BOBOwd$H$H$HN|H<..X-Z(Charles E. Miller, Jr.x HHHV,(hh hd'0F pfEfH(l/S"">@ ''pE\"HS1H  H@4(eDSET q,"N,ND,MMr0~M# !    1 V W R T        q   ) , > A X [ r t u        4 7 B E H K W [ x { | ~   A Q R S i      O          . ~  6 G Y Z [   7 P S   3 G V W X l o  01Vl !#R{$S%T & & & &{$ & & &$ &%* &+3 &48 &9d'g| & & & ( & &{$%C &Di &jy &z~ &##  &!' &(. &/1 &28 &9; &<G &HO &PR &SV &Wa &bi &jl &mp &q!r s%t! & ( & & &'  &  & & &* &+- &.2 &3|b)}~  &  &6 &7= &>!b)''''''' ' !'$*'-C &F (GJ &KO &PQ!Sg +h + + +#^ +_s +t, + ( + & +  +   .   +  % . & * + + G + H R! S ] + ^ w + x  +   +   +   +   +  / ,   +   +   +   +  30 4 E 1 F ,  ,  ,   +   +   +  # + $ - + . 4 + 5 9 + : F + G M + N $2      +  0  3   +   +   +   +   +  4  ! + " * + + @0 A C + D j4 k #   HM"M"4pllM"M"Rt{\{\{{{{{[{[rvvv5 7ddC fwwF I o#o#X ]#]#yr r |b[b[b{b{ b   %b   E3   Qss @ -M"M" L : j my y (NL8 PHY 1150 Doug Davis Chapter 15; Periodic Motion 15.9, 11, 12, 15, 20, 25, 31, 34, 35 15.9 This is an excellent one to solve with a spreadsheet:   15.11  where k in this equation is the "effective spring constant", the spring constant of a single spring that has the same effect as the combination of springs in the diagram.  With this arrangement, a displacement of x causes a force of F1= k1x due to the spring on the left and a force of F2 = k2x due to the spring on the right.  The net force, then, is Fnet = F1 + F2 = k1x k2x = (k1 + k2)x. Since the effective spring constant, keff is the constant is Fnet=keffx, we can see that keff = k1 + k2 The next arrangement of springs is a little more "tricky" or more interesting:  When the mass is moved a distance x, spring #1 stretches a distance x1 and spring #2 stretches a distance x2 with x = x1 + x2 That is, the two springs need not be stretched (or compressed) the same amount at all. However, the force exerted by each spring must be the same. That is, F1 = F2 or k1 x1 = k2 x2 or x1 = (k2/k1) x2 or x2 = (k1/k2) x1  To determine the "effective spring constant", we must write the force in the form of F = keff x We have F = k1 x1 and x = x1 + x2 x = x1 + (k1/k2) x1 x = [ 1 + (k1/k2)] x1  Therefore,    Therefore,  which can also be written as   The third arrangement of springs is actually fairly easy and straightforward. With this arrangement, a displacement of x causes a force of F1 = k1x due to spring #1 and a force of F2=k2x due to spring #2. The stretch of each spring is the same as the displacement of the mass.  