Partial Solutions Manual Ruina and Pratap - Andy Ruina

140 Pages·2013·22.11 MB·English

Partial Solutions Manual
Ruina and Pratap
Introduction to Statics and Dynamics
Thisdraft: January14,2013
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Note, the numbering of hand-written solutions is most-often wrong (corre-
spondingtoanoldnumberingscheme). Thehand-writtenproblemnumbers
shouldbeignored.
9.1.15 Consider a force F.t/ acting on cases.
a cart over a 3 second span. In case (a), F(t)
theforceactsintwoimpulsesofonesec- F
0
onddurationeachasshowninfig.9.1.15.
(a)
Incase(b),theforceactscontinuouslyfor
two seconds and then is zero for the last
second. Giventhatthemassofthecartis 0 1 2 3 t (sec)
10 kg,v.0/ 0,andF 10N,foreach F(t)
D 0 D
forceprofile,
F
0
a) Findthespeedofthecartattheend
of3seconds,and (b)
b) Find the distance travelled by the 0 1 2 3 t (sec)
cartin3seconds.
FPilrenoambe:lpefigmure9-91-f.c1om5pare
Comment on your answers for the two
2 IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
Chapter9.1.Forceandmotionin1D Problem9.1.16 3
9.1.16 Acarofmassmisacceleratedby
F(t)
applying a triangular force profile shown
infig.9.1.16(a). Findthespeedofthecar FT
at t T seconds. If the same speed is
D (a)
to be achieved at t T seconds with a
sinusoidalforceprofiDle,F.t/DFssin(cid:25)Tt, 0 T/2 T t
find the required force magnitude F . Is
s
thepeakhigherorlower?Why? F(t)
F
s
(b)
0 T/2 T t
FPilrenoambe:lpefigmure9-91-f.c1om6pare2
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
4 Chapter9.1.Forceandmotionin1D Problem9.1.22
9.1.22 A grain of sugar falling through
honey has a negative acceleration propor-
tionaltothedifferencebetweenitsveloc-
ity and its ‘terminal’ velocity, which is a
knownconstantv . Writethissentenceas
t
a differential equation, defining any con-
stantsyouneed.Solvetheequationassum-
ingsomegiveninitialvelocityv .
0
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
Chapter9.1.Forceandmotionin1D Problem9.1.26 5
9.1.26 A bullet penetrating flesh slows diameterd 5:7mm.
D
approximately as it would if penetrating
a) Plotthebulletpositionvstime.
water. The drag on the bullet is about
F c(cid:26) v2A=2 where (cid:26) is the den- b) Assume the bullet has effectively
siDtyoDfwatwer,v istheinstantwaneousspeed stoppedwhenitsspeedhasdropped
ofthebullet, Aisthecrosssectionalarea to 5m=s, what is its total penetra-
of the bullet, and c is a drag coefficient tiondistance?
which is about c 1. Assume that the c) Accordingtotheequationsimplied
(cid:25)
bullet has mass m D (cid:26)lAL where (cid:26)l is above, what is the penetration dis-
the density of lead, A is the cross sec- tanceinthelimitt ?
tionalareaofthebulletandListhelength !1
d) How would you change the model
ofthebullet(approximatedascylindrical).
to make it more reasonable in its
Assume m 2grams, entering velocity
D predictionsforlongtime?
v 400m=s,(cid:26) =(cid:26) 11:3,andbullet
0 D l w D
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
6 Chapter9.1.Forceandmotionin1D Problem9.1.26(continued)
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
Chapter9.1.Forceandmotionin1D Problem9.1.26(continued) 7
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
8 Chapter9.1.Forceandmotionin1D Problem9.1.26(continued)
Plot of position vs time
1.8
1.6
1.4
1.2
)
m 1
(

n
o
i
t
i
s 0.8
o
p
0.6
0.4
0.2
0
0 5 10 15 20
time (s)
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
Chapter9.2.Energymethodsin1D Problem9.2.3 9
9.2.3 A force F F sin.ct/ acts on a b) E ,
D 0 K
particle with mass m 3kg which has
D c) P,
positionx 3m, velocityv 5m=sat
t D 2s. FD0 D 4N and c DD2=s. At d) EPK,
t 2sevaluate(givenumbersandunits): e) therateatwhichtheforceisdoing
D
work.
a) a,
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.
10 Chapter9.2.Energymethodsin1D Problem9.2.3(continued)
IntroductiontoStaticsandDynamics,(cid:13)c AndyRuinaandRudraPratap1994-2012.