force_vectors

S_Q3

Find the magnitude and direction of force F which has scalar components: Fx = -2 N and Fy = -6 N

6.325 N, 251.6 degrees

force_vectors

S_Q4

Vector R has vector components A and B where A has a magnitude of 4 kN and points 35 degrees CW from the positive x-axis, and B has a magnitude of 6 kN pointing 85 degrees CCW from the positive x-axis. Find the magnitude and direction of force of the resultant R.

5.3 kN, 44.1 degrees CCW from the positive x-axis

force_vectors

S_Q5

Determine the resultant of three forces A, B and C, with scalar components of A (-6.26, 12.5 lbf), B (35.4, -35.4 lbf) and C (-36.8, 27.6 lbf).

9.06 lbf, 148.3 degrees

force_vectors

S_Q6

Three forces act on a particle. Force A is 600 N acting in the negative y direction. Force B acts 15 degrees CCW from the positive y-axis. Force C acts 60 degrees CCW from the positive x-axis. Determine the magnitudes of forces B and C.

B = 424.3 N, C = 219.6 N

force_vectors

S_Q7

A small massless ring is located at (0, 0m). A is located at (0, -4m), B is located at (-2, 6m) and C is located at (4, 3m). A force of 400 N is applied to the ring in the direction towards C. The ring is held in equilibrium by two inextensible cables, one connected to point A and one connected to point B. Determine the tension in cable A and cable B.

Cable A = 1200 N, Cable B = 1012 N

particle_equilibrium

S_Q8

A mass of 10 kg is held in equilibrium by two cables, cable A and cable B. The mass is located at (0, 0m). Point A is located at (-4.25, 3.5m) and point B is located at (4.5, 3.5m). Determine the tension in cable A and cable B.

Cable A = 79.36 N, Cable B = 77.61 N

particle_equilibrium

S_Q9

A 2 lb cylinder rests in a trough with smooth frictionless surfaces. The left side of the trough is angled at 35 degrees from the horizontal. The right side of the trough is angled at 60 degrees to the horizontal. Determine the reactions on the cylinder from the left side and the right side of the trough.

Left side = 1.74 lb, right side = 1.15 lb

particle_equilibrium

S_Q10

Cable ABC has a length of 8 ft and the cable supports a weight of 100 lbf riding on a frictionless pulley of negligible radius located at B. The cable is supported at A (0, 0 ft) and C (6.5, 2.5 ft). Determine the horizontal distance from A of the pulley and the tension in the cable, required for equilibrium.

Distance = 1.51 ft, tension = 85.8 lbf

particle_equilibrium

S_Q11

A 320 mm x 550 mm steel plate 35 mm thick is suspended from a point P located 350 mm above the center of the plate (in the z direction). Three cables support the plate at A, B and D. The four corners of the plate are E located at (0, 0mm), C located at (0, 550mm), B located at (320mm, 550mm) and A located at (320, 0mm). Point D is located 165 mm from E along the plate edge EC. The density of steel is 7970 kg per cubic metre. Determine the tension in cable A, cable B and cable D.

Cable A = 97.6 N, cable B = 227.8 N, cable D = 275.4 N

particle_equilibrium

S_Q12

A system of cables are suspended by connections at B, E and D. A weight of 75 lbf is located at A and a weight of 55 lbf is located at C. Point B is located at the left end of the system of cables, and cable B to A forms an angle of 10 degrees to the horizontal where A is below and to the right of B. Cable A to C forms an angle of 15 degrees to the horizontal and C is above and to the right of A. Two cables connect to C, a cable from C to E which forms an angle of 70 degrees to the horizontal where E is above and to the right of C, and cable C to D which is in line with the horizontal and D is to the right of C. Determine the tensions in cables AB, AC, CE and CD.

Cable AB = 171.4 lbf, cable AC = 174.8 lbf, cable CE = 106.7 lbf, cable CD = 132.3 lbf

particle_equilibrium

S_Q13

Two cylinders are placed in a 14 ft wide container. The cylinder that sits on the bottom of the container has a radius of 5 ft and weighs 3750 lbf. The second cylinder sits on top of the other cylinder and has a radius of 4.5 ft and weighs 3000 lbf. Assuming smooth contact surfaces determine the magnitude at the contact point between the two cylinders.

3406 lbf

particle_equilibrium

S_Q14

Point A is located at (-6, -2m), point B is located at (8, -4m) and point C is located at (0, 6m). A force of magnitude 40 kN acts along the line passing through points A and B. Determine to moment of the force about point C, with CCW moments positive.

350.7 kN.m

moments

S_Q15

A 16 in long control rod is connected to the ground at B and extends in a direction 100 degrees CCW from the positive x-axis to its other end at A. A force is applied at the end of the rod at A, in a direction of 45 degrees CW from the line of action of the rod. If the force creates a CW moment of 893.8 lbf.in about point B, calculate the magnitude of the force.

79 lbf

moments

S_Q16

An L-shaped bracket ABC is pinned at A (0,0). Point B is located at (-12, 0cm) and point C is located at (-12, 6cm). A cable extends from point B to a support at point D (-15, 11cm). A force of 45 N is applied to the bracket at B, in a direction 50 degrees CW from the positive x-axis. Determine the moment of the force about A and the tension in the cable CD required for equilibrium.