The net force, then, is Fnet = F1 + F2 = k1x k2x = (k1 + k2)x. Since the effective spring constant, keff is the constant is Fnet = keffx, we can see that keff = k1 + k2 15.12 W = (1/2) k x2 W = (1/2) (80 N/m) (0.15 m)2 W = 0.9 H 15.15 W = PE = (1/2) k A2 = (1/2) (50 N/m) (0.1 m)2 = 0.25 J As the mass moves through equilibrium, x = 0, so it has zero potential energy and its total energy is now KE, KE = (1/2) m v2 = (1/2) (0.2 kg) v2 = 0.25 J = E v2 = 2.5 m2/s2 v = 1.58 m/s 15.20 E = PEmax = (1/2) k A2 A = 12 cm = 0.12 m E = (1/2) ( k ) (0.12 m)2 E = KE + PE =  m v2 +  k x2 E = (1/2) m (0.2 m/s2 + (1/2) k (0.08 m)2 = (1/2) ( k ) (0.12 m)2 = E m (0.2 m/s)2 = k (0.12 m)2 k (0.08 m)2 (m/k)(0.2 m/s)2 = (0.12 m)2 (0.08 m)2 (m/k)(0.04) (m/s)2 = (0.0144 0.0064) m2 = 0.0080 m2 (m/k) = 0.2 (1/s2) T = 2  T = 2.8 sec 15.25 These multiple-data-set problems are good ones to solve with a spreadsheet. This is basically energy conservation, E = (1/2) k A2 = (1/2) k x2 + (1/2) m v2 k A2 = k x2 + m v2 m v2 = k (A2 x2) v2 = (m/k) (A2 x2)  So this will be the formula we put in the spreadsheet:  15.31  f2 = (1/4 2)  k = 4 2 f2 m k = 4 2 [2 (1/2)]2 (0.5 kg) k = 79 kg/s2 k = 79 N/m 15.34 T = 2.00 s T = 2  T2 = 4 2 (l/g) l = (1/42)T2 g l = 0.993 m 15.35 T = 2   T = 2.007 seconds (That is, this is almost a "seconds pendulum").ZNDSET.HN |"M96* |D"M M" "M"/ K"xprWEwdxpr" WEWEp"New Century SchlbkWE q"fWE q" WE q"=WE q" WE"WE,"New Century Schlbk" . K K*f = WE q"1 (1WEWE q"2WE q" WE q"(2 WE q" WE2"K###WE(' WE"9WEWE q"k(<kWEWE q"m(9mWEDSET.H< |l96* |DFFn n n"/ l 1j" >@ 8 i"#@" #" !i, Geneva .ll+. 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WE q"k)kWEWE q"2+ 2WEWEWEPWEWE q"k((kWEWE q"1+ 1WEWE q" WE q"+WE q" WEPWE(9 + WE q"k)kWEWE q"2+ 2WEDSET.H | r96* |D r r  r"/ p"xprWEwdxpr"WE WE" WEWEp"New Century SchlbkWE q"1,"New Century Schlbk" .pp+ 1WEWEPWEWE q"k(kWEWE q"eWE q"fWE q"f+ effWEWE q" WE q" WE q"=WE q" WE q" WE" 4WE( = WE q"1( 91WEWEPWEWE q"k(4kWEWE q"1+ 1WEWE q" WE q" WE q"+WE q" WE q" WE" ^WE(F + WE q"1( c1WEWEPWEWE q"k(^kWEWE q"2+ 2WEDSET.H> |[b96* |Ddd]d d] ]d"/d SPNTdSPNT Cp [b UUUU"-"d SPNT Cp"Zad SPNT "-d SPNT d SPNT d SPNT d SPNT  P0%Jd