NAME: ____________________________________

ME 270 – Fall 2012

Final Exam

Please review the following statement:

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

Signature: ______________________________________

INSTRUCTIONS

Begin each problem in the space provided on the examination sheets. If additional space is required,

use the white lined paper provided to you.

Work on one side of each sheet only, with only one problem on a sheet.

Each problem is worth 20 points.

Please remember that for you to obtain maximum credit for a problem, it must be clearly presented,

i.e.

The coordinate system must be clearly identified.

Where appropriate, free body diagrams must be drawn. These should be drawn separately

from the given figures.

Units must be clearly stated as part of the answer.

You must carefully delineate vector and scalar quantities.

If the solution does not follow a logical thought process, it will be assumed in error.

When handing in the test, please make sure that all sheets are in the correct sequential order

and make sure that your name is at the top of every page that you wish to have graded.

Instructor’s Name and Section:

Sections: J. Silvers 11:30am-12:20pm E. Nauman 8:30-9:20am J. Jones 9:30-10:20am

M. Murphy 9:00-10:15am K.M. Li 2:30-3:20pm K.M. Li 4:30-5:20pm

Last Name First Name

Problem 1 __________

Problem 2 __________

Problem 3 __________

Problem 4 __________

Problem 5 __________

Total _______________

1F (3 pts)

2F (2 pts)

NAME: ____________________________________

ME 270 – Fall 2012

PROBLEM 1 (20 points) – Prob. 1 questions are all or nothing.

PROBLEM 1A. (5 points)

FIND: In your own words, state each of Newton’s three laws of motion. Be sure to write legibly.

Unreadable definitions will be marked wrong.

PROBLEM 1B. (5 points)

FIND: Two force vectors are shown to the right. γ defines

the angle between F1 and the x-y plane; φ lies in the x-y

plane; and F2 lies in the x-z plane. Write F1 and F2 in vector

form. Values for F1, F2 and the angles are given to the right.

Last Name First Name

1st Law =

2nd Law =

3rd Law =

1

2

F 450 N

F 300 N

20

40

30

eqF (2 pts)

x (measured from wall on left) (3 pts)

NAME: ____________________________________

ME 270 – Fall 2012

PROBLEM 1C. (5 points)

FIND: The force P = 120 lb is applied to the L-shaped bar ABC and

acts in the direction of the line from C to D. Write P in vector form.

Determine the moment about point A due to the force P; express

your answer in vector form.

PROBLEM 1D. (5 points)

FIND: A cantilevered beam is subjected to the distributed

load shown. Determine the equivalent force and its

location, measured from the wall. Give your answers in

terms of A and/or L.

P (2 pts)

AM (3 pts)

Last Name First Name

NAME: ____________________________________

ME 270 – Fall 2012

PROBLEM 2. (20 points)

A mass-less boom is used to support a 180-lb load applied at point D. The boom is attached to the

wall at point O with a ball and socket and supported by cables at point A. Please place your

answers to the following questions in the boxes provided. You will be asked to:

a) Complete the free-body for the boom on the figure provided. (4 points)

b) Express the tension in cables in terms of their known unit vectors and unknown

magnitudes. (4 points)

c) Determine the magnitude of the tension in cables. (6 points)

d) Determine the reactions at point O. (6 points)

Last Name First Name

NAME: ____________________________________

PROBLEM 2B.

Express the tension in cables AB ACT and T in terms of their unit vectors and unknown magnitudes.

(4 points)

Problem 2C. Determine the magnitude of the tension in cables. (6 points)

Last Name First Name

NAME: ____________________________________

PROBLEM 2D. Determine the reactions at point O. (6 points)

Last Name First Name

NAME: ____________________________________

ME 270 – Fall 2012

PROBLEM 3. (20 points)

Last Name First Name

The frame is loaded with a 1000 lb load as shown.

Assume the mass of the members are negligible

compared to the load and that member DF can be

treated as a two-force member. DF has a cross-sectional

area of 10 in2.

Please place all answers in the box provided. All steps

of your work must be shown to earn credit.

a) The overall free-body diagram is provided. Complete

the individual free-body diagrams on the sketches

provided. (4 pts)

b) Determine the reaction forces at A and B. (4 pts)

c) Determine the load in DF and whether it is in tension

or compression. (4 pts)

d) Determine the axial stress in DF. (4 pts)

e) Assuming a single-shear design (see inset picture) and

pin cross-section of 0.1 in2, determine the shear-stress

at pin A. (4 pts)

a)

b) Ax = ____________ Ay = ____________ Bx = ____________

c) FDF = ___________________________Tension or Compression

d) σDF = ______________________________________________

e) τA = _______________________________________________

A

F

E E C

D

F

D

B

1,000

lbs

Bx

Ax

Ay

(Circle One)

1,000 lbs

single-shear

NAME: ___________________________________

Last Name First Name

NAME: ____________________________________

ME 270 – Fall 2012

Problem 4A (10 points)

A section has a trapezium shape with a circular opening as shown in the following diagram. The

diagram has a scale of 0.2 m corresponding to 1 unit. Take the bottom left corner as the origin.

