1 ME2100 Homework 1. Determine the gravitational force exerted by a. The moon on the earth, using the following data. Make sure that you show your work Mass of moon : m m = 7.35*10 22 kg Mass of earth : m e = 5.976*10 24 kg Radius of moon : r m = 1.738*10 6 m Radius of earth : r e = 6.731*10 6 m Distance between moon & earth : 3.844*10 8 m b. The sun on the earth, using the following data. Make sure that you show your work Mass of sun : m s = 1.990*10 30 kg Mass of earth : m e = 5.976*10 24 kg Radius of sun : r s = 6.960*10 8 m Radius of earth : r e = 6.731*10 6 m Distance between sun & earth : 1.496*10 11 m Goal: Given: Assume : Draw : Soln:
2 2. Determine the force of gravity acting on a satellite when it is in orbit 20.2 * 10 6 m above the surface of the earth. Its weight when on the surface of the earth is 8450 N. Use the data given in previous problem as needed. Goal: Given: Assume : Draw : Soln:
3 3. At what distance, in kilometers, from the surface of the earth on a line from center to center would the gravitational force of the earth on a body be exactly balanced by the gravitational force of the moon on the body? Use the data in problem as needed. (As a challenge first work out the problem using only symbols and then plug in the numbers at the end) Goal: Given: Assume : Draw : Soln:
4 4. Two forces are applied at point B of beam AB. Determine the magnitude and direction of the resultant using trigonometry. Challenge : use variables like P,Q for the forces and θ,φ for the angles and derive the solution for a whole class of problems. NOTE : the resultant will not be along line BC. GOAL : Given : P = 2kN, Q = 3kN, θ = 40,φ = 60 Find : R the resultant Solution : Draw the parallelogram with sides along P and Q. P C Hint : Use Sine and cosine law with parallelogram law of addition Q
5 5. The post is to be pulled out of the ground using two ropes A and B. Rope A is subjected to a force (tension) of 600 lb and is T lb directed at a known angle φ o from the vertical. If the resultant force acting on the post is to be always 1200 lb, vertically upward, B determine the force T in rope B and the corresponding angle θ o. Challenge : plot the value of T and θ as a function of φ. For students having difficulty use φ = 30 o θ φ A 600 lb
6 6. Using the figure shown a. Determine the magnitude and its direction of the resultant F r = F 1 + F 2 measured from the positive u- axis. b. Determine the components along the u and v axes of F 1 c. Determine the components along the u and v axes of F 2 F 1 = 150N
7 7. Three cables are attached to a tree as shown in figure. a. Represent the vectors AB, AC, and AD. b. Find the unit vector along u AB, u AC, and u AD c. Find the direction angles of each of the above vectors.
8 8. Three cables are attached to a block as shown in figure. a. Find the unit vector along u OA, u OB, and u OC b. Represent the vectors F 1, F 2, and F 3,. c. Find the direction angles of OA, OB, and OC
9 9. A cable is attached to B to the right angle pipe OAB in figure. The tension in the cable is 750 lb and a moment/torque of 10 lb.ft is applied at B as shown along line of action AB. a. Represent the tension in the cable as a vector F BC b. Find the unit vector u BC c. Find the direction angles of the vector F BC d. Represent the moment as a vector. M=10 lb.ft
10 10. Two forces F 1 and F 2 are applied to an eyelet as shown in figure. Determine the resultant F R = F 1 + F 2, and write in vector notation. Find the unit vector along the resultant and then use it to obtain the direction angles of the resultant.
11 11. The flex-headed ratchet wrench is subjected to a force P lb, applied perpendicular to the handle as shown. Find the moment or torque this imparts along the vertical axis of the bolt at A assuming θ as a given angle. θ
12 12. Determine a. Moment M 1 of F 1 about moment center A b. Moment M 2 of F 2 about moment center A c. Sum of M 1 and M 2 which we will call M 3 d. If F 1 and F 2 are parallel and opposite to each other will your answer in part c. Why OR why not? e. Find the scalar component of M 3 about the axis of the shaft AB. f. Would your answer in part e be different if M 1 and M 2 had been calculated for a moment center at B? (first reason your answer and than confirm by computing)
13 13. A 2 kn force acts on one end of the curved rod. Section AB of the rod lies in the xy plane and section BC lies in the zy plane. Determine the moment about moment center A and about moment center B Now find the magnitude about the line AB and then about line BC. Reason your answers. (For hints see on page 182)
14 14. For the beam pinned at A and acted by T BC, F B and M C, replace the loads by equivalent loading at A. Present your answers as vectors and also sketch the equivalent force and moment.
15 15. Find equivalent loading at A and represent as a vector and also as a diagram.
