RENSSELAER POLYTECHNIC INSTITUTE
TROY, NY
EXAM NO.2 INTRODUCTION TO ENGINEERING ANALYSIS
(ENGR-1100)
- Spring12
NAME:
Section:
RIN:
Wednesday,
March 28, 2012
8 :0 0- 9 :5 0
Pleasestate clearlv all assumntionsmade in order tbr lull credit to be given.
Problem
Points
I
25
2
25
a
J
25
4
25
Total
100
Score
[Sou urr(r^JI
GoodLuck
Problem#1 (25 7o)
For the matrices:
s el
or
ltl
['l
rrr
r-r
or
[e
o:l-'l,r=l
o
r o ol
LzJ
ll,E:l
l,'=l:;
il,o=1,
L
r
L
J
5
L s-l
L2
3l
determinethe following expressions.If an expressionis not valid, statethe reason.
a) AE
(5)
u) e'e
(s)
(s)
c) DtC
a; e'E
(5)
(5)
e) Determinantof E using the duplicatecolumnmethod.
Note: You should show all work to receive full credit
r]),
SOLUTION #1
[s-]:ztf;1
L
A'6 =
a ) N o h v a lA
A;s
L5l
3Yl 6*lt E b3r3
b)
+ z*5
sx[z) * (-t)xo
-to +to (D
=
c)t": I
d)
L a g 35 5\ l
:
+
c) D*C
--
e) 30
o
f r c- - z1J \\\
L- t
Lo
-\
u,
r l * .+ s Yt\)
=
t ;-",- ';6)
6 se
1
/-:\
I r:)
\
\ !-^:\
53
4)(
5 rl
JQ)
5X5- O+
r o)
/,
)
\!-/
4 r - lr /-\
fVt )
4.t-
t* o x ' * o,-f
o oxoo'(-z)tiJ
I
IOO
2
A)
-' ""J=Itoo\,l
fL z
I
l
(d) *o =
:1)
o
/ \
(l/
?t-,
)lJ
c( 5x e ?* + 0
]
Problem #2 QSo/o\
A containerof weight W=376N is suspendedfrom ring A as shownin the figure below. Cable
BAC passesthroughthe ring and is attachedto fixed supportsat B and C. Two forcesP=Pi and
Q=9k areappliedto the ring to maintainthe containerin equilibrium.
(2)
a) Idenitfy the particle to be analyzedin this problem.
b) Draw a completefree body diagrarnshowingthe particleto be analyzedand all the forces
(5)
actingon it.
(1s)
c) Write the equilibrium equationsof the particle.
(3)
d) Determinethe magnitudesof the forcesP andQ.
Note: You needto showall work to receivefull credit.
SOLUTION #2
a) Theparticleto be analyzedis: l:lorv't- k
b) Draw free body diagramof particle.
@
U{r,lr,..,
No ( ,
(o , - 4 o o ,o )
h11.J- a^^/tv\ -L
$".".f," O*,$<^ -f
c) Equilibriumequationsof particle:
:2:4' = -.r - jsjtP=o
?
-- 40- +
o
Tr ' Tut -37'= l
i-ZFz = fl r - T s ; + Q ; ( ) )
qET + P=-o
o'<-
%-'-
1
Th-- l, ?o52T - t](n o r
7F +- - -O,134 zT tQ = o )
d)F
/j t , 2 N
E
a g ,6 P
a
ti"^P4l-tg
--e
h
na= uof +4od+ t6c)
tlA6lt t
rl taf* Qtru}+f6pz 3
4
--i
etk =
klt
J
fln6il
)
'W
e
Tnr
4t0 raz,
O
-13 ?+4a?*16?
Ts'
45
TEd
O
, (-&?+#T * f,t)
<)
t Ab= , ea g =
-->
Fc=
,ffir i:
r-
43omn
l. AA
N
6afrt o
t
-J
\ E t l t=
()
rmfi
,ta il4
O
-" d n '= E
rfrx 4 - +ffil
,fn = r?o,= r(#i*
q= a - E: ? ,
F
f r= -sr(fro
/VO
i|f
Hil
q1*db,A t^not Qe4lxot^"tv
f^o+?p-+ fr +i*d=B
t r('ffi7+
ffiT-%r)
) r(#+ 5 *#l.f*")
\
a)
P i - 3TtT'8
*
* Y c,?
Sc;\r^ts*
ZFx =
2t,t -Zr*
=
P'"6'c^aO *4-T'T '?
pJ +P = 0
l3r
fiqe
du $A.Q'
)a
\^
o["
U rn +
, ffir-3* (=
b
r
LtS
?5i +Q =
o )o
'l
- o,S 95 r + P= t . l
1,aos7T-3TL*o
'0,t3QzT+Q=O
fo!)
gsl^rt
Z-F.1=o
l-"J"-F
-{r',r,- ZF* t t};
{",,^ ZFt= 0;
T--
#rt
22o'5N
f= 0 ,T q { T '
Q -- o ' I3 4 z T =
e
l3l,2N o
A g,6 Ft o
Problem#3 (25 7o)
Four forcesareappliedto a machinecomponentABDE as shown.
v
20 mrn
100rnm
a)
b)
c)
d)
Computethe momentof the force at E aboutA.
Computethe momentof the force at D aboutA.