MA = 413.7 N.cm, Cable CD = 57.43 N

moments

S_Q17

A rigid bent bar ABCD is pinned at B. The points are located at: A (0,0), B (1, 0ft), C (1, 1ft), D (4, 1ft). A force of 25 lbf is applied at A in a direction of 60 degrees CW from the positive x-axis. A force of 25 lbf is applied at C in a direction of 150 degrees CW from the positive x-axis. A force of 25 lbf is applied at D. Determine the direction (CW from the positive x-axis) of the force at D such that the sum of the moments about B are zero.

128.35 degrees

moments

S_Q18

A solid rectangle ABCD has corners A (0, 0m), B (0, 4m), C (10, 4m) and D (10, 0m). Two force couples act on the rectangle. One couple consists of 100 N acting in the negative x-axis direction at A and 100 N acting in the positive x-axis direction at C. The other couple consists of 35 N acting at B in a direction 120 degrees CW from the positive x-axis and 35 N acting at D in a direction 60 degrees CCW from the positive x-axis. Determine the magnitude of the resulting moment, with positive in the CW direction.

26.9 N.m

moments

S_Q19

An 8 m long cantilever beam has two vertical downwards forces applied, a force of 120 kN applied at the end of the cantilever and a force of 75 kN located at 5.2 m from the end of the cantilever. Determine the single equivalent resultant force and its location from the left end of the beam.

195 kN located 6 m from the left end

moments

S_Q20

An L-bracket has its corner located at C (0, 0 in), its horizontal leg extends 5 in to end A in the positive x direction and its vertical leg extends 4 in to end B in the positive y direction. A vertical downwards force of 150 lbf is applied at C. A force of 160 lbf is applied at A in a direction 60 degrees CCW from the positive x-axis. Replace the forces at A and C with an equivalent force-couple system acting at point B (force angle CCW from the positive x-axis and moment in the CCW direction).

80.8 lbf acting at -8.14 degrees, 1010 lbf.in

moments

S_Q21

A force couple systems acting at the origin (0, 0m) consists of a force of 90 N in a direction 20 degrees CCW from the positive x-axis and a couple of 369 N.m CW. Replace the force couple with a single force which has the same effect. Specify the magnitude and direction (CCW from the positive x-axis) of the resultant force, and the x coordinate of the point on the axis through which the line of action of the force passes.

90 N at 20 degrees, passing through the x-axis at -12 m

moments

S_Q22

The system consists of a rigid linkage. All dimensions are in inches. Point B is a pin support located at the origin, B = (0, 0). From point B, a rigid member extends upward and to the left to point A. Point A is located 24 in to the left of B and 20 in above B, giving A = (−24, 20). A cable connects point A to point C. Point C is a pin support located directly above point B at a vertical distance of 24 in, so C = (0, 24). A CCW moment M is applied at point B. Given that the tension in cable AC is 10 lbf, determine the magnitude of moment M.

236.7 lbf.in

rigid_body_equilibrium

S_Q23

A 75 kg pole is being raised by pulling a cable. Point A is a pin support located on the ground. The member AB is inclined upward from A at an angle of 70 degrees CCW from the positive x-axis. At the end of the pole at point B a cable is attached in order to raise the pole, and the cable extends towards the left creating an angle between the cable and the member AB of 74 degrees. Determine the tension in the cable.

130.9 N

rigid_body_equilibrium

S_Q24

The left end of a 1 m cantilever beam OABC is fixed at point O, which is located at O = (0, 0). Point A lies on the beam 0.2 m to the right of O. Point B lies a further 0.3 m to the right of A. Point C is located 0.5 m to the right of B, at the right end of the beam. An upward vertical force acts at point A. At point C, a force is applied in the direction 140 degrees CW from the positive x-axis. A CW moment is applied at B. Determine the reactions at O when A = 100 N, B = 325 N.m and C = 110 N.

MO = 375.7 N.m CCW, Ox = 84.3 N, Oy = -29.3 N

rigid_body_equilibrium

S_Q25

A 20 ft long beam ABC supports a frictionless pulley with radius 6 in at point B, located 13 ft from point A at the left end. Point A is a pin support. Point C is the right end of the beam, where the beam is supported by a roller on an inclined surface that makes an angle of 25 degrees with the horizontal. Point D lies on a vertical wall directly above A, and a cable runs from D through the pulley at B below which a weight of 150 lbf is supported. The cable BD is inclined such that it makes an angle of 65degrees with the vertical wall at point D. Determine the reactions at A and C.

Reaction C = 62.1 lbf at 115 degrees, Ax = 162.2 lbf, Ay = 30.3 lbf

rigid_body_equilibrium

S_Q26

A rigid V-shaped two-member linkage is pinned at A. From point A, one rigid member extends upward and to the left at an angle of 20 degrees above the horizontal, terminating at point B. A vertical downward force acts at point B. From point A, a second rigid member extends upward and to the right at an angle of 30 degrees above the horizontal, terminating at point C. At point C, a force is applied downward and slightly to the right, perpendicular to member AC. Member AB is 22 in long and AC is 22 in long. Determine the maximum force that can be applied at B if the magnitude of the reaction at A is not to exceed 950 lbf, and the force C that corresponds to this equilibrium state.

B = 507 lbf, C = 476.5 lbf

rigid_body_equilibrium

S_Q27

A car is parked on a 15 degree slope, with rear wheels locked by the parking brake. The vehicle dimensions (relative to when it is sitting horizontal) are: the wheelbase (distance between the front and rear wheels) is 2579 mm, the mass is 1291 kg, and the location of the center of gravity is 722 mm behind and 664 mm above the front wheel. Find the components of the forces acting on each of the front and rear wheels, parallel and perpendicular to the roadbed.