16 16. The wing of the jet aircraft is subjected to a thrust of T= 8,000N from its engine, and the moment of 6000 N*m due to the rotation and the resultant lift force L= 45,000N (This is normally a distributed force, but has been reduced to a concentrated force as shown). If the mass of the wing is 2,100 Kg and the mass center is at G, determine the components of reaction where the wing is fixed to the fuselage at A. GOAL: Given : Assume : Soln : Draw Free Body diagram The body of the aircraft provides a fixed support for the wing (meaning 6 components of reaction, 2 each in x,y,z co-ordinates) Write equations of equilibrium six equations and six unknowns M = 6kN-m
17 17. The pipe assembly supports the vertical loads shown. Determine the components of reaction at the ball and socket joint A and the tension in the supporting cables BC and BD. GOAL : Given : 100N/m Assume: Soln : Draw FBD. Reduce distributed load to conc. load and find its line of action (i.e. 3m*100N/m=300N force acting at the mid-point Already shown for you in FBD) A ball and socket joint does NOT allow translation in all directions, but lets free rotation in each direction. 300N 1.5m
18 18. Determine the reactions at the fixed wall A. The 150 N force is parallel to the z axis, the 200N force is parallel to the y axis and the 300N force is parallel to the x axis. The moment of a couple has a magnitude of 100 N-m and has direction angles of θ x =67.4ºand θ z =39.8º. GOAL : 100N-m 300N Given : Assume: Soln :
19 19. A 200-N force is applied to the handle of the hoist in the direction shown. The bearing at A is a thrust bearing, and the bearing at B is a journal bearing. If the hoist is in equilibrium, what forces act on the shaft at A? What forces acts on the shaft at B? What is the maximum mass m in kilogram that can be lifted? GOAL : Given : Find : m Soln : Draw FBD. A journal bearing does NOT permit translation and rotation in two axes. A thrust bearing does NOT permit translation in three directions and does NOT permit rotation in two axes. After showing the above we will have 10 unknowns which includes the mass. Now make the assumption that the bearings are perfectly aligned this will take care of 4 moment reactions, two each at the two bearings and leave you with six unknowns.
20 20. A shaft is loaded through a pulley and a lever that are fixed to the shown shown in figure. Friction between the belt and the pulley prevents the belt from slipping. The support at A is a journal bearing, and the support at B is thrust bearing. Determine a. the force P required for equilibrium b. the loads acting on the shaft at supports at A and at B. c. find the direction angles of the reaction at A and B d. It is known that the bearing at A fails if the force on it exceeds 1000N and bearing at B fails if force on it exceeds 400 N. Discuss the failure of this mechanism based on your answers in part b.
21 21. Determine the location of the centroid of the beam s crosssectional area. Neglect the size of the corner welds at A and B for the calculation. GOAL: Given : Assume:: Draw: Solution:
22 22. The gravity wall is made of concrete. Determine the location of the center of gravity G for the wall. GOAL: Given : Assume:: Draw: Solution :
23 23. Determine the location of the centroid C of the area. GOAL: Given : Assume:: Draw: Solution :
24 24. Determine the location of the center of gravity of the three-wheeler. The location of the center of gravity of each component and its weight are tabulated in the figure. If the three-wheeler is symmetrical with respect to the x-y plane, determine the normal reactions each of its wheels exerts on the ground. GOAL: Given : Assume:: Draw: Solution :
25 25. Replace the distributed loading by an equivalent resultant force and specify where its line of action intersects a. member AB, measured from A. b. member BC, measured from C. GOAL: Given : Assume:: Draw: Solution :
26 26. The beam is subjected to the distributed loading. Determine the length b of the uniform load and its position a on beam such that the resultant force and couple moment acting on the beam are zero. GOAL: Given : Assume:: Draw: Solution :
27 27. Replace the distributed loading by an equivalent resultant force, and specify its location on the beam, measured from the pin at C. GOAL: Given : Assume:: Draw: Solution :
28 28. The truss is made from five members, each having a length of 4m and a mass of 7kg/m. If the mass of the gusset plates at the joints and the thickness of the members can be neglected, determine the distance d to where the hoisting cable must be attached, so that the truss does not tip (not rotate) when it is lifted. GOAL: Given : Assume:: Draw: Solution :