Calculate the centroid ( , )X Y of the shaded section. Give your answers in correct units.

(Note: Take the bottom left corner of the section as the origin in your calculation.)

y

x

X _______________________ Y ________________________

Last Name First Name

NAME: ____________________________________

Problem 4B (10 points)

By symmetry, the centroid of the following shaded section passes through the axis Z.

(a) Determine the second moment of area of the shaded section passing through the centroid, IC

(b) Determine the second moment of area of the shaded section passing through the axis Z’-Z’, IB

You are required to give correct units for the answers.

The scale of the diagram is 1 unit = 1 inch.

Z’

Z Z

Z’

IC = __________________________ IB = __________________________

Last Name First Name

NAME: ___________________________________

Last Name First Name

NAME: ____________________________________

ME 270 – Fall 2012 Problem 5 For this problem, you may use the beam shown below (its rectangular cross-section is

shown below as well). You may use the following parameters: L = 6 ft., P = 450 lbs.,

MB = 250 ft.*lbs., b=4 in., and h = 7 in.

Problem 5a. Using an appropriate free body diagram, determine the reactions at points C and D. (5 points)

Problem 5b. Draw a free body diagram corresponding to a section cut through point E and

show that it is in a state of pure bending. (3 points) Problem 5c. Calculate the maximum tensile and compressive stresses and locate them on the section at point E. (3 points)

Last Name First Name

NAME: ____________________________________

Problem 5d. Draw the shear force and bending moment diagrams for the beam below. (9 points)

Last Name First Name

NAME: ___________________________________

Last Name First Name

ME 270 Final Exam Equations Spring 2013

Normal Stress and

Strain

σ� � F�A

σ��y � MyI

ε� � σ�E � ∆LL

ε� � ε� � �ε�

ε��y � yρ

FS � �����

Shear Stress and Strain

τ � VA

τ�ρ � TρJ

τ � Gγ

G � E2�1 % �

γ � δ'L' � π2 θ

For a rectangular cross-

section,

τ�y � 6VAh, -h,

4 y,/

τ0�� � 3V2A

Second Area Moment

I � 2 y,dA4

I � 112 bh6 Rectangle

I � π4 r@ Circle

IC � ID % AdDC,

Polar Area Moment

J � π2 �r�@ r�@ Tube

Shear Force and Bending

Moment

V�x � V�0 % 2 p�IdI�J

M�x � M�0 % 2 V�IdI�J

Buoyancy

= ρB

F gV

Fluid Statics

= ρp gh

( )=eq avg

F p Lw

Belt Friction

µβ=L

S

Te

T

Distributed Loads

( )= ∫L

eq0

F w x dx

( )= ∫L

eq0

xF x w x dx

Centroids

= ∫∫

cx dA

xdA

= ∫∫

cy dA

ydA

=∑∑

ci i

i

i

i

x A

xA

=∑∑

ci i

i

i

i

y A

yA

In 3D, =∑∑

ci i

i

i

i

x V

xV

Centers of Mass

ρ=

ρ∫∫

ɶcm

x dAx

dA

ρ

=ρ

∫∫

ɶcm

y dAy

dA

ρ=

ρ

∑∑

ɶ

cmi i i

i

i i

i

x A

xA

ρ=

ρ

∑∑

ɶ

cmi i i

i

i i

i

y A

yA

Fall 2012 Final Exam Answers

1A. Definitions

1B. 1F = -272 i - 324j +154k N 2F = 260 i - 150k N

1C. P = -80 i - 80j +40k lbs AM = 480 i - 440j +80k lbs-ft

1D. 4

eq

1F = AL

4

4x = L

5

2A. FBDs

2B. AB AB AB AB AB AB

2 6 3T = T m i + n j +p k = T i - j + k

7 7 7

AC AC AC AC AC AC

2 6 3T = T m i + n j +p k = T i - j + k

7 7 7

2C. ABT = 175 lbs ACT = 175 lbs

2D. O = 0i + 300j +30.0k lbs

3A. FBDs

3B. xA = 857 lbs yA = 1000 lbs xB = -857 lbs

3C. DFF 2500 lbs Compression

3D. DFσ = -250 psi

3E. A = 13.2 ksi

4A. X = 1.168 m Y = 0.624 m

4B. 4

cI = 178 in 4

BI = 400 in

5A. xC = 0 yC = 142 lbs yD = 308 lbs

5B. FBD

EV 0 & since the shear stress at point E equals zero, the beam is in static equilibrium of pure bending

5C. xσ = -91.8 psi

5D.