I certify that I have not given unauthorized aid nor have I received aid in the completion of this exam.

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1 ME 270 Fall 2012 Fnal Exam Please revew the followng statement: I certfy that I have not gven unauthorzed ad nor have I receved ad n the completon of ths exam. Sgnature: INSTRUCTIONS Begn each problem n the space provded on the examnaton sheets. If addtonal space s requred, use the whte lned paper provded to you. Work on one sde of each sheet only, wth only one problem on a sheet. Each problem s worth 20 ponts. Please remember that for you to obtan maxmum credt for a problem, t must be clearly presented,.e. The coordnate system must be clearly dentfed. Where approprate, free body dagrams must be drawn. These should be drawn separately from the gven fgures. Unts must be clearly stated as part of the answer. You must carefully delneate vector and scalar quanttes. If the soluton does not follow a logcal thought process, t wll be assumed n error. When handng n the test, please make sure that all sheets are n the correct sequental order and make sure that your name s at the top of every page that you wsh to have graded. Instructor s Name and Secton: Sectons: J. Slvers 11:30am-12:20pm E. Nauman 8:30-9:20am J. Jones 9:30-10:20am M. Murphy 9:00-10:15am K.M. L 2:30-3:20pm K.M. L 4:30-5:20pm Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Total

2 ME 270 Fall 2012 PROBLEM 1 (20 ponts) Prob. 1 questons are all or nothng. PROBLEM 1A. (5 ponts) FIND: In your own words, state each of Newton s three laws of moton. Be sure to wrte legbly. Unreadable defntons wll be marked wrong. 1 st Law = 2 nd Law = 3 rd Law = PROBLEM 1B. (5 ponts) FIND: Two force vectors are shown to the rght. γ defnes the angle between F 1 and the x-y plane; φ les n the x-y plane; and F 2 les n the x-z plane. Wrte F 1 and F 2 n vector form. Values for F 1, F 2 and the angles are gven to the rght. F 1 F N 300 N F 1 (3 pts) F 2 (2 pts)

3 ME 270 Fall 2012 PROBLEM 1C. (5 ponts) FIND: The force P = 120 lb s appled to the L-shaped bar ABC and acts n the drecton of the lne from C to D. Wrte P n vector form. Determne the moment about pont A due to the force P; express your answer n vector form. P M A (2 pts) (3 pts) PROBLEM 1D. (5 ponts) FIND: A cantlevered beam s subjected to the dstrbuted load shown. Determne the equvalent force and ts locaton, measured from the wall. Gve your answers n terms of A and/or L. F eq (2 pts) x (measured from wall on left) (3 pts)

4 ME 270 Fall 2012 PROBLEM 2. (20 ponts) A mass-less boom s used to support a 180-lb load appled at pont D. The boom s attached to the wall at pont O wth a ball and socket and supported by cables at pont A. Please place your answers to the followng questons n the boxes provded. You wll be asked to: a) Complete the free-body for the boom on the fgure provded. (4 ponts) b) Express the tenson n cables n terms of ther known unt vectors and unknown magntudes. (4 ponts) c) Determne the magntude of the tenson n cables. (6 ponts) d) Determne the reactons at pont O. (6 ponts)

5 PROBLEM 2B. Express the tenson n cables T AB and TAC n terms of ther unt vectors and unknown magntudes. (4 ponts) Problem 2C. Determne the magntude of the tenson n cables. (6 ponts)

6 PROBLEM 2D. Determne the reactons at pont O. (6 ponts)

7 ME 270 Fall 2012 PROBLEM 3. (20 ponts) The frame s loaded wth a 1000 lb load as shown. Assume the mass of the members are neglgble compared to the load and that member DF can be treated as a two-force member. DF has a cross-sectonal area of 10 n2. Please place all answers n the box provded. All steps of your work must be shown to earn credt. a) The overall free-body dagram s provded. Complete the ndvdual free-body dagrams on the sketches provded. (4 pts) b) Determne the reacton forces at A and B. (4 pts) c) Determne the load n DF and whether t s n tenson or compresson. (4 pts) d) Determne the axal stress n DF. (4 pts) e) Assumng a sngle-shear desgn (see nset pcture) and pn cross-secton of 0.1 n2, determne the shear-stress at pn A. (4 pts) a) sngle-shear Bx 1,000 lbs Ax Ay B E E C D F D A F b) Ax = Ay = Bx = c) FDF = Tenson or Compresson (Crcle One) d) σdf = e) 1,000 lbs τa =

