302 (S98) Final Exam

302 (S98) Final Exam - þ7########,###############...

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Unformatted text preview: þ7########,############### ÷##Ç@###############? ##################$##h##$##h##¨#####¨#####¨#####¨##‚##¯~####¯~####¯~####¯‚#R##¯è# F##°.####°.#P##°~#x##¨‚#î##°ö# ##±#####¶"#*##¶L#·##¨########³‚####±.#m## ´²####µÄ#^##Ç##=##¶"####¶"####¶"####¶"####¶"####¶"####Name #1. [30 points] An object is dropped from rest. After falling a distance R, it enters a frictionless, semicircular groove of radius R (as shown to the right). Find the magnitude of the normal force exerted by the groove on the object as a function of f, where f is measured from the vertical.2. (GRE problem) Circle the letter corresponding to the best answer. This is all or nothing! [5 points] A rock is thrown vertically upward with initial speed vo. Assume a friction (resistance) force proportional to -v, where v is the velocity of the rock, and neglect the buoyant force exerted by the air. Which of the following is correct? A. The acceleration of the rock is always g. B. The acceleration of the rock is equal to g only at the top of the flight. C. The acceleration of the rock is always less than g. D. The speed of the rock upon return to its starting point is vo. E. The rock can attain a terminal speed greater than vo before it returns to its starting point. 3. [20 points] In the presence of a resistive force of the form f = -mkv, we found the time of flight for a projectile on level ground to be: # to first order in the small expansion parameter k (where V=vosinq is the vertical component of the initial velocity of the projectile).A. [15 pts] Starting from the equation of motion, find an approximation to the range of the projectile to first order in k. Please show your work.B. [5 pts] For what values of k is this solution a good approximation to the range of the projectile? Express your answer in terms of givens in the problem: vo, q, m, g. Explain your reasoning. 4. [25 points] An object on a level surface is given an initial velocity vo > 0. Thereafter, it is acted on by a resistive force mbvn. In the text we learned that when n = 1, the stopping distance for the object was finite, but in class we learned that when n = 2, the stopping distance was infinite. For what value of n for 1 < n < 2 does the transition between a finite and an infinite stopping distance occur? Show your work.5. [25 points] Find the components of the acceleration vector # in plane polar coordinates (r,q). Show your work. (Hint: This was done in cylindrical coordinates as an example in the text. The answer is not very different in polar coordinates).Name Total Points Name Total Points Physics 301 (Fall 1997) Third Exam (October 31, 1997) #v#‚##v#‚#>#ü###‚#######®#É###r#########®#É#bÿÿÿÿ##È###ÿ #ÿÿÿÿÿÿ##ÿÿ###È##########¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚### ÿôÿô### ÿóÿó#¡#¼######### ######ÿÿÿÿÿÿ##### #########a#####‚#¬#´#Z#####¡#¶##################### ÿÿÿÿÿÿÿÿ#h#´#Z##ÿÿÿÿÿÿ# #########i#Z#Z###########¡#¶##################### ÿÿÿÿÿÿÿÿ#h#Z#Z###########¡#¶#############"#H###¸###########¡#¶####### #‚#q ###‚#^#‚#h#‚#c#‚#^#‚#c#‚#h#‚#c#######p###‚#^#‚#h#‚#c#‚#^#‚#c#‚#h#‚#c#"#‚#c#Å# #‚###### #####¡#¶#############"#################¡#¶#############"#H###############¡#¶#### ### #‚#q############ ##### ###### #######p############ ##### ###### #"### #ò# #‚###########¡#¶####### #‚#q###;###H###H# #;###;# #;###H# #######p###;###H###H# #;###;# #;###H# #"#;# #ò# #‚#¡#‚# ############# ÿþÿã### ###############"####### #‚#¡#‚##ÿò&######,# ## Helvetica########.#######*##R## #‚# #‚#####¡#‚# ############# ÿ´ÿ»##### #‚#¡#‚##ÿòk###W##)##R## #‚# #‚##### #N#b### ÿóÿó#¡#¼#"### ###¡#¶#############"#H#c'<#¡#‚# ############# ÿ‚ÿ ### ###############"####### #‚#¡#‚##ÿîl###r##,# ###Symbol######*##f## #‚# #‚##### #‚#ÿ###‚#######"#R###‚#########"#R#v#####"#R##¡#d##·xpr####### ## #‚## #####"#R#####"#"#R## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###T,# ###Symbol#####)##@###(# ###2##)##V(#####g#####"########(###+#1###)##-#####(# #=#kV##+###3##)##g"###<#######.######(# #'#é *##ë (###'#ê (# #L#ù *##û (###L#ú ¡#À#÷30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2624 div 1088 3 -1 roll exch div scale currentpoint translate 64 37 translate -12 603 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (T) show 327 603 moveto 384 /Symbol f1 (@) show 666 364 moveto 384 /Times-Roman f1 (2) show 864 364 moveto 384 /Times-Italic f1 (V) show 806 896 moveto (g) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 645 504 moveto 504 0 rlineto stroke 1333 603 moveto 384 /Times-Roman f1 (1) show 1576 603 moveto 384 /Symbol f1 (-) show 1903 364 moveto 384 /Times-Italic f1 (kV) show 1922 896 moveto 384 /Times-Roman f1 (3) show 2116 896 moveto 384 /Times-Italic f1 (g) show 1877 504 moveto 481 0 rlineto stroke 1213 355 moveto 384 /Symbol f1 (\351) show 1213 978 moveto (\353) show 1213 724 moveto (\352) show 2383 355 moveto (\371) show 2383 978 moveto (\373) show 2383 724 moveto (\372) show end #¿¡#d#UMATH###I#‚##### ##‚T#‚@######‚2#‚V###‚g########‚1#‚-######‚k#‚V###‚3#‚g####‚[#‚]###xc #‚ÿ###O####### # ###A######### # #1##### # ##¡#d##·xpr####### ## #‚## ##### # ,## Ó#MT Extra# Ó# .##ÿ###* # #####"# # ## #¾¡#À#currentpoint #¿"###### #¾(#####r ,#####Times#####(# ###a ¡#À#j30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 288 div 384 3 -1 roll exch div scale currentpoint translate 64 -3445 translate 32 3520 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /MT-Extra f1 (r) show -5 3765 moveto 384 /Times-Italic f1 (a) show end #¿¡#d##MATH#####6##### #2‚a#####t #‚ÿ###a##Equation#########I##### ##‚T#‚@######‚2#‚V###‚g########‚1#‚-######‚k#‚V###‚3#‚g####‚[# ‚]######'##Equation############### #2‚a#####££Prove that when particles are acted upon by central forces only, namely forces of the form #, motion will be in a plane############### ###e##Finder´####FNDRMACS#‚########ef##############Ûå##þ#®Ü;À¯Ä å###################################d###############ô#‚###Æ#l################### ##################################;###=###E###F###æ###ç###########n###o###²###³# ##í###î###[###\###b###c###ª###«###ê###ë###î###ï###‚###‚###‚###‚###########?###@## Á## Â## á## â## }## ‚## ‚## ‚## ‚## ‚## ‚## ‚## ‚## «## ¬## ## ®## ±## é## ì## í## î## ï## ò## ó## ö## ÷###ï###ð###ñ###ò###ù###T###U###p###qþ#ù#ò#ï#ï#é#þ#þ#þ#þ#þ#é#é#þ#þ#â#é#ï#é#ï#é#Ü #Õ#ï##ù#ù#þ#ù#ù#þ#þ#þ#þ#þ##Ïïï###È### #B###ø####+h# #@######### #B###ü####### # ######### #B###ä#####‚# # ###ü########### #@######################‚#O#######################>###? 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A spring of constant k is attached between the top of the rod and a nearby wall. The length of the spring is such that the equilibrium position for the rod is at qo = 0. The moment of inertia of the rod about its center of mass is ML23.#Tekton##9##OCRA##;Ï#BernhardMod BT##<# Impress ###q##+h##0)##0+##0-##0.##08##09##0;##0=##0>##0? ##0@##0A##0p##0q##0r##0s##0t##0‚##0‚##0‚##1T##1‚##1‚ ##1‚##1Ê##1Ë##1Ì##1Í##4###4W##4X##4^##4f##4g##4h##4i##4j##4ª##4ý##53##54##5A##5` ##5‚##5¬##5×##5ã##6###6###6)##6H##6n##6‚##6¾##6¿##6Å##6Í##6Î##6Ð##6Ò##6ë##6ì##7## #7###7 ##7"##7*##7+##7.##76##77##7: ##7B##7C##7F##7W##7X##7^##7f##7g##7z##7{##7| ##7}##7~##7‚##7‚##7‚##<#úú#####################÷ñ###ëú##êêê#####################ê êê###ñ##ñ##ñ##ñ##ñ###êêê##ã##êêêú #B###ê####7‚##‚ # ######### # ###ü########### #@########Z/12. What is the angular frequency of oscillation of this rod for small displacements.FourthNovember2 s The equation of motion of a rocket in free space can be written: where m is the rocketÕs mass, v is its velocity, t is time, and u is constant. [5 points] The constant u represents the speed of therocket at t=orocket after its fuel is spent.rocket in its instantaneous rest framerocketÕs exhaust in a stationary framerocketÕs exhaust relative to the rocket. [5 points] The equation can be solved to give v as a function of m.rocket at t=orocket after its fuel is spent.rocket in its instantaneous rest framerocketÕs exhaust in a stationary framerocketÕs exhaust relative to the rocket.FourthNovember2 If the rocket has m = mo and v = 0 when it starts, what is the solution?u mo/mu exp (mo/m)u sin (mo/m)u tan (mo/m)None of the aboveFourthNovember2For each problem, c#FourthNovember2#}####0#########Q###"###########Q#########Q##¡#d##·xpr####### ## #‚## #######Q#####"###Q## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###m(# ##dv*##dt#####"## #,# ###Symbol#####(#####+#####) #u(# #-#dm+###dt"##,#######(###@#=###) #0¡#À#×30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2592 div 896 3 -1 roll exch div scale currentpoint translate 64 43 translate -4 501 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 355 262 moveto (dv) show 381 794 moveto (dt) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 328 402 moveto 413 0 rlineto stroke 834 501 moveto 384 /Symbol f1 (+) show 1120 501 moveto 384 /Times-Italic f1 (u) show 1392 262 moveto (dm) show 1471 794 moveto (dt) show 1365 402 moveto 519 0 rlineto stroke 1992 501 moveto 384 /Symbol f1 (=) show 2301 501 moveto 384 /Times-Roman f1 (0) show end #¿¡#d#NMATH###B v##### ##‚m######‚d#‚v###‚d#‚t###‚+#‚u######‚d#‚m###‚d#‚t###‚=#‚0##ro #‚ÿ###Z##Equation##### ####B##### ##‚m######‚d#‚v###‚d#‚t###‚+#‚u######‚d#‚m###‚d#‚t###‚=#‚0##above FourthNovember278 FourthNovember2A rocket sits on a concrete launch pad on a planet where g = 10 m/s2. At t = 0 the rocket fires its engines and maintains a constant thrust force Fthrust = u a, where u = 103 m/s and a = -(dm/dt) = +900 kg/s. The rocke (with its fuel) has initial mass Mo = 105 kg, where 90% of this initial mass is fuel.The rocket does not lift off at t =0. Explain why not.B##·###¸###¹###º###»###¼###½###¾###¿###À###Á###Â###Ã###Ä###Å###Æ###Ç###È###‚ ####<###<###<! ##<'##<(##<.