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RENSSELAER POLY TECHNIC INSTITUTE TROY, NY
EXAM NO. 2 INTRODUCTION TO ENGINEERING ANALYSIS(ENGR-1100) – Fall 10
NAME: ______________________________ Section: ___________ RIN: _______________________________
Wednesday, October 20, 20108:00 – 9:50
Please state clearly all assumptions made in order for full credit to be given.
Problem Points Score
1 25
2 25
3 25
4 25
Total 100
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Problem #1 (25 %)For the following matrices:
A =[4 5 6],
32
14 D ,
101
210
115
C,
1
3
2
B
Determine the following expressions. If an expression is not valid just state the reason.a) AB (4)
b) BA (6)
c) B +C (2)
d) Determinant of C using the duplicate column method (4)
e) BtC (4)
f) Determinant of B (2)
g) D-1 (3)
7/27/1
14/114/3
14
42
13
D
142122x)1(3x4321-4 D g)
matrixsquareanotisBasdUndefine)f
7191-62032-1-010
1x)1(2x31x20x)1(1x3)1(x21x)1(0x35x2
101
210
11-5
132CB )e
21300102-5
1x0x)1(0x2x51x1x11x0x0(-1)x2x15x1x1
0
1
1-
1
0
5
101
210
11-5
C d)
sizesamethehavenotdomatricesthesincedUndefine)c
654
181512
12108
6x15x14x1
6x35x34x3
6x25x24x2
654
1
3
2
BA)b
1761586x(-1)5x34x2
1
3
2
654AB )a
1-
t
1
6
2
1
2
3
2
1
3
1
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Problem #2 (25%) Three cables (AB, AC and AD) are used to tether a balloon as shown. The force (tension) in thecable AC is 300 N.
1.
Identify the particle to be analyzed (3 points)2. Draw a complete and separate FBD showing the particle and all the forces acting on it. (5points)
3. Write the equilibrium equations (8 points)4. Determine the vertical force P exerted by the balloon. (3 point)5. Determine the magnitude of the force in the cables AB and AD (6 points)
Particle to be analyzed: A
FBD
kP P
k) je T 781.0625.0( ABABABAB T T
k jk ji
e 781.0625.0)5()4(
54022
AB
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k) jie T 745.0298.0596.0( ACACACAC T T
k jik ji
e 745.0298.0596.0
)5()2()4(
524
222
AC
k) jie T 857.00515.0( ADADADAD T T
k jik ji
e 857.00515.0)5()3(
50322
AD
N607.223443.89885.178(
N745.0298.0596.0(300
k) ji
k) ji T
AC
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Equilibrium
0P T T T ADACAB
0kk) ji
k) jik) j
P T
T
AD
AB
857.00515.0(
N607.223443.89885.178(781.0625.0(
0857.0(N607.223781.0(
0N443.89)625.0(
0)515.0(N885.178
P T T
T
T
ADAB
AB
AD
))
N054.633
N109.143
N350.347
P
T
T
AB
AD
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Problem #3 (25 %)
The beam fixed in the wall is subjected to forces T andF in addition to the weightW. Assumethat the beam is weightless. The forces T andF are given in the figure.
(1) Calculate the moment of T about O. (4)(2) Calculate the moment of F about O. (4)(3) Calculate the moment of G about O. (4)(4) Replace T, F, andG by an equivalent resultant forceR at point O and a coupleC.
ExpressR andC in Cartesian vector form. (9)
(5) Calculate the scalar component of the moment of T aboutz-axis (note thatz-axis followsthe right-hand rule). (4)Important: mark the senses of 2D moment and couple in your result.
Solution:
(1) M TO = Ty ×rOD=150 ×4 =600 Nm;
or: k
k ji
TrM OD TO 600
0150260
004
Nm;
M TO =600 Nm (eqn: 1; number: 1; unit: 1; direction: 1)
(2) MFO =Fy ×rOB=300 sin30° ×3 =150×3 =450 Nm;
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or: k
k ji
FrM OBFO 450
0150260
003 Nm
MFO =450 Nm (eqn: 1; number: 1; unit: 1; direction: 1)
(3) MGO =G×rOB=-500 ×6 =-3000 Nm;MGO =3000 Nm (eqn: 1; number: 1; unit: 1; direction: 1)
(4) R = T +F +G =( Tx+Fx+Gx)i +( Ty+Fy+Gy) j =(-260+260) i +(150+150-500) j ==-200 j N.
R =-200 j N (eqn: 1; number: 1; unit: 1; vector form: 1)
C =M TO +MFO +MGO =(600+450-3000)k =-1950k Nm;
C =1950 Nm (eqn: 1; number: 1; unit: 1; vector form: 1; direction: 1)
(5) 6001600 kMM TO TOzprojM Tz Nm
Or use triple scalar product:
600101504
0150260
004
100
k TrM OD TOzprojM Tz Nm
(eqn: 2; number 1; unit: 1)
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Problem #4 (25%)