Front parallel = 0 N, front perpendicular = 3982 N, rear parallel = 1639 N, rear perpendicular = 2134 N

rigid_body_equilibrium

S_Q28

A triangular plate ABC is secured by a pin at A and a roller at C. The interior angles of the triangle at the base are both 65 degrees. At the top point B a force is applied in the direction 60 degrees CCW from edge AB. Determine the reactions at the pin and the roller when the plate is subjected to a 500 lbf load at B.

C = 512.3 lbf, Ax = 286.8 lbf, Ay = 102.7 lbf

rigid_body_equilibrium

S_Q29

A truss is pinned at the left side at A, and sits on a roller at the right side at D. The bottom horizontal chord AFED consists of members AF, FE and ED which are all 2 m in length. The top horizontal chord is BC, where B is located 2.6 m above F and C is located 2.6 m above E. The truss has other members AB, BF, BE, EC and CD. Loads in the positive x direction are located at B of 3 kN and C of 5 kN. Vertical downloads loads of 1 kN are applied at F and also at E. Determine the reaction at D.

D = 4.5 kN upwards

trusses

S_Q30

A horizontal beam ABC is loaded with a force of 150 kN at B in the direction 30 degrees CW from the positive x-axis. Distance AB is 0.4 m and distance BC is 1.6 m. Left end A is pinned and right end C rests on a roller. Determine the magnitudes of the resultant reactions at A and C.

A = 143.1 kN, C = 15 kN

trusses

S_Q31

A cantilever truss is connected to a wall with pinned joints at E and vertically above E at D. The horizontal bottom chord of the truss extends a distance of X m from E to C, then a further 2X m from C to A. The diagonal top chord extends right and downwards from D to B, then B to A. The angle between EC and internal member DC is 60 degrees. The angle between CA and internal member CB is 60 degrees. The angle between CA and BA is 30 degrees. Vertical downwards forces are 400 kN applied at A, 950 kN applied at B and 750 kN applied at C. Use the method of joints to find the forces in all members. Indicate tension as positive and compression as negative.

AB = 800 kN, AC = -693 kN, BC = -823 kN, BD = 1275 kN, CD = 1689 kN, CE = -1949 kN

trusses

S_Q32

A triangle truss has horizontal bottom chord members AC and CD, and is pinned at A at the left and sits on a roller at D at the right. A vertical internal member extends vertically upwards from C to B. The angle between members AB and AC is 30 degrees. The angle between members CD and BD is 30 degrees. A vertical downwards force of 625 kN is applied at C. A force of 150 kN is applied at B in a direction 70 degrees CW from the positive x-axis. Determine the forces in all members. Indicate tension as positive and compression as negative.

BD = -796 kN, CD = 689 kN, AC = 689 kN, BC = 625 kN, AB = -736 kN

trusses

S_Q33

A symmetric triangular truss is pinned at the left side at A. The horizontal bottom chord consists of members extending from A to C, from C to E, from E to G and from G to H. Members AC, CE, EG and GH are all 6 ft long. The truss sits on a roller at H. Joint B is located 5 ft vertically above C. Joint D is located 10 ft vertically above E. Joint F is located 5 ft vertically above G. The other members of the truss are AB, BD, DF, FH, CB, EB, ED, EF and GF. Vertical downwards loads are applied as 500 lbf at A, 500 lbf at H, 400 lbf at B, 400 lbf at D and 400 lbf at F. Using the method of joints, determine the forces in all members. Indicate tension as positive and compression as negative.

AB = -937 lbf, AC = 720 lbf, BC = 0, BD = -625 lbf, BE = -312 lbf, CE = 720 lbf, DE = 400 lbf, DF = -625 lbf, EF = -312 lbf, EG = 720 lbf, FG = 0, FH = -937.2 lbf, GH = 720 lbf

trusses

S_Q34

A cantilever truss connects to a wall with pins at F and G, and F is 6.5 ft vertically above G. The top chord extends horizontally from F to D, from D to B and from B to A. Members FD, DB and BA are each 5 ft long. The diagonal bottom chord extends from G right and upwards towards A, including members GE where joint E is vertically below D, EC where joint C is vertically below B and CA. Other internal members are FE, DE, DC and BC. A force of 325 lbf is applied at A in a direction 100 degrees CW from the positive x-axis. Determine the forces in members DB, DC and EC. Indicate tension as positive and compression as negative.

DB = 682 lbf, DC = 0, EC = -805 lbf

trusses

S_Q35

A triangular Howe truss has overall dimensions of 36 ft wide and 9 ft high. The truss is pinned at A at the left and sits on a roller at G at the right. The horizontal bottom chord consists of members from left to right of AH, HJ, JK, KL, LM, MG and each of these members is 6 ft long. Joint B is located vertically above H. Joint C is located vertically above J. Joint D is located 9 ft vertically above K. Joint E is located vertically above L. Joint F is located vertically above M. Top chords extend right and upwards from A towards D, and left and upwards from G to D, and include members AB, BC, CD, DE, EF and FG. Internal members are HB, JB, JC, KC, KD, KE, LE, LF and MF. Vertical downwards forces applied are 250 lbf at C, 100 lbf at H and 350 lbf at J. Determine the forces in members DE, KE and KL. Indicate tension as positive and compression as negative.