##<6##<7##<8##<9##<: ##<@##<H##<I##={##=²##=³##B###BP##BQ##Bõ##Bö##B÷ ##Bø##Bù##Bú##C###C###C ##D ##DO##Db##D‚##DÄ##ET##EU##E[##Ec##Ed##E‚##E‚##E‚##E‚##E¡##E ¢##E¿##Eç##Eè##Eù##FW##FX##F‚##F©##F¬##Fô##F÷##Fø##GE##GF##GL##GT##GU##Nc##Nd##N e##Nj##Nl##Nr##Ns##Ny##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N²##NÆ##OD##OE##O‚##O‚##O‚##O ‚##O‚##O‚##O‚##O ####þþþ###þþþ###ø########þþþ#######þþþ#ñ#î#î########è#è###þþþø## #è##þþþ#þþþ#####á#î#îè#î## #B###ö####O´# # ###ü########### #B###ê####GX# #@##########‚#Y##<###<###<###<###<###< ##<! ##<(##<: ##Bú##EU##E‚##GF##Ns##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N‚##N ##N¡##N ¢##N£##N¤##N¥##N¦##N§##N¨##N©##Nª##N«##N¬##N##N®##N¯##N°##N±##N²##N³##N ´##Nµ##N¶##OC##OD##O‚##O‚##O‚##O‚##O£##Ui##Uj##Uk##Ul##U}##U‚##U‚##U‚##U‚##U‚##U‚ôôè èèèããããÚãããÑÑÑÑÑÑÑÑÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈȿȱ±±±ã±±±ã㦦¦¦¦¦### ###$‚###ÿL##Ð#þ‚#þè#####$‚# ##Ð#þ‚#ÿ#####h#Ð#8######$‚# ##Ð#ý0#ÿ######$‚# ##Ð#ý0#ÿ############Ð#ý0#ÿ############h#þ‚#þè###############$‚# ##h#þ‚#######¸######$‚####h#þ‚#þÀ####¸#AAt what time, t = tlo, does the rocket lift off the launch pad? Show your work.CWrite the equation of motion for the rocket after it lifts off the launch pad (t > tlo) such that there is no normal force exerted on the rocket by the launch pad. DEFGFourthNovember2Solve this equation of motion to find the velocity of the rocket as a function of t = t tlo. You may assume that all motion of the rocket is vertical, and that g = 10 m/s2 is a reasonable approximation throughout the rocketÕs trajectory. Show your work.The maximum velocity occurs at the instant the fuel is fully expended Why is this true?Find this maximum velocity. Show your work.How much higher will the rock(et) travel after burnout (Note: You are not asked to find the height of the rocket at burnout, just the ÒextraÓ height achieved from coasting after the fire is gone.)FourthNovember2A particle of mass m moves in a potential well# where a and b are both positive constants.What is the force acting on the particle? Show your work.Find the equilibrium points, and indicate whether each is stable or unstable. Show your work.CFor what range of energy can the particle undergo periodic motion?For an energy Etot in the range found in part C, find the turning points as functions of Etot.Find the frequency of small oscillations about any stable equilibrium points.FourthNovember2# ####‚#########}###‚###########}#z#######}##¡#d##·xpr####### ## #‚## #######}#####"###}## #¾¡#À#currentpoint #¿"###### #¾,# ###Symbol###### .##ÿ###+###y,#####Times#####) #(##)##x##)##)###)##º#####(# #'#U##) #(##)##x##)##)##(###/#m#####"# #'#######(###C#=##) #a###)##x####(###[#2#### +###-##)##b###)##x####(###x#3¡#À#ä30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4000 div 896 3 -1 roll exch div scale currentpoint translate 64 -986 translate -4 1530 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (y) show 304 1530 moveto 384 /Times-Roman f1 (\() show 463 1530 moveto 384 /Times-Italic f1 (x) show 656 1530 moveto 384 /Times-Roman f1 (\)) show 872 1530 moveto 384 /Symbol f1 (\272) show 1193 1291 moveto 384 /Times-Italic f1 (U) show 1493 1291 moveto 384 /Times-Roman f1 (\() show 1652 1291 moveto 384 /Times-Italic f1 (x) show 1845 1291 moveto 384 /Times-Roman f1 (\)) show 1452 1823 moveto 384 /Times-Italic f1 (m) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1190 1431 moveto 796 0 rlineto stroke 2094 1530 moveto 384 /Symbol f1 (=) show 2397 1530 moveto 384 /Symbol f2 (a) show 2651 1530 moveto 384 /Times-Italic f1 (x) show 2849 1359 moveto 224 /Times-Roman f1 (2) show 3079 1530 moveto 384 /Symbol f1 (-) show 3357 1530 moveto 384 /Symbol f2 (b) show 3586 1530 moveto 384 /Times-Italic f1 (x) show 3781 1359 moveto 224 /Times-Roman f1 (3) show end #¿¡#d#gMATH###[#_##### ##‚y#‚(#‚x#‚)#‚º######‚U#‚(#‚x#‚)###‚m###‚=#‚a#‚x########‚2## #‚-#‚b#‚x########‚3####nt #‚ÿ###s##Equation#########[##### ##‚y#‚(#‚x#‚)#‚º######‚U#‚(#‚x#‚)###‚m###‚=#‚a#‚x########‚2## #‚-#‚b#‚x########‚3####t#t = tbo that FourthNovember2FourthNovember2A. B. [5 points] The device pictured above is now immersed in some liquid such that the equation of motion aquires a damping term and becomes# Describe (qualitatively) the motion of the rod for the case where i) b < wo ii) b = wo FourthNovember2#####\#########Z###N###########Z#>#######Z##¡#d##·xpr####### ## #‚## #######Z#####"###Z## #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+###ú )##ú ##(# ###x ,# ###Symbol#####)##+###) #2#####)##b#####(###"#ú ##(# # #x #####) #+##) #w#####+ ##o##(###;#2### +###x#####) #=###) #0¡#À#â30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2880 div 544 3 -1 roll exch div scale currentpoint translate 64 -1997 translate 12 2362 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (\307) show 88 2362 moveto (\307) show 11 2381 moveto 384 /Times-Italic f1 (x) show 276 2381 moveto 384 /Symbol f1 (+) show 567 2381 moveto 384 /Times-Roman f1 (2) show 758 2381 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (b) show 1026 2362 moveto 384 /Times-Roman f1 (\307) show 987 2381 moveto 384 /Times-Italic f1 (x) show 1252 2381 moveto 384 /Symbol f1 (+) show 1538 2381 moveto 384 /Symbol f2 (w) show 1831 2477 moveto 224 /Times-Italic f1 (o) show 1851 2210 moveto 224 /Times-Roman f1 (2) show 2000 2381 moveto 384 /Times-Italic f1 (x) show 2280 2381 moveto 384 /Symbol f1 (=) show 2589 2381 moveto 384 /Times-Roman f1 (0) show end #¿¡#d#JMATH###> ø##### #2‚x####‚+#‚2#‚b2‚x####‚+#‚w#######‚o###‚2## #‚x#‚=#‚0## - #‚ÿ###V##Equation#########>##### #2‚x####‚+#‚2#‚b2‚x####‚+#‚w#######‚o###‚2## #‚x#‚=#‚0##10FourthNovember2#FourthNovember2 3 continuedFourthNovember2explain how you could findTimes#### Helvetica#####Courier###O ##O¡##O ¢##O£##O©##O±##O²##Uf##Uh##Uk##Ul##Ur##Uz##U{##U| ##U}##U‚##U‚##U‚##U‚##U‚##U‚##U‚##U¢##U£##U¥##U§##U¨##U©##U¬##U##U³##U»##U¼##U¾##UÃ##UÆ##Uà##Z###ZQ##ZR##ZX##Z`##Za##Zd##Ze##Zk##Zs##Zt##Zu##Zv ##Zw##Z}##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z‚##Z ¢##Z£##Zî##Zï##Zõ##Zý##Zþ##Zÿ##[###[###[###[###[###[###[###[###[ ##[! ##["##[###[$##[%##[+##[3##[4##[<##[=##[>##[? ##[@##[F##[Ný÷#öööð###ööö##ööö############ööö####ð##ööö##ööö###ööö########ööö##ö öö###ööö#ööö#####ööö#####öö# #@##########‚ # ###ü##########_##U‚##U‚##U‚##U¡##U¢##U£##U¤##U¥##U¦##U§##U¨##U©##Uª##U«##U¬##U##U½##U¾##U¿##UÀ##UÁ##UÂ##UÃ##UÄ##UÅ##UÆ##ZR##Zb##Zc##Zd##Ze##Zu##Zv##Zw##Z‚##Zï ##[###[###[%##[@##[P##^Æ##^Ö##^×##^Ø##` ##`##a###a###aB##aÜ##br##bÄ##v‚##v±##w‚##w‚##‚»##‚¼##‚5##‚6##‚©##¬¼##¿##Á##´[##´]## ´^##´_##´`##´a##´b##´c##´d##´e##´f##´g## ´hõõêõõõõõõõõõõõåÜÜÜÜÜÜÜÜÜÜåÜÜÜåÜÜååååååååååÐåÐåÐåååÐåÐåÐåÄåÐÐååååÐÐÐÐÐÐÐÐÐÐÐÐÐ# ######$‚####h#þ‚#þÀ####¸######$‚# ##h#þ‚#######¸######$‚# #ÿL##Ð#þ‚########### ########ÿL##Ð#þ‚#þè## ###$‚###ÿL##Ð#þ‚#þè#M (Note: you are not asked to find the turning points, just explain the process.)FourthNovember2FourthNovember2FourthNovember20555 total525 FourthNovember2 (Hint: A sketch of the potential energy could focus your thinking here.)FourthNovember246FourthNovember2FourthNovember24448 FourthNovember2acquires 0FourthNovember2FinalDecember17This section contains several GRE problems. In each case, circle the answer##á###â###ã###ä###å###æ###ç###è###é###ê###ë###ì###í###î###ï###ð###ñ###ò### ó###ô###‚####[N##[O##[P##[U####[_##[‚##[‚##[«##^###^Ä##^Å##^Æ##^Õ##^Ö##^×##^Ø## _‚##_¢##_à##_ê##_ô##_þ##`###` ##`###` ##`##`###a###a###a###a!##a"##a$##a %##a'##a(##a*##a,##a.##a/##a1##a2##a5##a7##a8##a: ##a;##a<##a? ##aA##aB##aÛ##aÜ##aÝ##b!##bl##bm##bn##bo##bp##bq##br##b‚##bÃ##bÄ##u_##u`##u ´##uµ##u÷##uø##v###vW##v^##vk##vx##v‚##v‚##v‚þ#þþþ###ø###############ò########### ##################ë####äÝÖ####ø##Ð#Ê#######Ê # ###ü##### # ######### #B###æ####n\# #B###æ####hà# #B###ê####f(# #B###ê####bÄ# # ### ##### #@##########‚#P that is most correct. This part is all or nothing! Some of the problems will also offer points for explaining your reasoning or showing your work. This is my attempt to provide partial credit.5Show your work.A 2-kilogram box hangs by a massless rope from a ceiling. A force slowly pulls the box horizontally to the side until the forizontal force is 10 Newtons. The box is then in equilibrium as shown below.The angle that the rope makes with the vertical is closest to arctan 0.5arcsin 0.5arctan 2.0arcsin 2.045o A 5-kilogram stone is dropped on a nail and drives the nail 0.025 meter into a piece of wood. Th the stone is moveing at 10 meters per second when it hits the nail, the average force exerted on the nail by the stone while the nail is going into the wood is most nearly10 NB100 NC1000 ND10,000 NE100,000 NA particle is initially at rest at the top of a curved frictionless track. The x- and y- coordinates of the track are related in dimensionless units by #, where the positive y-axis is in the vertical downward direction. As the particle slides down the track, what is itx tangential acceleration?0g### [Hint: A freebody diagram would help, as well as the concept of the gradient.] ###d#############V#############F##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###y,# ###Symbol#####)##=#####(#####x####(#####2# (#####4#####"#### #¡#À#A30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 960 3 -1 roll exch div scale currentpoint translate 64 47 translate 10 561 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (y) show 279 561 moveto 384 /Symbol f1 (=) show 640 322 moveto 384 /Times-Italic f1 (x) show 838 152 moveto 224 /Times-Roman f1 (2) show 710 854 moveto 384 ns (4) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 597 462 moveto 411 0 rlineto stroke end #¿¡#d#7MATH###+#µ##### ##‚y#‚=######‚x########‚2### ##‚4#### s #‚ÿ###¶#############¨#############‚##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #gx##+###2#####"### #¡#À#Ò30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 544 div 896 3 -1 roll exch div scale currentpoint translate 64 -1042 translate 29 1347 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (gx) show 116 1879 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 1487 moveto 425 0 rlineto stroke end #¿¡#d#'MATH#####ç##### #######‚g#‚x###‚2####su #‚ÿ# ###z#########+###l###########+#\#######+##¡#d##·xpr###### ## #‚## #######+#####"###+## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #gx(#####x####(#####2,# ###Symbol#### +###+###) #4#####"###################"#####ô###"###&#¡#À#230 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1376 div 960 3 -1 roll exch div scale currentpoint translate 64 -1042 translate 455 1347 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (gx) show 305 1929 moveto (x) show 