SPNT P>%Xd SPNT PJ%dd SPNT PW%qd SPNT Pb%|d SPNT Pp%d SPNT P{%d SPNT P&%@d SPNT P%d SPNT P%2d SPNT " d SPNT "d SPNT d SPNT d SPNT P-1OKd SPNT P-?OYd SPNT P-KOed SPNT P-XOrd SPNT P-cO}d SPNT P-qOd SPNT P-|Od SPNT P-'OAd SPNT P-Od SPNT P-O3d SPNT "> d SPNT "?d SPNT d SPNT ">d SPNT ")d SPNT ") d SPNT "+ d SPNT  """"1 P 8d SPNT d SPNT  UUUU"Td SPNT  SSZd SPNT d SPNT d SPNT d SPNT dSPNT &D[!d SPNT  b, > Vancouver> .[b[b+k [b"#d SPNT dSPNT &J_'d SPNT  b + 1 [b"%d SPNT d SPNT d SPNT  12UIx8d SPNT dSPNT &qd SPNT  Ab (>^k [b UUUU"P^d SPNT dSPNT &vd SPNT  Db + 2 [b"Qid SPNT d SPNT dSPNT &hkd SPNT  8b (5m [b"Gd SPNT dSPNTDSET.HN |{b96* |D }d d} }d"/d SPNTdSPNT o {b UUUU"k."d SPNT o"zad SPNT "k.d SPNT d SPNT d SPNT d SPNT dSPNT dSPNT iC P#5DTd SPNT dSPNT dSPNT $P#FDed SPNT dSPNT dSPNT CjP#TDsd SPNT dSPNT dSPNT P#dDd SPNT dSPNT dSPNT vP#qDd SPNT dSPNT dSPNT [5P#Dd SPNT dSPNT dSPNT iCP#Dd SPNT dSPNT dSPNT eP#*DId SPNT dSPNT  +dSPNT P#Dd SPNT dSPNT dSPNT PP#D8d SPNT dSPNT )4dSPNT QV"4 d SPNT dSPNT dSPNT xgxg"4d SPNT d SPNT d SPNT dSPNT dSPNT 8PM6oUd SPNT dSPNT dSPNT k"PMGofd SPNT dSPNT dSPNT -PMUotd SPNT dSPNT dSPNT <]PMdod SPNT dSPNT dSPNT 7CPMqod SPNT dSPNT dSPNT 8jPMod SPNT dSPNT dSPNT ePMod SPNT dSPNT dSPNT k] PM*oId SPNT dSPNT  +dSPNT NPMod SPNT dSPNT dSPNT hPMo9d SPNT dSPNT )4dSPNT `"^ d SPNT dSPNT dSPNT "_d SPNT d SPNT "^d SPNT d SPNT "4)d SPNT "I d SPNT d SPNT "K d SPNT  """"1)o  8d SPNT d SPNT  UUUU" Td SPNT  ssZd SPNT d SPNT  "!d SPNT "!d"SPNT (d SPNT "#"dSPNT d SPNT dSPNT &,Cd SPNT , > Vancouver> .{b{b+x UUUU""d SPNT d SPNT d SPNT d SPNT dSPNT &(;d SPNT  4b(1k {b"Cd SPNT dSPNT &2Ad SPNT  8b + 1 {b"Ed SPNT d SPNT d SPNT  1Rpi8d SPNT dSPNT &d SPNT  ab (^yk {b UUUU"pyd SPNT dSPNT &d SPNT  db + 2 {b"qd SPNT d SPNT dSPNT &d SPNT  Xb (Um {b"gd SPNT d SPNT dSPNTDSET.H< | 96* |D    "/  "xprWEwdxpr "WEWE" WEWEp"New Century SchlbkWE q"1,"New Century Schlbk" .  * 1WEWE q"2*2DSET.HD | >96* |D    "/  "xprWEwdxpr "WEWE" WEWEp"New Century SchlbkWE q"1,"New Century Schlbk" .  * 1WEWE q"2*2DSET.H | Mp96* |D  " " ""/  "xprWEwdxpr "WEWE2" ###WEWE" WEWEp"New Century SchlbkWE q"m,"New Century Schlbk" .  + mWEWE q"k+kWEDSET.H! |s96* |Ds s s"/ q"xprWEwdxpr" WEWEp"New Century SchlbkWE q"vWE q" WE q"=WE q" WE2"q###WE ,"New Century Schlbk" .qq*v = WE""WEWE q"k (%kWEWE q"m("mWE q" WE q"(WEPWE(2 (WE q"A) AWEWEWE q"2 ( F2WE q" WE q"WE q" WEPWE + WE q"x)xWEWEWE q"2 (d2WE q") +)WEDSET.HL |,96* |D<<  "/  1"f@ 8 "#@" #" !, Geneva .