Computethe momentof the forcesat B aboutA.
Replacethe systemof forcesby a singleforce(R) at A anda couple(M).
(s)
(s)
(s)
(10)
Note: ExpressALL answersin Cartesianvector form. You needto showall work to receive
full credit.
SOLUTION #3
a) M'a6= -1E,2 - tZ {
i
l./r^
b) iln" = 5,6 P."
i
c) Mt^d =*ty|
o '*:
5t'^^
-hz o?- l of
] \ {= 3 0 ,R
f
ZZt
R{bt N
Nnn
-?
12O z
--a
('
G)
c
4(
hr
*)
,,lM fr E
J
l6aR
" toCIt+
4
4
zao i
,n h
a)
=' ?ALxFr O
.-|
i?
\i
J
T-)
le"
I
0,16
- b,l
lo
,
,
' lr o ()
o
|
xo) -- d\
o , l x 0 - 0, t6
--t f
?
[-
+
-.r
( o, r x o- o, t 6r ( r z o) )
(-o,r)(- tz't
O,7J\ O
tu\
?
1')nr'
t27
| 9,2 A
0;
*)
(bl
f' b
- 25Dh_
4
,M'
,
--)
J
-..)
x ob
O
^l
+ t 6()k
z o a{ + C Ii
--)
-t
;
-)
"A D =
/rr)
r1'r
T^o X f-, \)'
-)
\i
--_\
k-'\._
i
to
\o,r
o 'tt
\*'
I
o'25o
lo
--r '
'[
O Y - Q 'tA)
I {r)
T ) - 0\0'
t ( .o x [z r
5r0
n
rD
N14
4
n
-3 0 0
"i
- o( o)
I o xo''L
_?
;)
r ^ f-
)u l
)'L
-)
2 aOt
o
lh
Iv
IU
?
P
K
*?
*> (t,r
vl
t'
1
I
I ,?00
-50 0l
I atZ
o
( !-'
v
$A(\
*)
_?
1'{ tF.,
b,) t)
N,
n
4rl vn
o
O rI
/f,o*u - lr't-lu} )
3l''L*a]T;1,*
r-*,,d
ro'-a'
"?
[-
G)
Rl'xl*a*
+r*^
R = Fr* Bu fB @
:
-+
* 3oo7 _ 5rrT
@
4zof- 5..oT zsDF N @
- tLo;
-s\
p_
- 2SD
Rg-^Jtr",.*tr(Tn^tlvtP.v-t^"
R = i l E+ R *+
&
+
to e C
=( *ts,r J'-, rt ) + @f S+ G raE) N,"-,
:
3 O,Bl
221;
r i' r r o' @
Problem #4 (25o/ol
A rod ABC is bentin the shapeof andarc of a circle of radiusR. It is connectedto a frictionless
hinge at C and is supportedby a smoothroller at B with F30o as shownin the figure below. A
force of magnitudeP : 100 N acts at point A in the verticalfy downwarddirection.For the
purposeof this problem,you may neglecfthe weightof thebffi& comparedto the forcesacting
on it.
a) Draw a completefreebody diagramof the rod ABC. Clearlylabelall forces,known and
( 10 )
unknown, and show the coordinate system.
b) Write downthe x- andy- forceequilibriumequationsfor the rod ABC.
6T
c) Choosea momentcenterandwrite downthe momentequilibriumequationaboutthat point.
(')
d)SolveforthereactionforcesatAandC.Expressyorrranswerin@.
(6)
Note: You needto show all work to receive full credit.
SOLUTION #4
a)FreebodydiagramUrj
[b )
fi I
()
LY
ZF* '
Lx
7ry =
- 8S, . nd'
ri-;
Ly
o
(i :
+ Bcc*o- {= 0
(c ) cAu,;x r^.'av's*l c'kfu6*l
a
,
w
(
.
H.' '.&,*q([s,.0)
r
" ';;i 't
ft(,n I- ',
-i?r'"oJ(:
-F'fr.
=o
G
{
(d ) Fr.orn. (/[Dr\r'e^f
A"W&'*
BRf;^gA,,,0
+8 Rg,'^P(
| -A6) * PP=o
cl t Bs*DC"S + B(i. g(t*t*0 _p*
o
(^, B = ? **
b) Forceequilibrium equations
Z F " - C-- 8 3;,^6=6
Z F , = Cy+ BC*S-,F=
9r"SLS*&,ng(t* 6^9")
=7
c) Moment equilibrium equation
=Pp
* Bt"0[A*
C*u" =@*d(nn,*s)
S',?o" ;
R&e)- pA=o
d) Reactionforcesat
pi^ g
= 2P
"
2Y lco = 21111
tl
$and C
-r)
6 - - t ooi + t + 3, 2j' N
rcalffii
uo,d= -(B('-.,30o)i*
,.A
- *) ,
C = lcr)i _ }I,-i,,.r"rt
= -t 6of , +I T 3, 2f
F". . n' - ZF* = ( t
O ',
C1 =
=
P- 13C".S
I ur:-
LOo4^go'
73, 2 Al @
o
-)
o o C:
t.-
lW
,-t
t
-+
YS ,Z6
^r
Ar
) C* = BA- S
= Zuo ( u3 0 t
= lr J Dt J @
F. rt ^Y 1 >
O