503 1759 moveto 224 /Times-Roman f1 (2) show 734 1929 moveto 384 /Symbol f1 (+) show 1032 1929 moveto 384 /Times-Roman f1 (4) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1213 437 384 32 1964 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 1487 moveto 1277 0 rlineto stroke end #¿¡#d#CMATH###7#Ë##### #######‚g#‚x#######‚x########‚2## #‚+#‚4########v #‚ÿ#ù###ß####### #0###Ñ######### #0#Á##### #0##¡#d##·xpr###### ## #‚## ##### #0#####"# #0## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###gx####(#####2### (#####x####(#####2,# ###Symbol#### +###+###) #16#####"###################"#####ô#"#"####+#¡#À#u30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1536 div 1024 3 -1 roll exch div scale currentpoint translate 64 -1039 translate 449 1408 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (gx) show 839 1238 moveto 224 /Times-Roman f1 (2) show 305 1990 moveto 384 /Times-Italic f1 (x) show 503 1820 moveto 224 /Times-Roman f1 (2) show 734 1990 moveto 384 /Symbol f1 (+) show 994 1990 moveto 384 /Times-Roman f1 (16) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1365 439 384 32 2027 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 1548 moveto 1429 0 rlineto stroke end #¿¡#d#SMATH###G#}##### #######‚g#‚x########‚2### ######‚x########‚2## #‚+#‚1#‚6######## e #‚ÿ###C##Equation#########+##### ##‚y#‚=######‚x########‚2### ##‚4#######3##Equation############### #######‚g#‚x###‚2#######O##Equation#########7##### #######‚g#‚x#######‚x########‚2## #‚+#‚4###########_##Equation#########G##### #######‚g#‚x########‚2### ######‚x########‚2## #‚+#‚1#‚6########sA particle of mass m that moves along the x-axis has potential energy U(x) = a + b x2, where a and b are positive constants. Its initial velocity is vo at x = 0. It will execute simple harmonic motion with a frequency determined by the value of b aloneb and a aloneb and m aloneb, a, and m aloneb, a, m, and voExplain your reasoning.A particle is constrained to move along the x-axis under the inluence of the net force F = - kx with amplitude A and frequency Ä, where k is a positive constant. When x = A/2, the particleÕs speed is###### ‚## ‚## ‚## x## y## Ñ## Ò## Ø## Ù## ß## à## æ## ç## í## î## ô## õ###b###c###d###e###¨###©###«###¬###ü###ý## 4## 5## J## K## o###v‚##v‚##v°##v±##w###w ##w###w###w0##w2##wT##wy##wz##w{##w| ##w}##w~##w‚##w‚##z###‚h##‚*##‚2##‚3##‚4##‚6##‚7##‚8##‚9##‚: ##‚§##‚¯##‚°##‚±##‚²##‚³##‚ ´##‚·##‚º##‚»##‚¼##‚4##‚6##‚? ##‚‚##‚‚##‚‚##‚‚##‚‚##‚‚##‚¨##‚©##¬###¬###¬‚##¬‚##¬¸##¬¹##¬º##¬»##¬¼##¬½##¬¾##¬÷#############þ#þ####øñêãÜ##ÖÖ##Ï########È########Ö###Áº¬###Ö####©################### #B###æ####§X# #B###æ####£¼# #B###è####‚t# #B###è####‚¬# #B###Ê####‚ð# #B###æ####‚¼# #@######### #B###ø####‚## #B###ø####‚|# #B###ø####‚ # #B###ø####|Ä###B###ø####z#‚#B###Â############# ´#############¤##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+# #2,# ###Symbol#####)##p###)##fA¡#À#Ý30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 448 3 -1 roll exch div scale currentpoint translate 64 -2201 translate -11 2521 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (2) show 200 2521 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) show 422 2521 moveto 384 /Times-Italic f1 (fA) show end #¿¡#d#!MATH#####‚##### ##‚2#‚p#‚f#‚A## r #‚ÿ8###Ü#########"###Î###########"#¾#######"##¡#d##·xpr####### ## #‚## #######"#####"###"## #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+ #3#####"# #################"# ###õ###,# ###Symbol#####)##p###)##fA¡#À#Ê30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1088 div 512 3 -1 roll exch div scale currentpoint translate 64 -2216 translate 246 2600 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (3) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 443 378 384 0 2637 16 sqr 440 2600 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) show 662 2600 moveto 384 /Times-Italic f1 (fA) show end #¿¡#d#+MATH#####Ê##### ######‚3#### #‚p#‚f#‚A##ne #‚ÿ###Ü#############Î#############¾##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+ #2#####"# #################"# ###õ###,# ###Symbol#####)##p###)##fA¡#À#Ê30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 512 3 -1 roll exch div scale currentpoint translate 64 -2216 translate 251 2600 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (2) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 465 373 384 0 2632 16 sqr 462 2600 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) show 684 2600 moveto 384 /Times-Italic f1 (fA) show end #¿¡#d#+MATH#####É##### ######‚2#### #‚p#‚f#‚A##ne #‚ÿ###‚#############|#############l##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,# ###Symbol###### .##ÿ###+# #p,#####Times###)##fA¡#À#¯30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 448 3 -1 roll exch div scale currentpoint translate 64 -2201 translate -3 2521 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) show 219 2521 moveto 384 /Times-Italic f1 (fA) show end #¿¡#d##MATH#####Ö##### ##‚p#‚f#‚A## - #‚ÿto###`#############R#############B##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times### ##.##ÿ###+###1*##3#####"######,# ###Symbol###### (#####p###)##fA¡#À#W30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 864 div 480 3 -1 roll exch div scale currentpoint translate 64 -2198 translate 19 2411 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 224 /Times-Roman f1 (1) show 23 2643 moveto (3) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 2451 moveto 151 0 rlineto stroke 212 2550 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (p) show 434 2550 moveto 384 /Times-Italic f1 (fA) show end #¿¡#d#/MATH#####w##### ########‚1###‚3## #‚p#‚f#‚A##3 #‚ÿ###-##Equation############### ##‚2#‚p#‚f#‚A#####7##Equation############### ######‚3#### #‚p#‚f#‚A#####7##Equation############### ######‚2#### #‚p#‚f#‚A#####*##Equation############### ##‚p#‚f#‚A#####;##Equation############### ########‚1###‚3## #‚p#‚f#‚A##The magnitude of the EarthÕs gravitational force on a point mass is F(r) where r is the distance from the EarthÕs center to the point mass. Assume the Earth is a homogeneous sphere of radius R.What is #?328421Suppose there is a very small shaft in the Earth such that the point mass can be placed at a radius of R/2. What is #?8421/21/4 ###4#########$###&###########$#########$##¡#d##·xpr###### ## #‚## #######$#####"###$## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #F##)##(##)##R##)##)##(#####F##)##()##2##)##R##)##)#####"# ####¡#À##30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1152 div 960 3 -1 roll exch div scale currentpoint translate 64 -2215 translate 139 2520 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (F) show 386 2520 moveto 384 /Times-Roman f1 (\() show 546 2520 moveto 384 /Times-Italic f1 (R) show 784 2520 moveto 384 /Times-Roman f1 (\)) show 38 3052 moveto 384 /Times-Italic f1 (F) show 285 3052 moveto 384 /Times-Roman f1 (\() show 422 3052 moveto 384 /Times-Roman f1 (2) show 648 3052 moveto 384 /Times-Italic f1 (R) show 886 3052 moveto 384 /Times-Roman f1 (\)) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 2660 moveto 1027 0 rlineto stroke end #¿¡#d#9MATH###-####### #######‚F#‚(#‚R#‚)###‚F#‚(#‚2#‚R#‚)####ub #‚ÿ###ª#######,#####‚#########,###‚#####,# ###¡#d##·xpr####### ## #‚## #####,#######"#,#### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #F##)##(##)##R##)##)##(#####F(#####R##*##2#####"######,# ###Symbol###.######(#####æ *#è (#####ö *##ø "#####¡#À#M30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 1408 3 -1 roll exch div scale currentpoint translate 64 -2215 translate 124 2520 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (F) show 371 2520 moveto 384 /Times-Roman f1 (\() show 531 2520 moveto 384 /Times-Italic f1 (R) show 769 2520 moveto 384 /Times-Roman f1 (\)) show 38 3232 moveto 384 /Times-Italic f1 (F) show 531 2993 moveto (R) show 537 3525 moveto 384 /Times-Roman f1 (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 8 th 487 3133 moveto 293 0 rlineto stroke 288 3055 moveto 384 /Symbol f1 (\346) show 288 3454 moveto (\350) show 792 3055 moveto (\366) show 792 3454 moveto (\370) show 16 th 0 2660 moveto 996 0 rlineto stroke end #¿¡#d#IMATH###=#T##### #######‚F#‚(#‚R#‚)###‚F###########‚R###‚2####‚(#‚)##### 3 #‚ÿ###E##Equation#########-##### #######‚F#‚(#‚R#‚)###‚F#‚(#‚2#‚R#‚)#######U##Equation#########=##### #######‚F#‚(#‚R#‚)###‚F###########‚R###‚2####‚(#‚)#####Two wedges, each of mass m, are placed next to each other on a flat floor. A cube of mass M is balanced on the wedges as shown below. Assume no friction between the cube and the wedges, but a coefficient of static friction m < 1 between the wedges and the floor. What is the largest M that can be balanced as shown without motion of the wedges?####EAll M will balance.## %###Æ#############¸#############¨##########¡#d##·xpr### ### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #m##+###2#####"###################"#####õ###"#####¡#À#¿30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 928 3 -1 roll exch div scale currentpoint translate 64 -3754 translate 128 4059 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 283 4591 moveto 384 /Times-Roman f1 (2) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 465 373 384 32 4623 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 4199 moveto 529 0 rlineto stroke end #¿¡#d#-MATH###!###### #######‚m#######‚2########g #‚ÿ#####H#############: #############*##########¡#d##·xpr### ### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,# ###Symbol###### .##ÿ###+# #m,#####Times###)##m##+###2#####"###################"#####õ###"# ####¡#À#+30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 672 div 928 3 -1 roll exch div scale currentpoint translate 64 -3754 translate 20 4059 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (m) show 253 4059 moveto 384 /Times-Italic f1 (m) show 295 4591 moveto 384 /Times-Roman f1 (2) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 465 373 384 44 4623 16 sqr /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 4199 moveto 553 0 rlineto stroke end #¿¡#d#0MATH###$####### #######‚m#‚m#######‚2######## 1 #‚ÿ###‚#############‚#############| ##########¡#d##·xpr###### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,# ###Symbol###### .