DE = -484 lbf, KE = 0, KL = 433 lbf

trusses

S_Q36

A rectangular Pratt truss has overall dimensions of 16 m wide and 4.5 m high. The top and bottom chords are both horizontal. The truss is pinned at A at the left and sits on a roller at G at the right. The bottom chord consists of members from A to H, from H to J, from J to K and from K to G. Members AH, HJ, JK and KG are each 4 m long. Joint B is located 4.5 m vertically above A. The top chord consists of members B to C, C to D, D to E and E to F. Members BC, CD, DE and EF are each 4 m long. Vertical members are AB, HC, JD, KE and GF and are each 4.5 m long. Diagonal members are HB, JC, JE and KF. Vertical downwards loads are 300 N at H and 150 N at K. A horizontal load of 425 N is applied at F in the positive x-axis direction. Determine the forces in members DE, JE and JK. Indicate tension as positive and compression as negative.

DE = 12.5 N, JE = 210 N, JK = 273 N

trusses

S_Q37

A trussed frame is connected to a wall at A and B. B (0, 0m) is a roller joint. A (0, 6m) is a pin joint. Joint F is located at (6, 6m) and joint C is located at (10, 10m). Truss members include AB, members from A right and upwards towards C and members from B right and upwards towards C. Joint D is at the midpoint between A and C, and creates two equal length members AD and DC. Joint E is at the midpoint between B and F and creates two equal length members BE and EF. Member FC connects F to C. Internal members are AE, AF and FD. A cable connects to the wall 4 m vertically above A and runs through a frictionless pulley at C, and there is a mass suspended vertically from the cable below C. Determine the maximum mass that may be supported by the cable without exceeding 60 kN in tension or 85 kN in compression in any member of the frame.

6120 kg

trusses

S_Q38

An A-frame consists of three members. Member AC is pinned at A (0, 0m) and connected to member BC at C (2, 4m) with a pin joint. Member BC is supported by a roller at B (4, 0m). Member DF is 1.5m long and is connected to AC at D (1.5, 3m) and BC at E (2.5, 3m) with pin joints. Member DF cantilevers out from E a distance of 0.5m to free end F. A vertical downwards load of 110 N is applied at F. Determine the planar components of the forces acting on member AC at joints A, D and C.

Ax = 0 N, Ay = 27.5 N, Dx = 82.5 N, Dy = 55 N, Cx = -82.5 N, Cy = -82.5 N

frames

S_Q39

A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (8, 2m). The two members are pin connected at B (3, 4m). A vertical downwards force of 250 N is applied at B. Determine the scalar components of the reactions at pins A and C.

Ax = 144.2 N, Ay = 192.3 N, Cx = -144.2 N, Cy = 57.7 N

frames

S_Q40

A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 3ft). The two members are pin connected at B (5, 3ft). A horizontal force of 175 lbf in the positive x direction is applied at the midpoint of AB. Determine the scalar components of the reactions at pins A and C.

Ax = -118.8 lbf, Ay = -18.8 lbf, Cx = -56.3 lbf, Cy = 18.8 lbf

frames

S_Q41

A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0ft). The right member is pinned at C (8, 0ft). The two members are pin connected at B (4, 5ft). A horizontal force of 475 lbf in the negative x direction is applied at the midpoint of AB. A CCW moment of 800 lb.ft is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.

Ax = 436.3 lbf, Ay = 248.4 lbf, Cx = 38.8 lbf, Cy = -248.4 lbf

frames

S_Q42

A frame consists of two members AB and BC. The left member AB is pinned at A (0, 0m). The right member is pinned at C (10, -1m). The two members are pin connected at B (5, 3m). A horizontal force of 75 N in the negative x direction is applied at the midpoint of AB. A CW moment of 100 N.m is applied at the midpoint of BC. Determine the scalar components of the reactions at pins A and C.

Ax = 44.6 N, Ay = 4.29 N, Cx = 30.4 N, Cy = -4.29 N

frames

S_Q43

Given the function (y = -1.5.x^2 + 2.x + 2.5), determine the area under the curve from the y-axis to the point where the curve crosses the positive x-axis. The x- and y- coordinates are in inches.

5.03 in^2

centroids

S_Q44

Use integration with horizontal strips to determine the area of the spandrel bounded by the function (y = h - k.x^0.5) and the lines y = h and x = b, and also find the coordinates of its centroid. Note: b and h are constants. Give your answers for the area and the centroid in terms of b and h.

Area = (b.h / 1.5), centroid x coordinate = (1.5.b / 2.5), centroid y coordinate = (2.5.h / 4)

centroids

S_Q45

Determine the coordinates of the centroid of the shaded area bounded by the y-axis, a parabola passing through the origin (0, 0m) and point A, and a straight line passing through points A and B. Point A is located at (9, 3m) and B is located at (0, 9m).

(3.2, 3.7m)

centroids

S_Q46

A 16 ft long beam is supported by a pin at A located 1 ft from the left end, and a roller at B located 2 ft from the right end. A downwards vertical uniformly distributed load of 20 lbf/ft is applied between 7 ft from the left end and 4 ft from the right end. The beam weight is negligible. Determine the reactions at A and B.

Ax = 0 lbf, Ay = 34.6 lbf, By = 65.4 lbf

beam_reactions

S_Q47

A 15 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (15, 0m). Point C is located at (1, 0m) and D is located at (12, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 45 kN/m at C to 60 kN/m at D. The beam self weight is 15 kN/m. Determine the reactions at A and B.

