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1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 5 "Work " }}{PARA 0 "" 0 "" {TEXT -1 63 "by Peter Stone, Mathematics Dept., Malaspina University-College" } }{PARA 0 "" 0 "" {TEXT -1 17 "Version: 8.4.2005" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 25 "The work done by a force " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 114 "Newton's second la w of motion states that the force on an object is the product of its m ass and its acceleration: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "F=m" "6#/%\"FG%\"mG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^ 2*x/(d*t^2);" "6#*(%\"dG\"\"#%\"xG\"\"\"*&F$F'*$%\"tGF%F'!\"\"" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 263 1 "x" }{TEXT -1 29 " is the displacement at time " }{TEXT 264 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 109 "If the mass is measured in k ilograms (kg) and the acceleration is measured in metres per sec. per \+ sec. ( m / " }{XPPEDIT 18 0 "s^2;" "6#*$%\"sG\"\"#" }{TEXT -1 30 "), t he force has the units kg " }{TEXT 267 1 "." }{TEXT -1 4 " m /" } {XPPEDIT 18 0 "s^2" "6#*$%\"sG\"\"#" }{TEXT -1 16 " or newtons, N. " } }{PARA 0 "" 0 "" {TEXT -1 89 "A force of 1 N acting on an object with \+ a mass of 1 kg produces an acceleration of 1 m / " }{XPPEDIT 18 0 "s^2 " "6#*$%\"sG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 40 "The weight of an object is the product " }{XPPEDIT 18 0 "M*`.`*g;" "6#*( %\"MG\"\"\"%\".GF%%\"gGF%" }{TEXT -1 14 " of its mass " }{TEXT 265 1 "M" }{TEXT -1 40 " and the acceleration due to gravity, g " }{TEXT 266 1 "~" }{TEXT -1 10 " 9.81 m / " }{XPPEDIT 18 0 "s^2;" "6#*$%\"sG\" \"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 51 "When a constant force is applied to an object, the " } {TEXT 259 4 "work" }{TEXT -1 1 " " }{TEXT 299 1 "W" }{TEXT -1 48 " don e is defined to be the product of the force " }{TEXT 268 1 "F" }{TEXT -1 24 " and the distance moved " }{TEXT 269 1 "d" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` W` = F*`.`*d;" "6#/ %#~WG*(%\"FG\"\"\"%\".GF'%\"dGF'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{TEXT 301 1 "F" }{TEXT -1 22 " is in newtons, N and \+ " }{TEXT 278 1 "d" }{TEXT -1 40 " is in metres, m, the unit for the wo rk " }{TEXT 300 1 "W" }{TEXT -1 40 " is the newton-metre, which is cal led a " }{TEXT 259 5 "joule" }{TEXT -1 5 ", J. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "The work done in lifting an object is the product of its weight and the distance that the obje ct is raised through. " }}{PARA 0 "" 0 "" {TEXT -1 128 "For example, t he work done in raising a 0.8 kg. book through 0.7 metres from the flo or to place it on a desk is its weight, 0.8 " }{TEXT 304 1 "x" }{TEXT -1 5 " 9.81" }{XPPEDIT 18 0 "`` = 7.848;" "6#/%!G-%&FloatG6$\"%[y!\"$ " }{TEXT -1 64 " newtons, times the distance lifted, 0.7 metres, that \+ is, 7.848 " }{TEXT 303 1 "x" }{TEXT -1 4 " 0.7" }{XPPEDIT 18 0 "`` = 5 .4936;" "6#/%!G-%&FloatG6$\"&O\\&!\"%" }{TEXT -1 1 " " }{TEXT 302 1 "~ " }{TEXT -1 6 " 5.5 J" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "For a variable force, where the force is a function " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 86 " of the displace ment of the object in a straight line, we define the work done to be: \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "W = Int(F(x),x = \+ a .. b);" "6#/%\"WG-%$IntG6$-%\"FG6#%\"xG/F+;%\"aG%\"bG" }{TEXT -1 2 " , " }}{PARA 257 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "x=a" "6#/% \"xG%\"aG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" } {TEXT -1 56 " are the initial and final displacements of the object. \+ " }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6# %\"xG" }{TEXT -1 63 " is positive (or at least never negative) in the \+ interval from " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 4 " to \+ " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 72 ", the work done i s represented by the area under the graph of the force " }{XPPEDIT 18 0 "F(x)" "6#-%\"FG6#%\"xG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "x=a" " 6#/%\"xG%\"aG" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG " }{TEXT -1 2 ". 