##ÿ###+# #m,#####Times###)##m##(#####1###)##-##) #m#####"#####¡#À#‚30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 896 div 960 3 -1 roll exch div scale currentpoint translate 64 -3754 translate 142 4059 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (m) show 375 4059 moveto 384 /Times-Italic f1 (m) show -10 4591 moveto 384 /Times-Roman f1 (1) show 233 4591 moveto 384 /Symbol f1 (-) show 522 4591 moveto 384 /Symbol f2 (m) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 4199 moveto 796 0 rlineto stroke end #¿¡#d#-MATH###!#z##### #######‚m#‚m###‚1#‚-#‚m####g #‚ÿ#####Ð#############Â#############²##########¡#d##·xpr###### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+# #2,# ###Symbol#####)##m###)##m##(#####1###)##-##) #m#####"# ####¡#À#»30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 896 div 960 3 -1 roll exch div scale currentpoint translate 64 -3754 translate 41 4059 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (2) show 243 4059 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (m) show 476 4059 moveto 384 /Times-Italic f1 (m) show -10 4591 moveto 384 /Times-Roman f1 (1) show 233 4591 moveto 384 /Symbol f1 (-) show 522 4591 moveto 384 /Symbol f2 (m) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 4199 moveto 796 0 rlineto stroke end #¿¡#d#0MATH###$#6##### #######‚2#‚m#‚m###‚1#‚-#‚m#### 1 #‚ÿ###9##Equation#########!##### #######‚m#######‚2###########<##Equation#########$##### #######‚m#‚m#######‚2###########9##Equation#########!##### #######‚m#‚m###‚1#‚-#‚m#######<##Equation#########$##### #######‚2#‚m#‚m###‚1#‚-#‚m####1.Find the gravitational field for all values of r due to a very long hollow circular cylinder of inner readius a and outer radius b. The density of the material is r.. 2Consider a thin wire circle of total mass M and radius a is lying in the x - y plane and centered at the origin. Calculate the gravitational field on the axis of the circle (z-axis) if the mass per unit length around the wire is: As always, show your work!# ##÷###ø###‚###‚###‚###‚###Î###Ï###Û###Ü###Ý###Þ###ß###ì###þ###ÿ#####s##¤##¾##¿##À##Á##°###´Z##´\##´e##´l##´m##´q##´r##´s##´t##´‚##´‚##¸###Àz##À{##À| ##À‚##À‚##À‚##À‚##À‚##À‚##À‚##À‚##Ç###Ç###Ç7##Ç8##ÇW##ÇX##ÇY##ÇZ##Ç[##Ðø##Ðý##Ñ###Ñ ###Ñ"##т##т##т##т##т##т##т##ÒG##ÒH##Òi##Òj##Òm##Òõ##Òö##Ù&##Ù¦##Ù§##Ù©##Ùª# #Ùá##Ùâ##Ùã##Ùä##ڂ##ڂ##àñ##àý##àþ##á ##á###á###á ##áf##ág##áh##ái####ù#óó######í###óó#####æ####ó#####ß##ó############Ù#Ò###ó##### ##Ë##ó############ #B###ø####ڂ# #@########Òø# # ######### #@########Ç\# #@########À‚###@########¸ #@###########B###ê####°#P#############,###############,#ð#######,##¡#d##·xpr#### ### ## #‚## #######,#####"###,## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #M##(#####4##)##a#####"######(#####sin,# ###Symbol#####)##f"##### "###*# ¡#À#Ü30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1408 div 896 3 -1 roll exch div scale currentpoint translate 64 44 translate 67 261 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (M) show 28 793 moveto 384 /Times-Roman f1 (4) show 236 793 moveto 384 /Times-Italic f1 (a) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 401 moveto 450 0 rlineto stroke 543 500 moveto 384 /Times-Roman f1 (sin) show 1040 500 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (f) show 522 184 moveto 0 434 rlineto stroke 1301 184 moveto 0 434 rlineto stroke end #¿¡#d#@MATH###4#¥##### #######‚M###‚4#‚a########‚s#‚i#‚n#‚f##‚a#‚b### d #‚ÿ###L##Equation#########4##### #######‚M###‚4#‚a########‚s#‚i#‚n#‚f##‚a#‚b### 3#in equilibrium as shown# ###Ð###‚###õ##ÿÿ#####÷##ÿÿ########ÿÿ########ÿÿ########ÿÿ########ÿÿ########ÿÿ#### # ##ÿÿ#####?P#ÿÿ### #####################Ý###Þ###ß###à###á###â###ã###ä###å###æ###ç###è###é###ê###ë## #ì###í###î###ï###ð###ñ###ò###ó###ô###õ###ö###÷###ï###ð###ò###ó###########! ###"###e###f###h###i###¹###º###ñ###ò###########,###-###Y###Z###‚###‚### ´###µ###¶###L###M###X###Y###j###k###|###}###‚##´h##´i##´j##´k##´l##´m##´n##´o## ´r##´t##´‚##À| ##À‚##À‚##À‚##À‚##À‚##À‚##À‚##À‚##À‚##Ç8##ÇX##ÇZ##Ç[##Ñ###т##т##т##т##Òã##Òä##Òå# #Òæ##Òç##Òö##Ù¦##Ù§##Ùª##Ùã##Ùå##Ùæ##Ùç##ڂ##àþ##á###ág##áh##ái##áj##á‚##á‚##á‚## á¤##á¥##á¦##á§##á¨##áª##á«##á¬##á##á®ôôôôïááïÒïïïôÃôôô·ôïïïÒïïïï輸ááááï‚ïïïáááïïïïááïïïïôôôôôôôïôï###$‚# ##Ð#þ‚#ÿ#####h#Ð#8############h#þ‚####h#Ð#############h#þ‚#######¸#######P####h#þ ‚####ÿø$#´####¸############h#þ‚####ÿø$#´####¸#####$‚# ##Ð#þ‚#ÿ#####h#Ð#8################$‚# ##h#þ‚#######¸#>###z#######¥#ï###l#########¥#ï#\ÿÿÿÿ#¤#î###ÿ #ÿÿÿÿÿÿ##ÿÿ###î###¤#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚### ÿôÿô### ÿóÿó#‚#â######### #########¡#¶################# # ### #½###########¡#¶#############"# ## ÷###########¡#¶#############"# ## ÷###########¡#¶#############"# #$ ÷###########¡#¶#############"# #6 ÷###########¡#¶#############"# #H ÷###########¡#¶#############"# #Z ÷###########¡#¶#############"# #l ÷###########¡#¶#############"# #~ ÷###########¡#¶#############"# #‚ ÷###########¡#¶#############"# #¢ ÷###########¡#¶#############"# #´ ÷###########¡#¶###########¡#¶#######¡#Ø####‚##¡#Ö####‚## #×#¡#´##### ############"# #6#~# #µ#¡#Ö####‚## #×# #Ù###############¡#¶#######¡#Ö### #‚##¡#Ø####‚## #¾# #Ù#######"# #6###"###6###"#-#6###"#? #6###"#Q#6###"#c#6###"#u#6###"#‚#6###¡#Ø####‚## #¿# #Ù# #×# #"D‚#"D‚#1#Q#l#v#‚###########¡#¶############# ÿÿÿÿÿÿÿÿ#8###########¡#¶####### #‚#q###^#Â#h#Ï#c#Ï#h#Â#c#Â#^#Â#c#Ï#######p ###^#Â#h#Ï#c#Ï#h#Â#c#Â#^#Â#c#Ï#"#c#‚)## #‚###########¡#¶#############"# #6HH#¡#‚# ############# ÿ‚ÿ‚### #######-#######"####### #‚#¡#‚##ÿö}#ÿÿ‚‚#,# ## Helvetica###### #.#######* #2 kg# #‚# #‚#####¡#‚# ############# ÿÓ#%### #######/##### #‚#¡#‚##ÿöX#ÿÿº‚#)##F = 10 N# #‚# #‚#####¡#‚# ############# #_#"### ############# #‚#¡#‚##ÿî;###M##,# ###Symbol#########*##q## #‚# #‚##### #‚#ÿhIf the stone is mov#######d#‚#######Ä###V###########Ä#Fÿÿÿÿ#b#####ÿ #ÿÿÿÿÿÿ##ÿÿ#######b#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#b##############ÿÿÿÿÿÿ# #########a##ÿ##¯#####E#####¡#¶##################### ÿÿÿÿÿÿÿÿ#h###E# #‚### ############¡#¶####### #‚#q###Ù##æ###æ###Ù##Ù###Ù###æ#########p###Ù##æ###æ###Ù# #Ù###Ù###æ### #Ù####### #‚###########¡#¶####### #‚#q#####ä###ñ##ñ###ä##ä###ä# #ñ#######p#####ä###ñ##ñ###ä##ä###ä##ñ# #####ä# #‚#¡#‚# ############# ÿý##### ###############"####### #‚#¡#‚##ÿò ####‚#,# ## Helvetica########.#######*##O## #‚# #‚#####¡#‚# ############# ##ÿ ##### #‚#¡#‚##ÿñé###‚#)##y## #‚# #‚#####¡#‚# ############# ÿ$#Ì##### #‚#¡#‚##ÿò##ÿÿé‚#)##x## #‚# #‚#####¡#‚# ############# ##ÿÒ### ############# #‚#¡#‚##ÿöI#ÿÿ×### #(# ###Track# #‚# #‚##### #‚# #‚#ÿ [Hint: Free-body diagrams would help a great deal!]balanced on the wedges as shown#### ‚#‚#####o#### ‚#########o## ~ÿÿÿÿ#‚#U###ÿ #ÿÿÿÿÿÿ##ÿÿ###U###‚#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#‚#U######### #########¡#¶################# #‚###‚#R###########¡#¶#############"#‚##H¸###########¡#¶##############HH######### ##¡#¶##############H¸###########¡#¶##############HH##ÿÿÿÿÿÿ# #########1###}#[#Ø###########¡#¶##################### ÿÿÿÿÿÿÿÿ#8#¡#‚# ############# ÿ]ÿÛ### ###############"####### #‚#¡#‚##ÿõ,‚ÿÿ¬##,# ## Helvetica########.#######*##M## #‚# #‚#####¡#‚# ############# #FÿÂ### ############# #‚#¡#‚##ÿöj###f### #(# ###m# #‚# #‚#####¡#‚# ############# ÿp####### #‚#¡#‚##ÿöi#ÿÿö##)##m## #‚# #‚##### #‚# #‚##### ÿ¼ÿ¢#G#ø#########¡#¶#############"#################¡#¶###############û# #‚# #‚# ÿp##### ÿ¼ÿ##G#h#######"###########¡#¶#############"#################¡#¶###############û # #‚# #‚# ÿÖÿÍ### ÿ‚þè###>#######"#######¡#‚# ############# ÿÿÿü### ############# #‚#¡#‚##ÿö#####‚#)##45# #‚# #‚#####¡#‚# ############# ÿó##### ############# #‚#¡#‚##ÿø#‚###### #(#####o# #‚# #‚##### #‚# #‚# #l##### ÿ‚ÿF###‚#####¡#‚# ############# ÿÿÿû### ############# #‚#¡#‚##ÿö ####‚## #*##45# #‚# #‚#####¡#‚# ############# ÿó##### ############# #‚#¡#‚##ÿø#‚###### #(#####o# #‚# #‚##### #‚# #‚# #@##### ÿ‚ÿx###Î#####¡#‚# ############# ÿÿÿü### ############# #‚#¡#‚##ÿö#####‚## #*##45# #‚# #‚#####¡#‚# ############# ÿó##### ############# #‚#¡#‚##ÿø#‚###### #(#####o# #‚# #‚##### #‚# #‚# #l##### ÿ‚ÿÖ###,#####¡#‚# ############# ##ÿü### ############# #‚#¡#‚##ÿö####‚## #*##45# #‚# #‚#####¡#‚# ############# ÿò##### ############# #‚#¡#‚##ÿø#‚###### #(#####o# #‚# #‚##### #‚# #‚#ÿthat that ow circular cylinder of inner rYou may assume that you are looking for the field at points that are far from the ends of the cylinder. 3 totalA modified AtwoodÕs machine is shown below. The table and pulley are both frictionless, but each block experiences a resistive force due to air resistance that has magnitude bv2, where b is a positive constant.#A. [5 points] Write down the equation of motion for block 1 (the hanging block) and the equation of motion for block 2.B. [5points] #####.#‚#####s##### #########s####ÿÿÿÿ#³#¯###ÿ #ÿÿÿÿÿÿ##ÿÿ###¯###³#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#³#¯######### ######### #"D‚#"D‚#1#8###B#k#####¡#¶############# ÿÿÿÿÿÿÿÿ#8##ÿÿÿÿÿÿ# #########Q###l#;#‚###########¡#¶##################### ÿÿÿÿÿÿÿÿ#X###########¡#¶#############"#9#j#ô##ÿÿÿÿÿÿ# #########1#####9#u###########¡#¶##################### ÿÿÿÿÿÿÿÿ#8###########¡#¶############# ###t###y###########¡#¶#############"#|#‚#±##ÿÿÿÿÿÿ# #########1#|#g#±############¡#¶##################### ÿÿÿÿÿÿÿÿ#8#¡#‚# ############# ÿÄÿì### ###############"####### #‚#¡#‚##ÿò#‚##H‚#,# ## Helvetica########.