Ax = 0 kN, Ay = 429.7 kN, B = 372.8 kN

beam_reactions

S_Q48

A 20 m long beam is supported by a pin at the left end A (0, 0m) and a roller at the right end B (20, 0m). Point C is located at (6, 0m), D is located at (12, 0m) and E is located at (19, 0m). A downwards vertical distributed load is applied to the beam and varies linearly between 0 kN/m at C to 60 kN/m at D, then remains constant at 60 kN/m to E. The beam weight is negligible. Determine the reactions at A and B.

Ax = 0 kN, Ay = 184.5 kN, By = 415.5 kN

beam_reactions

S_Q49

An 18 m long beam supports a vertical downwards load that varies uniformly from 50 N/m at the left end to 150 N/m at the right end. The beam is pinned at A (0, 0m) and sits on a roller at B (10, 0m). Determine the internal shear and bending moment at a section passing through point C (6, 0m).

V = -490 N, M = -1640 N.m

beam_internal_forces

S_Q50

A beam BC is 4 ft long and is pinned at the right end C (4, 0ft). At the left end B (0, 0ft) the beam is pin connected to member AB. A is located to the left and upwards from B, creating an angle of 25 degrees with the x-axis. Beam BC supports a vertical downwards uniformly distributed load of 15 lbf/ft. Determine the internal forces at point D located at the midlength of BC.

V = 0 lbf, N = 64.3 lbf, M = 30 lbf.ft

beam_internal_forces

S_Q51

A 12 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 500 kN at 8 m from A, and 650 kN at 10 m from A. Find the shear and bending moment equations as function of x for the three ranges: 0m < x < 8m, 8m < x < 10m, 10m < x < 12m.

(0m < x < 8m: V = 275 kN, M = 275.x kN.m), (8m < x < 10m: V = -225 kN, M = 275.x - 500.(x - 8)), (10m < x < 12m: V = -875 kN, M = 275.x - 500.(x - 8) - 650.(x - 10))

beam_internal_forces

S_Q52

A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards forces are 165 kN at 2 m from A, and 15 kN at 9 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

V (m, kN): (0, 0), (0, 157), (2, 157), (2, -8.3), (9, -8.3), (9, -23.3), (20, -23.3), (20, 0) M (m, kN.m): (0, 0), (2, 314), (9, 256), (0, 0)

beam_internal_forces

S_Q53

A 6 ft long cantilever beam is fixed at the left end A. Vertical downwards forces are 140 lbf at 3 m from A, and 240 lbf at 6 m from A. A CCW moment of 2340 lbf.ft is applied at 4 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

V (ft, lbf): (0, 0), (0, 380), (3, 380), (3, 240), (6, 240), (6, 0) M (ft, lbf.ft): (0, 0), (0, 480), (3, 1620), (4, 1860), (4, -480), (6, 0)

beam_internal_forces

S_Q54

A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards uniformly distributed loads are: 200 kN/m from A to 9 m from A, and 180 kN/m from 15 m from A to 20 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

V (m, kN): (0, 0), (0, 1508), (9, -293), (15, -293), (20, -1192), (20, 0) M (m, kN.m): (0, 0), (7.5, 5681), (9, 5468), (15, 3713), (20, 0)

beam_internal_forces

S_Q55

A 20 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards loads are 400 kN at 2 m from A and a uniformly distributed load of 300 kN/m from 12 m from A to 17 m from A. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

V (m, kN): (0, 0), (0, 773), (2, 773), (2, 373), (12, 373), (17, -1128), (20, -1128), (20, 0) M (m, kN.m): (0, 0), (2, 1545), (12, 5270), (13.2, 5501), (17, 3383), (20, 0)

beam_internal_forces

S_Q56

A 18 m long beam is pinned at the left end A and has a roller at the right end B. Vertical downwards linearly varying distributed loads are 0 kN/m at A varying linearly to 25 kN/m at 9 m from A, then varying linearly to zero at B. Find the shear and bending moment diagrams, expressed as a series of points at critical locations (x distance from the left end, value).

V (m, kN): (0, 0), (0, 113), (9, 0), (18, -113), (18, 0) M (m, kN.m): (0, 0), (9, 675), (18, 0)

beam_internal_forces

S_Q57

A 20 m long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards symmetric distributed parabolic loading w(x), given by the equation w(x) = -2.x^2 + 40.x kN/m. Find the maximum values of shear and moment and their locations in the beam.

V = 1333 kN at x = 0 m and x = 20 m, M = 8333 kN.m at x = 10 m

beam_internal_forces

S_Q58

A 20 ft long beam is pinned at the left end A and has a roller at the right end B. The beam is subjected to a vertical downwards distributed parabolic loading w(x), given by the equation w(x) = - 0.007.x^3 - 0.03.x^2 + 3.x lbf/ft. Find the bending moment function in terms of x.

M = (x^4)/400 + (x^5)/2857 - (x^3)/2 + 124.x

beam_internal_forces

S_Q59

A 220 kN horizontal force P is applied to a box with weight 200kN resting on a surface with an incline of 20 degrees from the horizontal. The line of action of P passes through the center of gravity of the box. The box is 4 m wide and 3 m tall, and the static and kinetic coefficients of friction between the box and the surface are 0.7 and 0.57, respectively. Determine the magnitude of the friction force acting on the box.

138.3 kN

friction

S_Q60

A block weighs 500 N and is resting on an incline of 30 degrees from the horizontal. The coefficient of friction between the block and the inclined surface is 0.4 . A force P is applied to the box. The line of action of P passes through the center of gravity of the box and is inclined 65 degrees from the horizontal. Determine the maximum value of P before the block starts to move up the incline (i.e. impending upwards motion), and the minimum value of P required to prevent the block from moving down the incline (i.e. impending downward motion).

