Exam 05: Chapters 10 and 11

PHYS 2311: STATICS!
SPRING 2014
Name: Exam 05: Chapters 10 and 11
• Select and solve five of the following problems to the best of your ability. You must choose two problem from
each column, and a final problem at your own discretion.
• Indicate below which five problems you wish to have graded. If you do not explicitly mark a problem to be
scored, it will not be scored. If you have worked on more than five problems, select only five to be
graded. I will not choose for you.
Choose At
Least Two
Grade this
one?
Choose At
Least Two
Score
Grade this
one?
Score
Problem 01
/15
Problem 06
/30
Problem 02
/25
Problem 07
/25
Problem 03
/15
Problem 08
/25
Problem 04
/15
Problem 09
/25
Problem 05
• You may use your calculator and the attached formula sheet.
• Read and follow the directions carefully.
• Solve using the method required by the problem statement. If you are not explicitly required to use a specific
technique, please be sure to show sufficient work so that your method is obvious.
• Show all your work. Work as neatly as you can. If you need scratch paper, blank sheets will be provided for you.
• It is permissible to use your calculator to solve a system of equations directly. If you do, state this explicitly.
• Express your answer as directed by the problem statement, using three significant digits. Include the
appropriate units.
EXAM 05!
PAGE 1
PHYS 2311: STATICS!
SPRING 2014
Problem 01
Use direct integration to determine the moment of inertia
with respect to the x-axis:
(Hint: A horizontal strip is the easiest area increment.)
dy
EXAM 05!
x
PAGE 2
PHYS 2311: STATICS!
SPRING 2014
Problem 02
A. Locate the centroid y of the channel’s crosssectional area.
B. Determine the moment of inertia with respect to the
x’ axis passing through the centroid.
y
x
A3
x’
C
A1
EXAM 05!
A2
PAGE 3
PHYS 2311: STATICS!
SPRING 2014
Problem 03
Use direct integration to determine the mass moment of inertia Ix
with respect to the x-axis. The density of the zinc alloy used is
6800 kg/m3.
For a disk-shaped mass increment dm:
HINT: If y2 = 50x, then to be dimensionally consistent, the
constant must have units!! (y mm)2 = (50mm)(x mm). Be doubleplus extra careful with your units.
EXAM 05!
PAGE 4
PHYS 2311: STATICS!
SPRING 2014
Problem 04
The pendulum consists of a 3-kg slender rod attached to a 5-kg thin plate.
A. Determine the location ȳ of the center of mass G of the pendulum.
B. Find the mass moment of inertia of the pendulum about an axis perpendicular to
the page and passing through G.
y1
ȳ
G1
d1
d2
y2
G2
EXAM 05!
PAGE 5
PHYS 2311: STATICS!
SPRING 2014
Problem 05
Determine the moment of inertia Iz of the frustum of the cone which has a
conical depression. The alpha bronze used to make the part has a density of
8470 kg/m3.
Cone 1: Complete solid
h
Cone 2: Negative tip
0.8
0.2
Cone 3: Negative depression
EXAM 05!
0.4
PAGE 6
PHYS 2311: STATICS!
SPRING 2014
Problem 06
Determine the angles θ for equilibrium of the 4-lb disk
using the principle of virtual work. Neglect the weight of
the rod. The spring is unstretched when θ = 0° and always
remains in the vertical position due to the roller guide.
Disk at A:
Spring at C:
yC
yA
EXAM 05!
PAGE 7
PHYS 2311: STATICS!
SPRING 2014
Problem 07
The spring has a torsional stiffness of k = 300
N·m/rad and is unstretched when θ = 90°. Use
the method of virtual work to determine the
angles θ when the frame is in equilibrium. Ignore
the masses of the frame members.
HINT: MB for the torsion spring is k(∆α), where
α is the angle at B. You’ll need the relationship
between angles θ and α. The trig equation factors
easily, for two solutions.
α
Torque at A:
Torsion spring at B:
Virtual Work:
EXAM 05!
PAGE 8
PHYS 2311: STATICS!
SPRING 2014
Problem 08
The spring is unstretched when θ = 45° and has a stiffness
of k = 1000 lb/ft. Use the potential energy method to
determine the angle θ for equilibrium if each of the
cylinders weighs 50 lb. Neglect the weight of the
members.
Spring at E:
Potential Function:
yB = yC
Masses at B and C:
EXAM 05!
PAGE 9
xE
PHYS 2311: STATICS!
SPRING 2014
Problem 09
The uniform bar AB weighs 100 lb. If both springs
DE and BC are unstretched when θ = 90°. Both
springs always remain in the horizontal position due
to the roller guides at C and E.
xB
A. Determine the angle θ for equilibrium using
the principle of potential energy.
B. Evaluate the stability of the equilibrium
position.
HINT: Units!! All inches or all feet–pick one! Trig
equation solves easily by factoring, yields two values
for θ.
xD
yG
Potential Energy of Bar:
Potential of Springs B and D:
Potential Function:
Stability of Equilibrium: Neither position is stable!
EXAM 05!
PAGE 10

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