#######*##m##+###2# #‚# #‚#####¡#‚# ############# þ½ÿ‚##### #‚#¡#‚##ÿñ‚‚ÿÿ‚‚#(#####m#+###1# #‚# #‚##### #‚#ÿEliminate the tension force (due to the string) between the two equations of motion to find the single equation for the system: Two blocks are connected by a massless string to form aa#C. [15 points] Solve this equation of motion for the system to find the velocity of block 2 when it has traveled a distance d. Assume that the system starts from ###æ#########x###Ø###########x#È#######x##¡#d##·xpr####### ## #‚## #######x#####"###x## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #m####+###1### (# ##g,# ###Symbol#####)##-###) #2##)##bv####(###1#2#### +###=###) #(##)##m####+###1#### (# #U#+#####) #m####+ ##2# (# #l#)(###q#ú )##ú ##(# #q#x ¡#À#"30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3840 div 512 3 -1 roll exch div scale currentpoint translate 64 -967 translate -4 1351 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (m) show 250 1447 moveto 224 /Times-Roman f1 (1) show 367 1351 moveto 384 /Times-Italic f1 (g) show 639 1351 moveto 384 /Symbol f1 (-) show 929 1351 moveto 384 /Times-Roman f1 (2) show 1134 1351 moveto 384 /Times-Italic f1 (bv) show 1514 1181 moveto 224 /Times-Roman f1 (2) show 1760 1351 moveto 384 /Symbol f1 (=) show 2060 1351 moveto 384 /Times-Roman f1 (\() show 2204 1351 moveto 384 /Times-Italic f1 (m) show 2458 1447 moveto 224 /Times-Roman f1 (1) show 2671 1351 moveto 384 /Symbol f1 (+) show 2969 1351 moveto 384 /Times-Italic f1 (m) show 3241 1447 moveto 224 /Times-Roman f1 (2) show 3400 1351 moveto 384 /Times-Roman f1 (\)) show 3553 1332 moveto (\307) show 3629 1332 moveto (\307) show 3552 1351 moveto 384 /Times-Italic f1 (x) show end #¿¡#d#sMATH###g#`##### ##‚m#######‚1### #‚g#‚-#‚2#‚b#‚v########‚2## #‚=#‚(#‚m#######‚1### #‚+#‚m#######‚2### #‚)2‚x#####sl #‚ÿ###‚##Equation#########g##### ##‚m#######‚1### #‚g#‚-#‚2#‚b#‚v########‚2## #‚=#‚(#‚m#######‚1### #‚+#‚m#######‚2### #‚)2‚x#####rest at t=0.and show that is along the tablethat d is small enough that the block does not strike the pulley, and , inextensible Show your work. 4678910111010152530105 [Note: ] 5.6.7.influence8.9.m ########ÿÿ########ÿÿ#####¦##ÿÿ########ÿÿ#########ái##áj##áx##á‚##á‚##á‚##á‚##á‚##á ‚##á‚##á‚##á‚##á‚##á‚##á‚##á‚##á‚##á‚##á‚##á¡##á¢##á£##á¨##á©##á«##á¬##á##á®##á¶##á·##á¸##á¹##á»##á½##á¿##áÈ##áÊ##áÌ##áÍ##áÎ##áÏ##áÐ##áÑ##æ###æ9##æ: ##æ; ##æ<##æH##æN##æQ##æf##æp##æq##æ{##æ|##æÃ##æÄ##æÊ##æË##æÌ##ç###ç! ##ç"##ç###ç$##ç(##ç)##ç0##ç5##çJ##çK##çV##çW##ç‚##ç‚##çÙ##çÜ##çÝ##çß##çà##çá##çã# #çå######################################ü###öõî#######í#í#ç##ç##à#######Ù###Ò## ####### #B###ü####û # #B###þ####øÐ# #B###Ò####ò‚# # ###ü######@ #@########él##‚ #@################S##á®##á¸##áÍ##áÐ##áÑ##æ9##æ;##ç ##ç! ##ç###ç$##çÚ##çÛ##çÜ##çÝ##çÞ##çß##çà##çá##çâ##çã##çä##çå##è¦##è§##è¨##è©##èª##è« ##è¬##è##è®##è¯##è°##è±##è²##éh##éi##éj##þ‚##ÿ‚##ÿ‚##ÿÔ##ÿÕ##ÿÖ##ÿ×##ÿØûûûûòçÞÞÕÍ······ ·····¬¤¤¤¤¤¤¤¤¤¤¤¤Â¤ûû‚‚uuuuu####$‚# ##Ð#þ‚#ÿ### ####h#Ð#8### ###$‚######ð####h#Ð#8##ì#############h#þ‚#ÿ###ð####h#Ð#8##ì######$‚# ##h#þ‚## ###$‚###ÿL##Ð#þ‚#þè## ###$‚###ÿL##Ð#þ‚#þè## ########ÿL##Ð#þ‚#þè############h#þ‚##############h#þ‚############h#þ‚#þè# #########h#þ‚#ÿø$#´###ô#########!À#$‚@##########.Physics 301 (Fall 1997) Final Exam (December 17, 1997)#2. [20 points total] A thin uniform plate has sides a and mass m. If you choose the origin at the center of the plate and orient the x1- and x2-axes parallel to the sides of the plate, as shown, the inertia tensor is given by: #A. [7 pts] Find the angular momentum # about the #center of the plate when the plate is rotating with angular velocity # which is directed along a diagonal of the plate (as shown).B. [6 pts] Use SteinerÕs parallel axis theorem to find the moment of inertia (not the tensor) for rotation about an axis perpendicular to the page that passes through one corner of the plate.C. [6 pts] If you suspend the plate about a corner and let it oscillate in the plane of the plate, what oscillation frequency do you expect for small oscillations. Show your work.µ### ##‚#####¦#¬## ##########¦#¬#þÿÿÿÿ#Ü#è###ÿ #ÿÿÿÿÿÿ##ÿÿ###è###Ü#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#Ü#è######### ##### #‚#########¡#¶#################0#H###Ù#‚###########¡#¶####### #‚#q###‚#È#‚#Õ #‚#Õ#‚#È#‚#È#‚#È#‚#Õ#######p###‚#È#‚#Õ#‚#Õ#‚#È#‚#È#‚#È#‚#Õ# #‚#H#‚#È# #‚###########¡#¶####### #‚#q#####C###M###H###M###H###C###H#######p##### C###M###H###M###H###C###H#"#‚#H#‚# #‚###########¡#¶####### #‚#q#####Ì###Û###Û###Õ ###Ð###Ì###Û#######p#####Ì###Û###Û###Õ###Ð###Ì###Û#######"#<#‚2Î# #‚###########¡# ¶###########¡#¶#######¡#Ø####‚##¡#Ö####‚## #×#¡#´##### ############ #Ø###I#‚# #µ#¡#Ö####‚## #×# #Ù###############¡#¶#######¡#Ö####‚##¡#Ø####‚## #¾# # Ù#######"#Ø###ú#"#Ë# #ú#"#¿###ù#"#²#&#ù#"#¥#3#ú#"#‚#@#ú#"#‚#L#ù#"#‚#Y#ú#"#r#f#ú#"#e#s#ú#"#Y#‚#ù#"#L#‚#ý #¡#Ø####‚## #¿# #Ù# #×#¡#‚# ############# ÿlÿl### #######$#######"####### #‚#¡#‚##ÿñ‚#ÿÿ¦##,# ## Helvetica########.#######*##a/2## #‚# #‚#####¡#‚# ############# #f#]##### #‚#¡#‚##ÿò@###@##)##a/2## #‚# #‚#####¡#‚# ############# ÿöÿ±### ############# #‚#¡#‚##ÿñ‚###D##)##O## #‚# #‚#####¡#‚# ############# ÿ‚#w### ############# #‚#¡#‚##ÿé#‚ÿÿ¬##,# ###Symbol#########* #w## #‚# #‚##### # ##### ÿÿÿÿ#Ü#è#"#######¡#¶####### #‚#q#####¦###¯###¯###¦###¦###¦###¯#######p#####¦###¯ ###¯###¦###¦###¦###¯#"###‚ ## #‚# #‚#¡#‚# ############# ÿ2ÿj### ###############"####### #‚#¡#‚##ÿñ¡#ÿÿÚ#########(#####x####+ ¡#‚# ##1# #‚# #‚##### ############# #‚#‚##### #‚#¡#‚##ÿò ###<#####(#####x####+ ##2# #‚# #‚##### #‚#ÿ###F#######4#e###8#########4#e#(#### #4#e##¡#d##·xpr####### ## #‚## #####4#e#####"#4#e## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###I##+###ij,# ###Symbol###### (#####=#####(#####ma####(###(#2# (#&###12#####"######(###5#1)##0)##0(###5#0)##1)##0(#/#5#0)##0)##2###.######(###/ #æ *##è (###/#ç *##ç (##^#ö *!#ø (###^#÷ *##÷ ¡#À#N30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3232 div 1664 3 -1 roll exch div scale currentpoint translate 64 -689 translate 9 1617 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (I) show 136 1713 moveto 224 ns (ij) show 402 1617 moveto 384 /Symbol f1 (=) show 748 1378 moveto 384 /Times-Italic f1 (ma) show 1233 1209 moveto 224 /Times-Roman f1 (2) show 853 1910 moveto 384 ns (12) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 720 1518 moveto 683 0 rlineto stroke 1641 1041 moveto (1) show 2199 1041 moveto (0) show 2756 1041 moveto (0) show 1642 1617 moveto (0) show 2198 1617 moveto (1) show 2756 1617 moveto (0) show 1642 2193 moveto (0) show 2199 2193 moveto (0) show 2755 2193 moveto (2) show 1452 1083 moveto 384 /Symbol f1 (\346) show 1452 2196 moveto (\350) show 1452 1534 moveto (\347) show 1452 1905 moveto (\347) show 2950 1083 moveto (\366) show 2950 2196 moveto (\370) show 2950 1534 moveto (\367) show 2950 1905 moveto (\367) show end #¿¡#d#‚MATH###‚#‚##### ##‚I#######‚i#‚j### #‚=######‚m#‚a########‚2### ##‚1#‚2#################‚1###‚0###‚0###‚0###‚1###‚0###‚0###‚0###‚2####‚(#‚)###17 #‚ÿ @###O######### ###A########### #1####### ##¡#d##·xpr####### ## #‚## ####### ,## Ó#MT Extra# Ó# .##ÿ###*## #####"### ## #¾¡#À#currentpoint #¿"###### #¾(#####r ,#####Times#####(####L ¡#À#j30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 320 div 448 3 -1 roll exch div scale currentpoint translate 64 -2477 translate 63 2566 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /MT-Extra f1 (r) show 12 2893 moveto 384 /Times-Italic f1 (L) show end #¿¡#d##MATH#####!##### #2‚L#####t #‚ÿö###‚######## ###‚########## #r###### ##¡#d##·xpr####### ## #‚## ###### ,## Ó#MT Extra# Ó# .##ÿ###*# #####"## ## #¾¡#À#currentpoint #¿"###### #¾(#####r ,# ###Symbol#####(#####w ¡#À#ª30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 384 div 416 3 -1 roll exch div scale currentpoint translate 64 -3148 translate 77 3234 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /MT-Extra f1 (r) show -16 3500 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (w) show end #¿¡#d##MATH#####=##### #"‚w#####t #‚ÿ###‚##Equation#########‚##### ##‚I#######‚i#‚j### #‚=######‚m#‚a########‚2### ##‚1#‚2#################‚1###‚0###‚0###‚0###‚1###‚0###‚0###‚0###‚2####‚(# ‚)######'##Equation############### #2‚L########'##Equation############### #"‚w#####3. [25 points total] A particle of mass m rests on a smooth plane. The plane is raised to an inclination q at a constant rate a (q=0 at t=0), causing the particle to move down the plane. The Lagrangian L for this motion is given by: #A. [10 pts] What is the Hamiltonian H for this motion? Show your work.B. [15 pts] ##‚###Ù¤##‚###Ù§##‚#####çå##çæ##çé##çê##çñ##è###è2##èW##è‚##è‚##è¥##è¦##è³##è¶##è· ##è¾##è×##èÞ##èå##ég##éh##éi##éj##þ‚##þ‚##þ‚##þ±##þñ##ÿ‚##ÿ‚##ÿ‚##ÿ‚##ÿ‚##ÿÓ##ÿé### ####: ###<###=###>###? ###B###C###O###X###Z###a###d###g###t###‚###‚###‚###‚###‚###‚### ## ‚## ‚## ‚## ‚## ¹## º## Ò## î## ï#######L###M###‚###‚###‚###‚###‚#######y###z###‚###‚###0###1###H###L###‚###‚###‚### ‚###Ì###Ï###Ð###‚######################ú######ó####ú#####################úì##### ã##æ#####ú#########æ#####ú###### # ###ü##### #@#########Ð# #B###ø##### # #@########ZWhat are HamiltonÕs equations of motion? Show your work.C. [15 pts] Is E = H? Explain.B. [15 pts] Is either E or H conserved? Explain. ###X#########‚###J###########‚#: #######‚##¡#d##·xpr####### ## #‚## #######‚#####"###‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #L,# ###Symbol#####) #=#####(#####1*##2#####"######### (# ###m##)##((###+#ú ##(# #*#r ####(###/#2#### +###+##) #a#######(###H#2### +###r####(###R#2# +###)###)##-#####) #mgr##)##sin)##(#####)##a###)##t##)##)¡#À#O30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5088 div 512 3 -1 roll exch div scale currentpoint translate 64 63 translate 12 321 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 321 moveto 384 /Symbol f1 (=) show 669 182 moveto 224 /Times-Roman f1 (1) show 669 414 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 222 moveto 164 0 rlineto stroke 867 321 moveto 384 /Times-Italic f1 (m) show 1149 321 moveto 384 /Times-Roman f1 (\() show 1326 302 moveto (\307) show 1288 321 moveto 384 /Times-Italic f1 (r) show 1464 152 moveto 224 /Times-Roman f1 (2) show 1695 321 moveto 384 /Symbol f1 (+) show 1982 321 moveto /f2 {findfont matrix dup 2 .22 put makefont dup /cf exch def sf} def 384 /Symbol f2 (a) show 2271 152 moveto 224 /Times-Roman f1 (2) show 2400 321 moveto 384 /Times-Italic f1 (r) show 2576 152 moveto 224 /Times-Roman f1 (2) show 2735 321 moveto 384 /Times-Roman f1 (\)) show 2936 321 moveto 384 /Symbol f1 (-) show 3233 321 moveto 384 /Times-Italic f1 (mgr) show 3897 321 moveto 384 /Times-Roman f1 (sin) show 4354 321 moveto (\() show 4487 321 moveto 384 /Symbol f2 (a) show 4741 321 moveto 384 /Times-Italic f1 (t) show 4872 321 moveto 384 /Times-Roman f1 (\)) show end #¿¡#d#‚MATH###‚#h##### ##‚L#‚=#######‚1###‚2## #‚m#‚(2‚r###########‚2## #‚+#‚a########‚2## #‚r########‚2## #‚)#‚-#‚m#‚g#‚r#‚s#‚i#‚n#‚(#‚a#‚t#‚)##32 #‚ÿ###‚##Equation#########‚##### ##‚L#‚=#######‚1###‚2## #‚m#‚(2‚r###########‚2## #‚+#‚a########‚2## #‚r########‚2## #‚)#‚-#‚m#‚g#‚r#‚s#‚i#‚n#‚(#‚a#‚t#‚)###4. [25 points total] A block of mass M1 sits on a frictionless incline that makes an angle q with the horizontal and is connected by a massless, inextensible string to a block of mass M2 that hangs over a frictionless pulley, as shown. A. [20 pts] ‚##<###U‚##´h##á®##þ‚####### #,#[#r##########################!