Maximum = 404 N, minimum = 130 N

friction

S_Q61

A block with mass 10 kg rests on a horizontal surface. A block with mass 35 kg rests on top of the 10 kg block, and is restrained from moving horizontally. The coefficient of friction between all surfaces is 0.2. A horizontal force P is applied to the 10 kg block with a line of action through its center of gravity. Determine the smallest force P required to start the 10 kg block moving.

157 N

friction

S_Q62

A 2D object is defined by 5 vertices in the x-y space with straight lines between them. The vertices are: (-1, 0cm), (-7, 0cm), (0, 7cm), (1, 7cm), (1, 6cm). Calculate the moment of inertia and radius of gyration with respect to the x-axis.

Ix = 278.4 cm^4, kx = 3.3 cm

moments_of_inertia

S_Q63

A square has vertices in the x-y space of (0, 6in), (-4, 6in), (-4, 10in) and (0, 10in). Calculate the moments of inertia of the object about both the x- and y-axes.

Ix = 1045 in^4, Iy = 85.3 in^4

moments_of_inertia

S_Q64

A 2D object is defined by 6 vertices in the x-y space with straight lines between them. The vertices are: (0,0ft), (0, 9ft), (1, 9ft), (1, 3ft), (4, 3ft), (4, 0ft). Calculate the moments of inertia of the object about both the x- and y-axes.

Ix = 270 ft^4, Iy = 66 ft^4

moments_of_inertia

S_Q65

A circle has a diameter of 8 in and is positioned in the x-y space such that its centre is located at (5, 4in). Determine the moment of inertia and radius of gyration about the y-axis.

Iy = 1458 in^4, ky = 5.4 in

moments_of_inertia

S_Q66

A rectangle has vertices in the x-y space of (0, 0cm), (0, 8cm), (10, 8cm), (10, 0cm). Inside this rectangle is a rectangular void with vertices of (1, 6cm), (1, 7cm), (9, 7cm), (9, 6cm). Determine the moment of inertia of the composite shape about the horizontal neutral axis of the composite shape.

370.4 cm^4

moments_of_inertia

S_Q67

A 2D object is defined by 8 vertices in the x-y space with straight lines between them. The vertices are: (0, 2ft), (7, 2ft), (7, 8ft), (8, 8ft), (8, -8ft), (7, -8ft), (7, -2ft), (0, -2ft). Calculate the moment of inertia of the object about the vertical neutral axis of the object.

278.6 ft^4

moments_of_inertia

S_Q68

Two steel channel sections are placed back to back to each other with a distance of 120 mm separating them. They are placed upright such that their large dimension (depth) is in line with the vertical y-axis. A built up beam is constructed by welding two plates to these channels, symmetrically placed, one on the top and one on the bottom. The plates are both 300 mm wide and 35 mm thick. The channel sections have properties: Area = 4740 mm^2, depth = 305 mm, flange width = 77.5 mm, Ixx = 59900000 mm^4, Iyy = 1850000 mm^4, and the horizontal distance to the section centroid is 17.1 mm. Determine the moment of inertia and radius of gyration of the built up beam with respect to the x- and y-axes.

Ix = 728800000 mm^4, kx = 154.6 mm, Iy = 217600000 mm^4, ky = 84.5 mm

moments_of_inertia

M_Q1

A pedestal has a triangular cross section. If it is subjected to a compressive force P of 500 lb, specify the x and y coordinates for the location of point P(x, y), where the load must be applied on the cross section, so that the average normal stress is uniform. Compute the stress. Vertices of the triangular cross section: Vertex 1: (0 in., 0 in.), Vertex 2: (0 in., 9 in.), Vertex 3: (12 in., 3 in.)

x = 4in, y = 4in, stress = 9.26psi

stress

M_Q2

A metal rod specimen failed in a tension test at an angle of 52 degrees (to the longitudinal axis) when the axial load was 19.80 kip. If the diameter of the specimen is 0.5 in., determine the average normal and average shear stress acting on the area of the inclined failure plane. Also, what is the average normal stress acting on the cross section when failure occurs?

Average normal stress on cross section: 100.9 ksi, Average normal stress on inclined failure plane: 62.7 ksi, Average shear stress on inclined failure plane: 48.9 ksi

stress

M_Q3

Rods AB and BC have diameters of 4 mm and 6 mm, respectively. The geometry: Point A: Pin support on left, Point B: Ring connection where both rods meet, Point C: Pin support on right, Rod AB: Horizontal, connects A to B, Rod BC: Inclined at theta = 60 degrees above horizontal, connects B to C. If a vertical downwards load of 8 kN is applied to the ring at B, determine the average normal stress in each rod if theta = 60 degrees.

AB = 368MPa, BC = 327MPa

stress

M_Q4

The bars of the truss each have a cross-sectional area of 1.25 in squared. Geometry: Joint A: (0 ft, 0 ft), Joint B: (4 ft, 3 ft), Joint C: (8 ft, 3 ft), Joint E: (4 ft, 0 ft), Joint D: (8 ft, 0 ft), Joint C: Pinned to wall, Joint D: Pinned to wall. Members: AB, AE, ED, EB, BC, BD. Force P = 8 kip acting downward at joint A, Force 0.75P = 6 kip acting downward at joint E. Determine the average normal stress in each member due to the loading P = 8 kip. State whether the stress is tensile (T) or compressive (C).