++ A"C )9B" )FC" )FD" )FE"[ )FF"#"!!( 1"#*2"7*3"K*4"_*5"s*6"*7"*8"*9"(10"*11  UUUU CC    [[ ## 77 KK __ ss     2# #,"New Century Schlbk" ( '=SQRT((C4/B4)*(D4*D4-E4*E4)) 2# #)'=SQRT((C4/B4)*(D4*D4-E4*E4)) 2[#\ #\)'=SQRT((C4/B4)*(D4*D4-E4*E4)) \#)'=SQRT((C4/B4)*(D4*D4-E4*E4)) (4Tmass);spg cnst )C Amplitude )Iposition)Ivelocity(HU( kg ))?( N/m ))L( m ) )H( m ))A( m/s )(\*a)/0.18)J3.5)F0.2)F0.1):0.7638(p*b)/0.18)J3.5)F0.2)B0.15)>0.5833(+c).0.28)J3.5)F0.2)B0.18)>0.3082(*d)/0.35)P5)<0.25)F0.05)>0.9258(+e).0.35)J5.5)B0.25)J0.1):0.9083(,f)10.5)F5.5)B0.25)J0.1):0.7599(*g)30.5)F5.5)B0.25)F0.15)>0.6633(*h)30.5)F5.5)F0.3)B0.15)>0.8617 "DSET.H* |"M96* |D"M M" "M"/ K"xprWEwdxpr" WEWEp"New Century SchlbkWE q"fWE q" WE q"=WE q" WE"WE,"New Century Schlbk" . K K*f = WE q"1 (1WEWE q"2WE q" WE q"(2 WE q" WE2"K###WE(' WE"9WEWE q"k(<kWEWE q"m(9mWEDSET.HOT |96* |D  "/ "xprWEwdxpr"WEWE" WEWEp"New Century SchlbkWE q"k,"New Century Schlbk" .+ kWEWE q"m(mDSET.Ht |x96* |D""  "/ "xprWEwdxpr "WEWE2"###WEWE" WEWEp"New Century SchlbkWE q"l,"New Century Schlbk" .+ lWEWE q"g(gWEDSET.H |)P96* |D""  "/ "xprWEwdxpr "WEWE2"###WEWE" WEWEp"New Century SchlbkWE q"l,"New Century Schlbk" .+ lWEWE q"g(gWEDSET.H | yN96* |D y y  y"/ w"xprWEwdxpr " WE WEp"New Century SchlbkWE q"TWE q" WE q"=WE q" WE q"2WE q" WE q"WE q" WE2"w###WE,"New Century Schlbk" .ww*T = 2 WE";9WEWE q"1WE q".WE q"0WE q" WE q"m (D1.0 mWEWE q"9WE q".WE q"8WE q" WE q"mWE q"/WEPWE(;9.8 m/WE q"s)+sWEWEWE q"2 (m2WEDSET.H]l6*]lDSET| .HNSSUSp %S @ SSVp & @ 6*]lDSET.HPk`02*`0000000DSETT )?@5)5 8) 8 ' *"X*PHY 1150, Homework, Chapter 15, page  ZNFNTMCUTSDSUM/Charles E. Miller, Jr.HDNISTYL*@STYL5)t`pl        |                    ! # $  %  5&  ' ( )  *+ ,  . !/ "0 #1 $2 ! %3 "&4 #'5  (85 )9  *HASHd* d* d*d*d( $d*!d*d*d*d*d+d+d+"d,d,d,d,e* e*%e*&e*#d( %x)%p& Af*(C('Dd*Dd*De*De*$De*De*De+Fd( L:Qhjazo}Bod$ CHAR   "     S W l@ @@ @W{@ @ b         $HASH           "# #!akzR CELL>("HASH  GRPHv&fn.HASHog& l RULRv@....@......@..}@..@..X HASH\^\^@@^ B C\ E^: LKUP  "*-6 !"#$%&'7() $NAMEDefault Default SSHeaderBodyFooterFootnoteFootnote Index DFNTM HelveticaGeneva"New Century SchlbkNew YorkETBL@FNTMCUTSDSUMHDNI?STYLIETBLم