#"#####Ñ#Ó#Õ#Ö#Ú#Û#ö## Ó#(#)(#(‚(ö)#33.9#;Ï<#<s<t##ÿØ##ÿÙ##ÿÚ##ÿÛ##ÿÜ###: ###;###<###=###>###? ###@###A###B###c###d###e###f###g###‚###‚###‚###‚### ## ‚###‚###‚###1###2###3###4###5###6###7###8###9###: ###;###<###=###>###? ###@###A###B###C###D###E###F###G###H###I###J###K###L###M###N###O###P###Q###R###S ###ÏðððððððððàððððððððððððÛÌû°§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§‚######$‚# ##Ð#þ‚#ÿ######$‚# ##Ð#þ‚#ÿ## ########ÿ°##Ð#þ‚#ÿ####+#########$##############h#þ‚#ÿ#############ÿL##h#þ‚##Á#ÿü$ #´######################Ð#þ‚#ÿ### ####h#Ð#8#######$‚# ##Ð#þ‚#ÿ### ####h#Ð#8##>Use LagrangeÕs method of undetermined multipliers to find the tension in the string and the acceleration of the system. #(Note: All of us can find the tension and the acceleration fairly simply with NewtonÕs laws, so no points will be awarded for correct answers. Only the method used will be graded.)B. [5 pts] What is the minimum magnitude of M2 that will accelerate downward? Show your work or explain your reasoning. ###°#######w#ô###¢#########w#ô#‚ÿÿÿÿ#v#ó###ÿ #ÿÿÿÿÿÿ##ÿÿ###ó###v#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#v#ó######### ######ÿÿÿÿÿÿ##### #########Q###¿###Ò#####¡#¶##################### ÿÿÿÿÿÿÿÿ#X###########¡#¶#############"###Á#ô##ÿÿÿÿÿÿ# #########1#L#Ä#t#â###########¡#¶##################### ÿÿÿÿÿÿÿÿ#8###########¡#¶#############"#L#Ñ#¾###########¡#¶############# #U#####¿###########¡#¶#############"#]###õ###########¡#¶#############"#X###ô#### #######¡#¶#############"#R#)#ô###########¡#¶#############"#L#: #ô###########¡#¶## ###########"#F#K#õ###########¡#¶#############"#@#\#õ###########¡#¶#############" #: #m#õ###########¡#¶#############"#5#~#ô###########¡#¶#############"#/#‚#ô###### #####¡#¶#############"#)# #ô###########¡#¶#############"###±#õ#"±à±àÿÿ# ÿüÿÞ### ÿÝÿû#T#ï#######"#######¡#Ø####@##¡#Ö####@## #×#####"###&###¡#Ê###########¡#È#### #U#U#Y##ÿÿÿÿÿÿ# #########A#####%#J#####¡#¶##################### ÿÿÿÿÿÿÿÿ#H# #É# ¡#Ö####@## #×# #Ù# ###"### ÿÿÿÿ#v#ó#####"±à±àÿÿ#¡#Ö####@##¡#Ø####@## #¾# #Ù###########¡#¶########ÿÿÿÿÿÿ# #########q#######Q#P###E#: #P#Q###/#####E#"###E###¡#¶##################### ÿÿÿÿÿÿÿÿ#p#######Q#P###E#: #P#Q###/#####E#¡#Ø####@## #¿# #Ù# #×###########¡ #¶#############"#*#KxÖ#"±à±àÿÿ# ÿäÿÕ### ÿÔÿã#K#×#######"#######¡#Ø####@##¡#Ö####@## #×#####"# #####¡#Ê###########¡#È#####U#U#Y# #É#¡#Ö####@## #×# #Ù# ###+### ÿÿÿÿ#v#ó#####"±à±àÿÿ#¡#Ö####@##¡#Ø####@## #¾# #Ù#¡#Ø####@## #¿# #Ù# #×# ÿäÿÕ### ÿÔÿã#K#×#"#######¡#‚# ###U###U#Y##### ############# #‚#¡#‚##ÿþ####‚##,# ## Helvetica########.#######*##M##+###1# #‚#"######### ÿøÿû###.#"#######‚##ÿìÿú####ÿìÿú######ÿü################################‚###‚# #‚###À##‚À##‚À##Æ`##v`##4`##>0##‚###‚###‚###‚###################‚##ÿîÿú# #ÿþ# #ÿîÿú# ###################################‚###À###À###À###`###`###`###0###0###0######## ################################### #‚# ###+### ÿÿÿÿ#v#ó#######"±à±àÿÿ# ÿäÿÕ### ÿÔÿã#K#×#######"#######¡#Ø####@##¡#Ö####@## #×#####"# #####¡#Ê###########¡#È#####U#U#Y#"#######"####### #É#¡#Ö####@## #×# #Ù# ###+### ÿÿÿÿ#v#ó#####"±à±àÿÿ#¡#‚# ############# ÿ9ÿª### #######*#"####### #‚#¡#‚##ÿòa‚ÿÿÜ##(#####M#+###2# #‚# #‚##### #‚#ÿ3. [20 points total] mbedded_Object_20#Embedded_Object_21#Embedded_Object_3#Embedded_Object_2#Embedde d_Object_5#Embedded_Object###Ï###Ð###‚###‚###‚###‚###‚###‚###‚###‚###‚###1###2###3# ##ê###ë###ì###í###î###ï###ð###ñ###ò###ó###õ##$‚##$‚##$‚##$‚##$‚##$‚##$‚##$ ##$¡##$ ¢##$£##$¤##$¥##$¦##$§##$¨##$©##$ª##$«##$¬##$##$®##$¯##.l##.zûòûêêêêêêêáÖûɼ¼¼¼¼¼¼¼¼¬‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚ûûû####################### ##############$‚# ##Ð#ý0#ÿ############Ð#ý0#ÿ## #########h#þ‚####h#Ð#8####### È#Ð##h#þ‚#ÿø$#´####h#Ð#8### ###$‚# ##h#þ‚####h#Ð#8### # ÿ$‚# ##h#þ‚####h#Ð#8### ###$‚# #ÿL##Ð#þ‚#ÿ#############h#þ‚#þè#####$‚# ##h#þ‚############h#þ‚#ÿ###########1Suppose that it is possible to measure the location of the center of mass and the period of rotation of a double star system. Find formulas for the masses, m1 and m2, of the stars in terms of these data. The orbits may be considered to be circular. (Note: Knowing the location of the center of mass means that we know the distances r1 and r2 of each of the stars from the center of mass.) 2. [20 points] A red ball of mass m and speed u1 = 6##x###‚###‚###Ñ###Ò###Ô###Õ###%###&###]###^###s###t###‚#####‚###‚ ###‚###‚#######‚###‚###¥###¦###N###O###Q###R###X###Y###]###e###‚###‚###‚###‚###‚###‚ ###‚###¢###Â###Ã###Æ###Ç##"###"E##"F##"h##"i##"j##"‚##"‚##"²##"³##"Å##"Í##"ú### ###0###1###2###3###6###B###\###]###d###l###x###‚###‚###ª###é###ñ###ó###ô###õ###ø# #$###$!##$ +##$6##$8##$9##$;##$<##$B##$C##$‚##$‚##$‚##$‚##$®##$¯##.k##.l##.m##.s##.t##.w##.x ##.y###ú#ô#ô###ô#ô###########ô##ú###î#í####è#í#######á####í####Ú#######ô#ô###### ##ú#íííííí## #@########'8# #B###ô####$°############‚ # ######### # ###ü##### #@########V0 m/s undergoes a glancing blow with a stationary green ball of mass 4m. If the red ball scatters by 90o in the center of mass frame, and if the coefficient of restitution is 0.75, find the final velocity (magnitude and direction) of the green ball in the lab frame after the collision. Show your work.1. [5 points] Write the general form of # for a general non-inertial frame (not necessarily the earth centered frame.) Sorry that this problem offends your sense of justice Trevor.#2. [10 points] Four identical point masses of mass M lie in the x1-x2 plane as shown in the diagram at the right. Calculate the inertia tensor. Show your work.####‚ #############w#############g##########¡#d##·xpr####### ## #‚## ########,## Ó#MT Extra# Ó# .##ÿ###*## #####"###### #¾¡#À#currentpoint #¿"###### #¾(#####r ,#####Times#####(####F ##+###eff¡#À#‚30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 608 div 608 3 -1 roll exch div scale currentpoint translate 64 63 translate 74 26 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /MT-Extra f1 (r) show 6 353 moveto 384 /Times-Italic f1 (F) show 231 450 moveto 224 ns (eff) show end #¿¡#d#-MATH###!#;##### #2‚F##########‚e#‚f#‚f#####g #‚ÿ#‚####ú#‚#####Ð#‚###ì#########Ð#‚#Üÿÿÿÿ#5#Í###ÿ #ÿÿÿÿÿÿ##ÿÿ###Í###5#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚### ÿôÿô### ÿóÿó#)#Á######### #########¡#¶####### #‚#####q#########'###"###'###"#######"#######p#########'###" ###'###"#######"# ###"###"# #‚###########¡#¶####### #‚#q###ü#‚###‚###‚###‚###‚#ü#‚###‚#######p###ü#‚## #‚###‚###‚###‚#ü#‚###‚#"###"h## #‚#Q# %#f#.#o###########¡#¶#############X#Q#ü#f###o###########¡#¶#############X#Q# %###.#'###########¡#¶#############X#Q#ü#####'###########¡#¶#############X#¡#‚# ############# ÿÿÿà### ###############"####### #‚#¡#‚##ÿò)#####,# ## Helvetica########.#######*##3a# #‚# #‚#####¡#‚# ############# ÿîÿ###### #‚#¡#‚##ÿò ######)##O## #‚# #‚#####¡#‚# ############# ÿÿû##### #‚#¡#‚##ÿò####r##)##a## #‚# #‚#####¡#‚# ############# ÿÌ# ### ############# #‚#¡#‚##ÿò #ÿÿ¦##)##x#####+ ##1# #‚# #‚#####¡#‚# ############# #p#ý##### #‚#¡#‚##ÿò ###6#####(#####x####+ ##2# #‚# #‚##### #‚#ÿ###9##Equation#########!##### #2‚F##########‚e#‚f#‚f#####2Spring8May482Spring8May48#A capital letter D is formed from wire having uniform line density l. It is then hung on the wall with a frictionless nail that passes through the pivot point P, as shown. 2Spring8May48###Ó###Ô###Õ###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚###‚# ##‚###‚###‚###‚###‚###‚###‚### ###¡### ¢###£###¤###¥###¦###§###¨###©###K###L###M###N###O###P###Q###R###S###T###U####### .y##.z##.{##.‚##.‚##.‚ ##.‚##.‚##.‚##.‚##/5##/6##/7##/=##/>##/A##/B##/C##2###86##87##88##8‚##8‚##8ß##8à## 8î##8ï##9I##9J##9K##9Q##9R##9U##9V##9W##9j##9k##9l##9r##9s##9v##9w##9x##9y##9z## 9‚##9‚##9‚##9‚##9‚##9‚##9‚##<###C: ##C;##C<##C| ##D###D###D<##D‚##D‚##D‚##D‚##D‚##D‚##D‚##D‚##D ##D¶##D·##M]##M^##M`##N###N‚##N‚##N‚ ##N‚##N ¢##N£#þþþþþþø###þþþþþþòò##########þþþþþþ##þþþþþþ#þþþþþþì#òò##########å##Þ###ò### ###×#Ð####### #B###è####U # #B###è####Nü# #B###ú####HÌ# #B###ü####D¸###@########< #@###########@########2#‚#Q###6#‚#####y#¼###(#########y#¼##ÿÿÿÿ#³#####ÿ #ÿÿÿÿÿÿ##ÿÿ#######³#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚### ÿôÿô### ÿóÿó#§########### ##### #‚# ##ÿë### ÿÞÿó#‚########ÿÿÿÿÿÿ##### #########aÿ‚###‚###Z#Z#####¡#¶##################### ÿÿÿÿÿÿÿÿ#h#Z#Z##ÿÿÿÿÿÿ# #########i#´#Z###########¡#¶##################### ÿÿÿÿÿÿÿÿ#h#´#Z# #‚# ####### ÿóÿó#§#############¡#¶############# #######ý#Q###z###‚###########¡#¶#############X#¡#‚# ############# ÿ‚ÿÿ### ###############"####### #‚#¡#‚##ÿò #ÿÿ‚‚#,# ## Helvetica########.#######*##P## #‚# #‚#####¡#‚# ############# ÿÒÿ´##### #‚#¡#‚##ÿòV#ÿÿ±‚#)##R## #‚# #‚##### #¦#M### ÿóÿó#§###"### ###¡#¶####### #‚#q###c#É#p#Ö#p#Ö#j#É#g#Í#c#Ð#p#Ö#######p###c#É#p#Ö#p#Ö#j#É#g#Í#c #Ð#p#Ö#"###‚LL# #‚# #‚#ÿFind the moment of inertia of the D about this pivot. Show your work.B. [10 pts] Find the location of the center of mass of the D. Show your work.C. [13 pts] Find the frequency of oscillation for small oscillations. Show your work.2Spring8May48[30 points total] 2Spring8May48 2Spring8May48###$‚# ##$‚# ##$‚####$‚# ##$‚# ##$‚# ##$‚# ##$‚# ##$‚# ##$‚# ##$‚# ##$‚####$‚####$‚####$‚####$‚####$‚####$‚####$‚####$‚####.z##.‚##/6##87##88##8‚##8‚## 8‚##8‚##8‚##8‚##8‚ ##8‚##8‚##8‚##8‚##8‚##8‚##8‚##8Ü##8Ý##8Þ##8ß##8à##8á##8â##8ã##8ä##8å##8æ##8ç##8è##8 é##8ê##8ë##8ì##8í##8î##8ï##8ð##8ñ##8ò##9J##9k##9y##9‚##C;##D###D###D! ##D<##D=##D‚##D‚##D ##D·##N‚##N‚ñìã×ÌÌÌÌÌÁÁÁÁÁÁÁÁÁÁÁÁÁÁÌÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁìììì쵩© ©©©ì‚ì‚©###########h#þ‚#þÀ####¸######$‚# ##h#þ‚#######¸######$‚####h#þ‚#þÀ####¸######$‚####h#þ‚#þÀ####¸## ###$‚# #ÿ°##Ð#þ‚#ÿ### ########ÿ°##Ð#þ‚#ÿ#############h#þ‚#######¸############h#þ‚#ÿ#################### ##h#þ‚####ÿø$#´####¸#9###: #######‚#è###,#########‚#è##ÿÿÿÿ#‚#ç###ÿ #ÿÿÿÿÿÿ##ÿÿ###ç###‚#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚### ÿôÿô### ÿóÿó#‚#Û######### #########Q###‚###¡#####¡#¶#############X#Q#g#‚#m#¡###########¡#¶#############X#Q# ##################¡#¶#############X#Q#f###l#############¡#¶#############X#¡#‚# ############# ÿÿÿç### ####### ########### #‚#¡#‚##ÿö ######,# ## Helvetica###### #.