AB=10.7ksi T, AE=8.53ksi C, ED=8.53ksi C, EB=4.8ksi T, BC=23.5ksi T, BD=18.7ksi C

stress

M_Q5

A beam and hanger assembly is used to support a distributed loading of w = 0.8 kip/ft. Beam: Pinned to the wall at C (0, 0ft) and extends horizontal for 6ft. Hanger rod AB: Pinned to the wall at B (0, 3ft) and connected to the beam with a bolt at A (4, 0ft). Uniformly distributed load w = 0.8 kip/ft acting downward over the 6 ft beam length. Determine the average shear stress in the 0.40 in. diameter bolt at A and the average tensile stress in rod AB, which has a diameter of 0.5 in. If the yield shear stress for the bolt is 25 ksi, and the yield tensile stress for the rod is 38 ksi, determine the factor of safety with respect to yielding in each case.

Bolt: shear stress=23.9ksi, FOS=1.05. Rod: stress=30.6ksi, FOS=1.24

stress

M_Q6

A rod BC is made of steel having an allowable tensile stress of 155 MPa and supports the left end of beam AB. Joint A: (4.5, 0 m) right end, pinned support, Joint B: (0, 0 m) left end, connection point on beam for rod BC, Joint C: Above point B, upper connection of rod to support. Distributed downwards load increases from zero at B to w = 15 kN/m at (3, 0m) then decreases to zero at A. Determine its smallest diameter so that it can support the load.

11.1 mm

stress

M_Q7

A thin strip of rubber has an unstretched length of 15 in. If it is stretched around a pipe having an outer diameter of 5 in., determine the average normal strain in the strip.

0.0472 in/in

strain

M_Q8

A bar CA has length L m and is supported at its right end A by a wire AB. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at B which is L m vertically above C. When the wire AB is unstretched it makes an angle of theta = 45 degrees below the horizontal at B. If a load is applied to the bar AC which causes the angle to change to theta = 47 degrees, determine the normal strain in the wire.

0.0343 m/m

strain

M_Q9

Two wires are connected together at A with a ring. Wire CA: Connects C to A, length = 300 mm, oriented at 30 degrees above horizontal (measured from horizontal at A). Wire BA: Connects B to A, length = 300 mm, oriented at 30 degrees below horizontal (measured from horizontal at A). B and C are connections to a wall and C is vertically above B. A horizontal force P is applied at A. If the force P causes point A to be displaced horizontally 2 mm away from the wall, determine the normal strain developed in each wire.

Strain in both wires = 0.0058 mm/mm

strain

M_Q10

A right-angled triangular plate ABC has vertices: A (0, 0mm), B (0, 800mm), C (800, 0mm). Edge BC is fixed, and its apex A is given a displacement of 5 mm in the direction perpendicular to edge BC. Determine the shear strain at A.

0.0088 rad

strain

M_Q11

A rectangular plastic block is glued at its top and bottom to rigid plates. Point A: Bottom-left corner (0, 0). Point D: Bottom-right corner. Point B: Top-left corner (0, 2in). Point C: Top-right corner. A horizontal force P is applied to the top plate in the positive x direction, and causes the material to deform so that its sides are described by the equation y = 3.56x^(1/4). Determine the shear strain in the material at its corners A and B.

A = 0, B = 0.199 rad

strain

M_Q12

The stress - strain diagram for an elastic fibre has coordinates: (0 in/in, 0 psi), (2 in/in, 11 psi), (2.25 in/in, 55 psi). Determine the modulus of elasticity and estimate the modulus of toughness and modulus of resilience.

modulus of elasticity = 5.5psi, modulus of toughness = 19.25psi, modulus of resilience = 11 psi

mechanical_properties

M_Q13

The stress-strain (s-e) diagram for a bone is described by the equation e = 0.45(10^-6)s + 0.36(10^-12)s^3, where s is in kPa. Determine the modulus of toughness and the amount of elongation of a 200 mm long region just before it fractures if failure occurs at e = 0.12 mm/mm.

613MPa, 24mm

mechanical_properties

M_Q14

A specimen is originally 1 ft long, has a diameter of 0.5 in., and is subjected to a force of 500 lb. When the force is increased to 1800 lb, the specimen elongates 0.9 in. Determine the modulus of elasticity for the material if it remains elastic.

88.3ksi

mechanical_properties

M_Q15

A bar BC has length 10 ft and is supported at its right end B by a wire AB with diameter 0.2 in made of A-36 steel. At the left end the bar is attached to a wall by a pin at C. The wire is attached to the wall at A which is vertically above C. Wire AB makes an angle of 30 degrees above the horizontal at B. A downwards uniformly distributed load is applied to the full length of the rod BC. Determine the magnitude of the distributed load if the end B is displaced by 0.75 in downwards.

0.246kip/ft

mechanical_properties

M_Q16

An aluminum block has a rectangular cross section and is subjected to an axial compressive force of 8 kip. If the 1.5 in. side changed its length to 1.500132 in., determine Poisson's ratio and the new length of the 2 in. side. The modulus of elasticity is 10(10^3) ksi.

Poissons ratio = 0.33, side length = 2.00018 in

mechanical_properties

M_Q17

An 8 mm diameter brass rod has a modulus of elasticity of 100 GPa. If it is 3 m long and subjected to an axial load of 2 kN, determine its elongation. What is its elongation under the same load if its diameter is 6 mm?