29 29. The wall crane supports a load of 1000lb as shown. Determine the horizontal and vertical components of reaction at the pins A and D. Also what is the force in the cable at the winch? The jib ABC has a weight of 100lb and member BD has a weight of 40lb. Each member is uniform and has a center of gravity at its center. Outline : First consider the pulley at E as shown below and find the tension in the cable. Use this tension and find the reactions at C as shown in the diagram below. Equivalent system of forces Similarly reduce the pulley at B into an equivalent system of forces. Now consider the structure with pulleys removed and equivalent forces shown for their point of attachment. (Remember Newton s third law). Do not forget the weight of the members. Now draw the FBD of structure which will now consist of ABC and BD. Write equilibrium equations Next break it into members ABC and BD. Again do not forget the weight of the members 4 3 P=1000lb T E T T 3 P=1000lb 4 Pulley E R CX T T 3 4 Pulley C R CY T R B1 Pulley B T R B2
30 30. Determine the reactions at the supports of the frame shown. The pin attached to member BCD, passes through a smooth slot in member AB. (Hint : Frame has two members AB and BD. Draw their FBD and apply equilibrium conditions. You will need to reduce distributed load to concentrated load) Outline Determine support reactions (FBD of whole structure, Eqm equations) Breakup structure and draw FBD, Eqm equations. Do not forget that pin attached to BCD behaves like a roller reactions are normal to surface of contact. Do not forget external moment at A when drawing FBD of AB or FBD of structure. 100 lb-ft
31 31. The two-bar mechanism consists of a lever arm AB and smooth link CD, which has a fixed collar at its end C and a roller at the other end D. a) Determine the force P needed to hold the lever in the position θ. The spring has a stiffness k and unstretched length 2L. Assume that the roller contacts either the top or bottom portion of the horizontal guide at D. b) Using any software of your choice (Matlab, Mathcad), draw a graph of θ vs P. (θ = 30 to 60). From the graph determine the maximum P required and at what angle does this occur. For part b, assume L = 1m, k = 100N/m. Challenge : Assume that P is acting at an angle β from the horizontal measured in clockwise direction instead of as shown in figure.
32 32. Determine the force in each member of the truss. First solve using P 1 and P 2 as the forces and the given length. Then substitute values a) P 1 = 240lb, P 2 = 100lb. state if the members are in tension OR compression. b) Determine the largest permissible load P 2 if P 1 = 0lb. No member should exceed 500lb in tension and 350 lb in compression.
33 33. Determine the force in each member of the truss in terms of the load P and state if the members are in tension OR compression. Challenge Use the result to solve P a. Members AB and BC can each support a maximum compressive force of 800lb, and members AD, DC, and BD can support a maximum tensile force of 1500lb. If a = 10ft, determine the greatest load P the truss can support. b. Members AB and BC can each support a maximum compressive force of 800lb, and members AD, DC, and BD can support a maximum tensile force of 2000lb. If a = 6ft, determine the greatest load P the truss can support.
34 34. Determine the force in a) members GF, GD and CD b) members BG, BC, and HG using the method of sections. c) using method of joints now solve for forces in AB, AH, BH, CG, DF, FE, DE. In all cases state if the members are in tension OR compression.
35 35. Draw the axial force, shear force and bending moment for the structure shown. OUTLINE : First obtain the support reactions R Ax, R Ay and R By by drawing R By Fx = 0 R Ax = 0N r M A = i 2000 j i = Fy = i 4000 j + + 8i R By j 2000 j i 2000 j i the FBD of whole structure and using equilibrium equations j i 4000 j + Now recognize that we have seven regions in the beam so you will have to draw seven FBD and write 7*3=21 equilibrium equations to obtain axial force, shear force and bending moments.
36 36. Draw the axial force, shear force and bending moment diagrams. (Even though not asked you will have to find the support reactions before proceeding) OUTLINE : First obtain the support reactions R Ax, R Ay and R By by drawing the FBD of whole structure and using equilibrium equations. Remember to reduce distributed load to concentrated load which will be area under curve area of R By Fx = 0 R Ax = 30 lb r M A = 0 6i 225 j + 9i R By j 200 k = Fy = 0 triangle. 225lb acting at 6ft from A. Now realize that the beam has two regions. So you will write 3*2 = 6 equations to obtain the axial force, shear force and bending moment. Be sure to consider the area of triangle as we did in ICE lb
37 37. Draw the shear force and bending moment OUTLINE : Find support reactions. Identify regions we did this in class for this problem. Write eqm. equations for each section. (There are five regions in this beam most often people will make a mistake of not identifying point E where a concentrated moment is applied). E
38 700lb 38. The beam consists of two segments pinconnected at B. Draw the axial force, shear force and bending moment diagrams for the E D 300lb beam. Assume that the moment is applied at point E for the analysis. (HINT : We have solved 300lb*ft a similar problem in class) CLUE : once you A find the reaction in pin B, you can actually look at 8ft 4ft the two members independently for the axial force, shear force and bending moment diagrams. 200lb/ft B 6ft C
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