#######* #a## #‚# #‚#####¡#‚# ############# ÿþÿ«### ############# #‚#¡#‚##ÿúr#########(#####O# #‚# #‚#####¡#‚# ############# ÿì#m### ####### ##### #‚#¡#‚##ÿö######## #*##x### #)##2## #‚# #‚#####¡#‚# ############# ÿXÿ‚##### #‚#¡#‚##ÿöw#ÿÿÅ### #(# ###x## #)##1## #‚# #‚##### #¿#p### ÿóÿó#‚#Û#"# #####¡#¶####### #‚#q###e#µ#o#Â#j#Â#o#µ#j#µ#e#µ#j#Â#######p###e#µ#o#Â#j#Â#o#µ#j#µ #e#µ#j#Â# #j##j#µ# #‚###########¡#¶####### #‚#q########################### #######p############################"#j##¤# #‚#¡#‚# ############# ÿhÿ‚### ###############"####### #‚#¡#‚##ÿöw#ÿÿ¡‚## #(# ###2a## #‚# #‚##### #‚#ÿ Two observers, O and OÕ, observe two events, A and B. The observers have a constant relative speed of 0.8c. In units such that the speed of light is 1, observer O obtained the following coordinates: Event A: x=3 y=3 z=3 t=3 Event B: x=5 y=3 z=1 t=5 What is the length of the space-time interval between these two events, as observed by OÕ?1#23#Explain your reasoning Î###############################õ##########¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+ #2#####"# #################"# ###õ###¡#À#(30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 576 div 448 3 -1 roll exch div scale currentpoint translate 64 43 translate 251 341 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (2) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 465 373 384 0 373 16 sqr end #¿¡#d#!MATH#####ø##### ######‚2###### r #‚ÿ ###4#############&########################¡#d##·xpr####### ## #‚## #############"###### #¾¡#À#currentpoint #¿"###### #¾,#####Times#### .##ÿ###+# #2)##3#####"# #################"# # #õ###¡#À#B30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 768 div 480 3 -1 roll exch div scale currentpoint translate 64 43 translate -11 341 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Roman f1 (2) show 449 341 moveto (3) show /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 443 378 384 203 378 16 sqr end #¿¡#d#$MATH#####µ##### ##‚2#####‚3######ll #‚ÿ###-##Equation############### ######‚2#########0##Equation############### ##‚2#####‚3###### 5.A cylindrical tube of mass M can slide on a horizontal wire. Two identical pendulums, each of mass m and length b, hang from the ends of the tube, as shown. For small oscillations of the pendulums in the plane of the paper, the eigenfrequencies of the normal modes of oscillation of this system are 0, # , and #####Descibe (qualitatively) the normal mode associated with this third eigenfrequency.#####"####### #C############# #C####### #C##¡#d##·xpr### ### ## #‚## ##### #C#####"# #C## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g##)##(##)##M,# ###Symbol#####)#+###) #2##)##m##) #)##(#####bM#####"### 5#"############# #####"#####ä#7#¡#À#Ï30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 2144 div 1024 3 -1 roll exch div scale currentpoint translate 64 -596 translate 291 997 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 485 997 moveto 384 /Times-Roman f1 (\() show 642 997 moveto 384 /Times-Italic f1 (M) show 1061 997 moveto 384 /Symbol f1 (+) show 1352 997 moveto 384 /Times-Roman f1 (2) show 1562 997 moveto 384 /Times-Italic f1 (m) show 1851 997 moveto 384 /Times-Roman f1 (\)) show 864 1529 moveto 384 /Times-Italic f1 (bM) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 1137 moveto 1730 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 2024 909 384 0 1565 16 sqr end #¿¡#d#BMATH###6#Ð##### ###########‚g#‚(#‚M#‚+#‚2#‚m#‚)###‚b#‚M########iv #‚ÿ#####´####### #####¦######### ###‚##### ####¡#d##·xpr### ### ## #‚## ##### #######"# #### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g(### #b#####"### ##"############# #####"#####ä###¡#À#30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 640 div 1024 3 -1 roll exch div scale currentpoint translate 64 -1737 translate 291 2138 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 287 2670 moveto (b) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 2278 moveto 241 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 535 909 384 0 2706 16 sqr end #¿¡#d#-MATH###!#B##### ###########‚g###‚b########g #‚ÿ###®####### #7### ######### #7#‚##### #7##¡#d##·xpr### ### ## #‚## ##### #7#####"# #7## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g(### #b#####"### ##(# ###M,# ###Symbol#####)#+#####) #m(#####M"#### #"############# #####"#####ä#+#¡#À#_30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 1024 3 -1 roll exch div scale currentpoint translate 64 -2925 translate 291 3326 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 287 3858 moveto (b) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 3466 moveto 241 0 rlineto stroke 608 3326 moveto (M) show 1027 3326 moveto 384 /Symbol f1 (+) show 1325 3326 moveto 384 /Times-Italic f1 (m) show 938 3858 moveto (M) show 567 3466 moveto 1058 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1657 909 384 0 3894 16 sqr end #¿¡#d#BMATH###6#Î##### ###########‚g###‚b########‚M#‚+#‚m###‚M########iv #‚ÿ¾###-####### ############### ######### ####¡#d##·xpr### ### ## #‚## ##### #######"# #### #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g(### #b#####"### ##(# ###m*##M"#### #"############# #####"#####ä###¡#À##30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1120 div 1024 3 -1 roll exch div scale currentpoint translate 64 -2925 translate 291 3326 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 287 3858 moveto (b) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 3466 moveto 241 0 rlineto stroke 630 3326 moveto (m) show 608 3858 moveto (M) show 567 3466 moveto 399 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 998 909 384 0 3894 16 sqr end #¿¡#d#<MATH###0#9##### ###########‚g###‚b########‚m###‚M########11 #‚ÿ######®####### #7### ######### #7#‚##### #7##¡#d##·xpr### ### ## #‚## ##### #7#####"# #7## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g(### #b#####"### ##(# # #m(#####M,# ###Symbol#####)#+#####) #m"#### #"############# #####"#####ä#+#¡#À#_30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1760 div 1024 3 -1 roll exch div scale currentpoint translate 64 -2925 translate 291 3326 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 287 3858 moveto (b) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 3466 moveto 241 0 rlineto stroke 960 3326 moveto (m) show 608 3858 moveto (M) show 1027 3858 moveto 384 /Symbol f1 (+) show 1325 3858 moveto 384 /Times-Italic f1 (m) show 567 3466 moveto 1058 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1657 909 384 0 3894 16 sqr end #¿¡#d#BMATH###6#î##### ###########‚g###‚b########‚m###‚M#‚+#‚m########iv #‚ÿ#####ä####### #>###Ö######### #>#Æ##### #>##¡#d##·xpr### ### ## #‚## ##### #>#####"# #>## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #g(### #b#####"### ##(# ###m(#####M,# ###Symbol#####)#+###) #2##)##m"####'#"############# #####"#####ä#2#¡#À#‚30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1984 div 1024 3 -1 roll exch div scale currentpoint translate 64 -2925 translate 291 3326 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (g) show 287 3858 moveto (b) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 262 3466 moveto 241 0 rlineto stroke 1061 3326 moveto (m) show 608 3858 moveto (M) show 1027 3858 moveto 384 /Symbol f1 (+) show 1318 3858 moveto 384 /Times-Roman f1 (2) show 1528 3858 moveto 384 /Times-Italic f1 (m) show 567 3466 moveto 1261 0 rlineto stroke /sqr { 3 index div /thick exch def gsave translate dup dup neg scale dup 4 -1 roll exch div 3 1 roll div 0 setlinewidth newpath 0 0 moveto dup .395 mul 0 exch lineto .375 .214 rlineto dup thick add dup .375 exch lineto 2 index exch lineto dup thick 2 div sub dup 3 index exch lineto .6 exch lineto .375 0 lineto clip thick setlinewidth newpath dup .395 mul 0 exch moveto .15 .085 rlineto .375 0 lineto thick 2 div sub dup .6 exch lineto lineto stroke grestore } def 1860 909 384 0 3894 16 sqr end #¿¡#d#EMATH###9#ª##### ###########‚g###‚b########‚m###‚M#‚+#‚2#‚m########10 #‚ÿ###N##Equation#########6### ## ###########‚g#‚(#‚M#‚+#‚2#‚m#‚)###‚b#‚M###########9##Equation#########!##### ###########‚g###‚b###########N##Equation#########6##### ###########‚g###‚b########‚M#‚+#‚m###‚M###########H##Equation#########0##### ###########‚g###‚b########‚m###‚M###########N##Equation#########6##### ###########‚g###‚b########‚m###‚M#‚+#‚m###########Q##Equation#########9##### ###########‚g###‚b########‚m###‚M#‚+#‚2#‚m######## A particle of mass m on the EarthÕs surface is confined to move on the parabolic curve y=ax2, where y is up. Which of the following is a Lagrangian for the particle?##### J###########‚#: #######‚##¡#d##·xpr####### ## #‚## #######‚#####"###‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #L,# ###Symbol#####) #=#####(#####1*##2#####"######### (# ###m##)##((###+#ú ##(# #*#r ####(###/#2#### +###+##) #a#######(###H#2### +###r####(###R#2# +###)###)##-#####) #mgr##)##sin)##(#####)##a###N£##N¤##N¥##N¦##N§##Nù##Nú##r###r###r`##ra##r«##r¬## r##r®##r¯##r°##r±##x###‚é##‚ê##‚ë##‚ì##‚í##¢d##¢u##¢v##¢w## ¢x##£###£Ë##£Ì##£Í##£Î##£Ñ##£Ó##£Ú##£à##£ã##£ô##£ö##¤‚##¤‚##¤‚##¤‚##¤‚##¤ ##¤ ¢##¤§##¤¨##¤¬##¤®##¤µ##¤¾##¤Ç##¤É##¤Ï##¤Ñ##¤Ö##¤×##¤Þ##¤á##¤â##¤ú##¤ý##¤þùòëä##Þ ##Ø#ÒËĽ¶#ÞÞ¯###Þ#################¬############¦#¦########### # ###ü########### #@########‚ð# #B###ø####‚ì# #B###ô####‚Ü# #B###è####‚h# #B###ä####~´###B###ä####x # ######### #@######### #B###è####jd# #B###è####d´# #B###è####_‚# #B###è####YÔA##N‚##Nú##r±##‚ë##‚í##¢y##¢z##¢{##£Ì##£Í##¤‚##¤‚##¤‚##¤‚##¤‚##¤‚##¤‚##¤ ##¤¡##¤¢##¤£##¤¤##¤ ¥##¤¦##¤§##¤¨##¤û##¤ü##¤ý##¤þ##¤ÿ##¥###¥###¥###¥###¥###¥###¥###¥###¥###¥ ##¥ ##¥###¥U##¥V##¥W##¥X##¥Y##¥Z##¥\##¥k##¥l##¥z##¥‚##¥‚ ##¥‚##¥‚##¥‚ûûîûãããûûÚÑÉÉÉÉÉÉÉÉÉÉÉÉÉɾ¾¾¾¾¾¾¾¾¾¾³¾¾¾¾¾¾¾¾¾¾ûûûûûûû¾ûû############ ######################################################### ########ÿL##Ð#þ‚#þè## ###$‚###ÿL##Ð#þ‚#þè######$‚# ##h#þ‚######$‚####h#þ‚#þè###########h#þ‚#þè# ###$‚# #ÿL##Ð#þ‚#ÿ### ###########h#þ‚#######¸###########9### ´#######"#‚###¦#########"#‚#‚#####"#‚##¡#d##·xpr####### ## #‚## #####"#‚#####"#"#‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###L,# ###Symbol#####) #=#####(#####1*##2#####"######### (#####m##(###&#ú ##(###%#y ####(###+#2# + ##1###)##+###(# #M#1(###G#4##)##ay"###F#######.