For 8mm diameter = 1.19mm, for 6mm diameter = 2.12m

mechanical_properties

M_Q18

An assembly consists of a steel rod CB of length 3 m and an aluminum rod BA of length 2 m, each having a diameter of 12 mm. The rod is fixed at C and free at A and is subjected to axial loadings at A and at B: 6 kN acting at B in the negative x-direction, 18 kN acting at end A in the positive x-direction. Determine the displacement at the free end A. Modulus of elasticity (steel) is 200 GPa and Modulus of elasticity (aluminum) is 70 GPa.

6.14mm

axial_load

M_Q19

A truss is made of three A-36 steel members AB, BC and CA, each having a cross-sectional area of 400 mm squared. Joint A: (0 m, 0 m), Joint C: (1.4 m, 0 m), Joint B: (0.8 m, 0.8 m). Joint A is pinned and joint C is a roller. Forces: Force P acting downward at joint B, force 5 kN acting horizontally to the right at joint B. Determine the magnitude P required to displace the roller to the right 0.2 mm.

21.7kN

axial_load

M_Q20

A linkage is made of three pin-connected A-36 steel members DA, CA and AB, each having a cross-sectional area of 0.730 in squared. Joint D: (0 ft, 0 ft), Joint C: (6 ft, 0 ft), Joint A: (3 ft, -4 ft), Joint B: (3 ft, -10 ft). Joints C and D are pinned. A vertical force of P = 50 kip is applied to the end B of member AB. Determine the vertical displacement of point B.

0.281in

axial_load

M_Q21

A 500 mm long steel pipe is filled with concrete and subjected to a compressive force of 80 kN. The steel pipe outer diameter is 80 mm and inner diameter is 70 mm. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (concrete) is 24 GPa. Determine the stress in the concrete and the steel due to this loading.

steel stress = 48.8MPa, concrete stress = 5.85MPa

axial_load

M_Q22

A 10 mm diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. The assembly is subjected to a compressive load P. The yield stress for the steel is 640 MPa and the bronze is 520 MPa. Modulus of elasticity (steel) is 200 GPa, Modulus of elasticity (bronze) is 100 GPa. Determine the magnitude of the largest elastic load P that can be applied to the assembly.

126kN

axial_load

M_Q23

A tapered member is fixed connected at its ends A and B and is subjected to a downwards load P. The member is 2 in thick and 60 in long. The height tapers linearly from 6 in at A to 3 in at B. Determine the distance from A of the load position and its greatest magnitude, if the allowable normal stress for the material is 4 ksi.

28.9in, 60.4kip

axial_load

M_Q24

A post is made from 6061-T6 aluminum and has a diameter of 50 mm and length 0.5 m. It is fixed supported at A and B and at its center C there is a coiled spring with stiffness 200 MN/m attached to a rigid collar. If the spring is originally uncompressed, determine the compression of the spring when a load of P = 50 kN is applied to the collar at C.

0.039mm

axial_load

M_Q25

A thermo gate consists of two 6061-T6 aluminum plates that have a width of 15 mm and are fixed supported at their ends. One plate is 600 mm length, the other is 400 mm length, and both plates have thickness 10 mm. If the gap between them is 1.5 mm when the temperature is 25 degrees C, determine the temperature required to just close the gap. The coefficient of thermal expansion is 24(10^-6)/degrees C.

87.5 degrees C

axial_load

M_Q26

A member is to be made from a steel plate that is 0.25 in thick. If a 1 in hole is drilled through its center, determine the approximate width of the plate so that it can support an axial force of 3350 lb. The allowable stress is 22 ksi.

2.49in

axial_load

M_Q27

A 0.25 in diameter steel rivet having a temperature of 1500 degrees F is secured between two plates such that at this temperature it is 2 in long and exerts a clamping force of 250 lb between the plates. Determine the approximate clamping force between the plates when the rivet cools to 70 degrees F. Assume that the heads of the rivet and the plates are rigid and use a coefficient of thermal expansion of 8(10^-6)/degrees F, and modulus of elasticity of 29(10^3) ksi.

16.5kip

axial_load

M_Q28

A copper pipe has outer diameter 40 mm and inner diameter 37 mm. It is fixed at a wall at A and the axis is the longitudinal axis, positive away from the wall. Three torques are applied, sequentially moving away from the wall: -30 Nm, 20 Nm then -80 Nm. Determine the maximum shear stress in the pipe.

26.7MPa

torsion

M_Q29

A steel tube having an outer diameter of 2.5 in. is used to transmit 350 hp when turning at 27 rev/min. Determine the inner diameter of the tube if the allowable shear stress is 10 ksi.

2.48in

torsion

M_Q30

Two steel 20 mm diameter steel shafts are connected using a brass coupling. If the yield point for the steel is 100 MPa and for the brass is 250 MPa, determine the required outer diameter of the coupling so that the steel and brass begin to yield at the same time when the assembly is subjected to a torque T. Assume that the coupling has an inner diameter of 20 mm.

21.9mm

torsion

M_Q31

The drive shaft of an automobile is made of a steel having an allowable shear stress of 8 ksi. If the outer diameter of the shaft is 2.5 in. and the engine delivers 200 hp to the shaft when it is turning at 1140 rev/min, determine the minimum required thickness of the shaft's wall.

0.174in

torsion

M_Q32

A 0.75 in. diameter shaft for an electric motor develops 0.5 hp and runs at 1740 rev/min. Determine the torque produced and compute the maximum shear stress in the shaft. The shaft is 6 in long and supported by ball bearings at its ends.

1.51lbft, 219psi

torsion