Partial Solutions Manual Ruina and Pratap Introduction to Statics and Dynamics Thisdraft: January14,2013 Haveasuggestion? Wanttocontributeasolution? [email protected]: SolutionsManual 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.
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