######(###0#æ *##è (###0#ç (##Z#ö *##ø .##ÿ###(###b#-#####) #mgy¡#À#Â30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4128 div 1088 3 -1 roll exch div scale currentpoint translate 64 -4089 translate 12 4729 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 4729 moveto 384 /Symbol f1 (=) show 669 4590 moveto 224 /Times-Roman f1 (1) show 669 4822 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 4630 moveto 164 0 rlineto stroke 867 4729 moveto 384 /Times-Italic f1 (m) show 1178 4711 moveto 384 /Times-Roman f1 (\307) show 1144 4729 moveto 384 /Times-Italic f1 (y) show 1331 4560 moveto 224 /Times-Roman f1 (2) show 1643 4729 moveto 384 ns (1) show 1887 4729 moveto 384 /Symbol f1 (+) show 2403 4490 moveto 384 /Times-Roman f1 (1) show 2217 5022 moveto (4) show 2425 5022 moveto 384 /Times-Italic f1 (ay) show 2189 4630 moveto 621 0 rlineto stroke 1486 4481 moveto 384 /Symbol f1 (\346) show 1486 5022 moveto (\350) show 1486 4803 moveto (\347) show 2822 4481 moveto (\366) show 2822 5022 moveto (\370) show 2822 4803 moveto (\367) show 3086 4729 moveto (-) show 3383 4729 moveto 384 /Times-Italic f1 (mgy) show end #¿¡#d#vMATH###j#Ç##### ##‚L#‚=#######‚1###‚2## #‚m2‚y###########‚2## ######‚1#‚+######‚1###‚4#‚a#‚y####‚(#‚)##‚-#‚m#‚g#‚y##at #‚ÿ### ´#######"#‚###¦#########"#‚#‚#####"#‚##¡#d##·xpr####### ## #‚## #####"#‚#####"#"#‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###L,# ###Symbol#####) #=#####(#####1*##2#####"######### (#####m##(###&#ú ##(###%#y ####(###+#2# + ##1###)##-###(# #M#1(###G#4##)##ay"###F#######.######(###0#æ *##è (###0#ç (##Z#ö *##ø .##ÿ###(###b#-#####) #mgy¡#À#Â30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4128 div 1088 3 -1 roll exch div scale currentpoint translate 64 -4089 translate 12 4729 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 4729 moveto 384 /Symbol f1 (=) show 669 4590 moveto 224 /Times-Roman f1 (1) show 669 4822 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 4630 moveto 164 0 rlineto stroke 867 4729 moveto 384 /Times-Italic f1 (m) show 1178 4711 moveto 384 /Times-Roman f1 (\307) show 1144 4729 moveto 384 /Times-Italic f1 (y) show 1331 4560 moveto 224 /Times-Roman f1 (2) show 1643 4729 moveto 384 ns (1) show 1886 4729 moveto 384 /Symbol f1 (-) show 2401 4490 moveto 384 /Times-Roman f1 (1) show 2215 5022 moveto (4) show 2423 5022 moveto 384 /Times-Italic f1 (ay) show 2187 4630 moveto 621 0 rlineto stroke 1486 4481 moveto 384 /Symbol f1 (\346) show 1486 5022 moveto (\350) show 1486 4803 moveto (\347) show 2820 4481 moveto (\366) show 2820 5022 moveto (\370) show 2820 4803 moveto (\367) show 3084 4729 moveto (-) show 3381 4729 moveto 384 /Times-Italic f1 (mgy) show end #¿¡#d#vMATH###j#É##### ##‚L#‚=#######‚1###‚2## #‚m2‚y###########‚2## ######‚1#‚-######‚1###‚4#‚a#‚y####‚(#‚)##‚-#‚m#‚g#‚y##at #‚ÿ###q#########‚###c####### ####‚#S#######‚##¡#d##·xpr### ### ## #‚## #######‚#####"###‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+###L,# ###Symbol#####) #=#####(####1*##2#####"######### (#####m##(###&#ú ##(###%#x ####(# #+#2# + ##1###)##+###(# #M#1(###G#4##)##ax"###F#######.######(###0#æ *#è (##Z#ö *##ø .##ÿ###(###c######) #mgx¡#À#‚30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 960 3 -1 roll exch div scale currentpoint translate 64 -4082 translate 12 4658 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 4658 moveto 384 /Symbol f1 (=) show 669 4519 moveto 224 /Times-Roman f1 (1) show 669 4751 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 4559 moveto 164 0 rlineto stroke 867 4658 moveto 384 /Times-Italic f1 (m) show 1183 4639 moveto 384 /Times-Roman f1 (\307) show 1144 4658 moveto 384 /Times-Italic f1 (x) show 1342 4489 moveto 224 /Times-Roman f1 (2) show 1654 4658 moveto 384 ns (1) show 1898 4658 moveto 384 /Symbol f1 (+) show 2420 4419 moveto 384 /Times-Roman f1 (1) show 2228 4951 moveto (4) show 2436 4951 moveto 384 /Times-Italic f1 (ax) show 2200 4559 moveto 632 0 rlineto stroke 1497 4481 moveto 384 /Symbol f1 (\346) show 1497 4880 moveto (\350) show 2844 4481 moveto (\366) show 2844 4880 moveto (\370) show 3108 4658 moveto (-) show 3405 4658 moveto 384 /Times-Italic f1 (mgx) show end #¿¡#d#vMATH###j#Ä##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## ######‚1#‚+######‚1###‚4#‚a#‚x####‚(#‚)##‚-#‚m#‚g#‚x##te #‚ÿ#2#### #########‚###ÿ###########‚#ï#######‚##¡#d##·xpr####### ## #‚## #######‚#####"###‚## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #L,# ###Symbol#####) #=#####(#####1*##2#####"######### (# ###m##(###&#ú ##(# #%#x ####(###+#2# +###1###)##+###) #4##)##a####(###Q#2### +###x####(###\#2#### ###### ##(##0#()1#)#########(# #g#+#####) #mgx¡#À##30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4320 div 576 3 -1 roll exch div scale currentpoint translate 64 -4094 translate 12 4478 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 4478 moveto 384 /Symbol f1 (=) show 669 4339 moveto 224 /Times-Roman f1 (1) show 669 4571 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 4379 moveto 164 0 rlineto stroke 867 4478 moveto 384 /Times-Italic f1 (m) show 1183 4459 moveto 384 /Times-Roman f1 (\307) show 1144 4478 moveto 384 /Times-Italic f1 (x) show 1342 4309 moveto 224 /Times-Roman f1 (2) show 1597 4478 moveto 384 ns (1) show 1841 4478 moveto 384 /Symbol f1 (+) show 2139 4478 moveto 384 /Times-Roman f1 (4) show 2347 4478 moveto 384 /Times-Italic f1 (a) show 2555 4309 moveto 224 /Times-Roman f1 (2) show 2704 4478 moveto 384 /Times-Italic f1 (x) show 2902 4309 moveto 224 /Times-Roman f1 (2) show 1492 4521 moveto /f3 {findfont 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1533 /Symbol f3 (\() show 3061 4521 moveto (\)) show 3260 4478 moveto 384 /Symbol f1 (+) show 3558 4478 moveto 384 /Times-Italic f1 (mgx) show end #¿¡#d#‚MATH###x#Þ##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## ######‚1#‚+#‚4#‚a########‚2## #‚x########‚2### #‚(#‚)##‚+#‚m#‚g#‚x##an #‚ÿ##‚###c#########x###U###########x#E#######x##¡#d##·xpr## ##### ## #‚## #######x#####"###x## #¾¡#À#currentpoint #¿"###### #¾,#####Times###### .##ÿ###+# #L,# ###Symbol#####) #=#####(#####1*##2#####"######### (# ###m##(###&#ú ##(# #%#x ####(###+#2#### +###+#####(###=#1*##2"###<##### (# #C#m##(###M#ú ##(# #L#y ####(###R#2#### +###+#####) #mgy¡#À#‚30 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3840 div 512 3 -1 roll exch div scale currentpoint translate 64 -4063 translate 12 4447 moveto /fs 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {findfont dup /cf exch def sf} def /ns {cf sf} def 384 /Times-Italic f1 (L) show 325 4447 moveto 384 /Symbol f1 (=) show 669 4308 moveto 224 /Times-Roman f1 (1) show 669 4540 moveto (2) show /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 4348 moveto 164 0 rlineto stroke 867 4447 moveto 384 /Times-Italic f1 (m) show 1183 4428 moveto 384 /Times-Roman f1 (\307) show 1144 4447 moveto 384 /Times-Italic f1 (x) show 1342 4278 moveto 224 /Times-Roman f1 (2) show 1573 4447 moveto 384 /Symbol f1 (+) show 1901 4308 moveto 224 /Times-Roman f1 (1) show 1901 4540 moveto (2) show 1875 4348 moveto 164 0 rlineto stroke 2099 4447 moveto 384 /Times-Italic f1 (m) show 2410 4429 moveto 384 /Times-Roman f1 (\307) show 2376 4447 moveto 384 /Times-Italic f1 (y) show 2563 4278 moveto 224 /Times-Roman f1 (2) show 2794 4447 moveto 384 /Symbol f1 (+) show 3092 4447 moveto 384 /Times-Italic f1 (mgy) show end #¿¡#d#xMATH###l#À##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## #‚+#######‚1###‚2## #‚m2‚y###########‚2## #‚+#‚m#‚g#‚y## 6 #‚ÿ###‚##Equation#########j##### ##‚L#‚=#######‚1###‚2## #‚m2‚y###########‚2## ######‚1#‚+######‚1###‚4#‚a#‚y####‚(#‚)##‚-#‚m#‚g#‚y#####‚##Equation#########j##### ##‚L#‚=#######‚1###‚2## #‚m2‚y###########‚2## ######‚1#‚-######‚1###‚4#‚a#‚y####‚(#‚)##‚-#‚m#‚g#‚y#####‚##Equation#########j##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## ######‚1#‚+######‚1###‚4#‚a#‚x####‚(#‚)##‚-#‚m#‚g#‚x#####‚##Equation#########x##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## ######‚1#‚+#‚4#‚a########‚2## #‚x########‚2### #‚(#‚)##‚+#‚m#‚g#‚x#####‚##Equation#########l##### ##‚L#‚=#######‚1###‚2## #‚m2‚x###########‚2## #‚+#######‚1###‚2## #‚m2‚y###########‚2## #‚+#‚m#‚g#‚y###: ######t#‚#####l#Î###f#########l#Î#Vÿÿÿÿ#‚#%###ÿ #ÿÿÿÿÿÿ##ÿÿ###%###‚#######¡#À#y%PostScript Hack by Mike Brors 12/7/90/DisableNextSetRGBColor { userdict begin /setrgbcolor { pop pop pop userdict begin /setrgbcolor systemdict /setrgbcolor get def end } def end } bind def/bcarray where { pop bcarray 2 { /da 4 ps div def df setfont gsave cs wi 1 index 0 ne{exch da add exch}if grestore setcharwidth cs 0 0 smc da 0 smc da da smc 0 da smc c gray { gl} {1 setgray}ifelse da 2. div dup moveto show }bind put } if%% Used to snap to device pixels, 1/4th of the pixel in./stp { % x y pl x y % Snap To Pixel, pixel (auto stroke adjust) transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform} bind def/snapmoveto { % x y m % moveto, auto stroke adjust stp moveto} bind def/snaplineto { % x y l % lineto, auto stroke adjust stp lineto} bind def## #‚##### ÿÿÿÿ#‚#%######### #########¡#¶################# ####### ###########¡#¶#############0###3# ### #‚# ÿÓÿá### ÿàÿÒ#}#ø#######"###########¡#¶#############"#k###‚#Q#k###y#############¡#¶###### #######X# #‚# #‚# ÿ2##### ÿàÿ##}#*#######"###########¡#¶#############"#k###‚#Y###########¡#¶#############X # #‚#¡#‚# ############# #ÂÿÊ### ###############"####### #‚#¡#‚##ÿò^###D##,# ## 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[20 points total] Two clocks located at the origins of the K and KÕ systems (which have a relative speed v) are synchronized when the origins coincide. At some later time an observer at the origin of the K system observes the KÕ clock through a telescope and observes a reading of T. What does the K clock read? Show your work.3. [30 points total] Three equal masses m are constrained to move along a circle of radius r. The masses are equally spaced around the circle and are connected by identical springs of force constant k.A. [10 pts] Find the tensors mjk and Ajk for this system. Please show your work.B. [10 pts] Find the eigen-frequencies for this system. 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