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9 ME 270 Fall 2012 Problem 4A (10 ponts) A secton has a trapezum shape wth a crcular openng as shown n the followng dagram. The dagram has a scale of 0.2 m correspondng to 1 unt. Take the bottom left corner as the orgn. Calculate the centrod ( XY, ) of the shaded secton. Gve your answers n correct unts. y x (Note: Take the bottom left corner of the secton as the orgn n your calculaton.) X Y

10 Problem 4B (10 ponts) By symmetry, the centrod of the followng shaded secton passes through the axs Z. (a) Determne the second moment of area of the shaded secton passng through the centrod, I C (b) Determne the second moment of area of the shaded secton passng through the axs Z -Z, I B You are requred to gve correct unts for the answers. Z Z Z Z The scale of the dagram s 1 unt = 1 nch. I C = I B =

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12 ME 270 Fall 2012 Problem 5 For ths problem, you may use the beam shown below (ts rectangular cross-secton s shown below as well). You may use the followng parameters: L = 6 ft., P = 450 lbs., M B = 250 ft.*lbs., b=4 n., and h = 7 n. Problem 5a. Usng an approprate free body dagram, determne the reactons at ponts C and D. (5 ponts) Problem 5b. Draw a free body dagram correspondng to a secton cut through pont E and show that t s n a state of pure bendng. (3 ponts) Problem 5c. Calculate the maxmum tensle and compressve stresses and locate them on the secton at pont E. (3 ponts)

13 Problem 5d. Draw the shear force and bendng moment dagrams for the beam below. (9 ponts)

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15 ME 270 Fnal Exam Equatons Sprng 2013 Normal Stress and Stran σ F A σ y My I ε σ E L L ε ε ε ε y y ρ FS σ σ Shear Stress and Stran τ V A τ ρ Tρ J τ Gγ G E 2 1 γ δ L π 2 θ For a rectangular crosssecton, τ y 6V h Ah 4 y τ 3V 2A Second Area Moment I y da I 1 12 bh Rectangle I π 4 r I I Ad Crcle Polar Area Moment J π 2 r r Tube Shear Force and Bendng Moment V x V 0 p d M x M 0 V d Buoyancy F =ρgv B Flud Statcs p=ρgh ( ) F = p Lw eq avg Belt Frcton T T L S = e µβ Dstrbuted Loads F eq xf eq = L 0 = L ( ) w x dx 0 Centrods x x= = xda c da y ( ) x w x dx x A = c A y= In 3D, x= y da c da y A c A x V c V Centers of Mass xɶ = xɶ = x ρda cm ρda yɶ = x ρa cm ρa yɶ = y ρda cm ρda y ρa cm ρa

16 Fall 2012 Fnal Exam Answers 1A. Defntons 1B. F 1 = j +154k N F 2 = k N 1C. P = j +40k lbs M A = j +80k lbs-ft 1D. F = eq 1 AL 4 4 x = 4 5 L 2A. FBDs T = T m + n j +p k = T - j + k B. AB AB AB AB AB AB T AC = TAC mac + nac j +p ACk = TAC - j + k C. T AB = 175 lbs T AC = 175 lbs 2D. O = j +30.0k lbs 3A. FBDs 3B. A x = 857 lbs A y = 1000 lbs B x = -857 lbs 3C. FDF 2500 lbs Compresson 3D. σ DF = -250 ps 3E. A = 13.2 ks 4A. X = m Y = m 4B. I = 178 n c 4 I = 400 n B 4

17 5A. C x = 0 C y = 142 lbs D y = 308 lbs 5B. FBD VE 5C. x 0 & snce the shear stress at pont E equals zero, the beam s n statc equlbrum of pure bendng σ = ps

18 5D.