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++++]#*!#5FgfmF[cmF^dm-%%TEXTG6%7$Ffz$!#:FfcmQ\"x6\"F^dm-F^gm6%7$$Fbgm F)$\"#TF)Q%F(x)FdgmF^dm-F^gm6%7$Fa[lF_fmQ&x~=~aFdgmF^dm-F^gm6%7$$\"#bF )F_fmQ&x~=~bFdgmF^dm-F^gm6%7$$\"$v#Ffcm$\"$v\"FfcmQ1area~=~work~doneFd gmF^dm-%+AXESLABELSG6%FcgmQ!Fdgm-%%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%N ONEG-%%VIEWG6$;FhgmFfz;$!\"%F)Fafm" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "To explain why this is a reasonable definition think of \+ the integral as a limit of a sum of the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Limit(``,Delta*x = 0);" "6#-%&LimitG6$% !G/*&%&DeltaG\"\"\"%\"xGF*\"\"!" }{XPPEDIT 18 0 "Sum(F(x)*Delta,x = a \+ .. b);" "6#-%$SumG6$*&-%\"FG6#%\"xG\"\"\"%&DeltaGF+/F*;%\"aG%\"bG" } {TEXT 270 1 "x" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{TEXT 271 1 "x" }{TEXT -1 42 " ranges over a sequence of values bet ween " }{TEXT 272 1 "a" }{TEXT -1 5 " and " }{TEXT 273 1 "b" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Imagine that the interval fro m " }{TEXT 274 1 "a" }{TEXT -1 4 " to " }{TEXT 275 1 "b" }{TEXT -1 35 " is subdivided into a large number " }{TEXT 276 1 "n" }{TEXT -1 38 " \+ of small subintervals of equal width " }{XPPEDIT 18 0 "Delta*x;" "6#*& %&DeltaG\"\"\"%\"xGF%" }{TEXT -1 225 ". Suppose that the force is appr oximately constant while it acts over each of these small intervals. T hen using the simpler concept of work for a constant force, the work d one in passing through each subinterval has the form " }{XPPEDIT 18 0 "F(x)*`.`*Delta*x;" "6#**-%\"FG6#%\"xG\"\"\"%\".GF(%&DeltaGF(F'F(" } {TEXT -1 7 ", where" }{TEXT 277 2 " x" }{TEXT -1 221 " is chosen somew here in the interval. The total work done is obtained by adding terms \+ of this form. Taking progressively more and shorter subintervals produ ces the given integral as the limit of a sum as indicated above. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 " The work done in extending or compressing a spring " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 161 "Consider the system consisting of a spring atached to a rigid support at one e nd and at the other end to an object resting on a horizontal frictionl ess surface. " }}{PARA 0 "" 0 "" {TEXT -1 13 "According to " }{TEXT 259 11 "Hooke's law" }{TEXT -1 12 ", the force " }{XPPEDIT 18 0 "F(x) " "6#-%\"FG6#%\"xG" }{TEXT -1 49 " required to hold the spring extende d a distance " }{TEXT 279 1 "x" }{TEXT -1 73 " from its natural (unext ended) state is proportional to the displacement " }{TEXT 280 1 "x" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "F(x)=k*x" "6#/-%\"FG6#%\"xG*&%\"kG \"\"\"F'F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " } {TEXT 281 1 "k" }{TEXT -1 26 " is a constant called the " }{TEXT 259 15 "spring constant" }{TEXT -1 2 ". 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3@$)HC=?F-7$$\"3I;VJ$)Q'*4?F-$\"3PC-='4+g.#F-7$$\"38\"GV#>f!*R?F-$\"33 y@\"zSgw<#F-7$$\"33$pa0GE\\1#F-$\"3JKOX)e`CD3#F-$\"3= -.:<1UfFF-7$$\"3%H8+7Go/4#F-$\"3u=.QTjSyJF-7$$\"3o\"**yY_xo3#F-$\"3M`8 =gU]jOF-7$$\"3c]o[b&*Gq?F-$\"3sI'QHU3r>%F-7$$\"3$G&)R%=PrR?F-$\"3[;Hu7 C&)fZF-7$$\"3%Gnd%Gqm%*>F-$\"37z1TBAJJ`F-7$$\"3b'G#**3=.Q>F-$\"3#3?)*e s%GneF-7$$\"3#G\"=!G*=mo=F-$\"3-$*R\\?TTujF-7$$\"3s!ecNWnuy\"F-$\"31(e 7.\"\\$e$oF-7$$\"3#HQl4(=v&p\"F-$\"3eF&Q+s,iB(F-7$$\"3#)Ga!\\]lw[\"F-$ \"3avs(p+()H!yF-7$$\"3#H7?0%=xg7F-$\"3u\\F!>,*****zF-7$$\"34'*4jN5()H7 F-$\"3_#e'e-(pi*zF-7$$\"3?V=>@#=!*>\"F-$\"3-9p&GdH`)zF-7$$\"3!eke3\\g# o6F-$\"37H%zF-7$$\"3%z[FFjCq2 \"F-$\"3;F3m,hCqyF-7$$\"3eb\"3*pS_<5F-$\"3!=Og@CR4x(F-7$$\"3(GK]MJ&)G. *!#>$\"3Mu\\QWIc$\\(F-7$$\"3)=o0,v2f(zFces$\"3%3!\\&yHf/7(F-7$$\"3#\\F \"RyrbFqFces$\"3E5r/09&fm'F-7$$\"3\">$>5#f7p?'Fces$\"3yl$GE'y[ZhF-7$$ \"3*>Y'[p.:GbFces$\"3')\\F-;C(\\e&F-7$$\"3)e)***R/+++&Fces$\"3'f.6t&** ****\\F--%&COLORG6&%$RGBG\"\"!$\"\"'!\"\"Fcgs-%*THICKNESSG6#\"\"#-F$6% 7%7$$FcgsFcgs$\"\"\"Fcgs7$F_hsF_hs7$$\"33+++++++@F*F_hs-F`gs6&Fbgs$Fjg sFfgsFhhsFhhsFggs-F$6%7$7$F_hs$\"3++++++++]F-7$$\"3')*************H&Fc esF]isF_gsFggs-F$6%7$7$$\"33+++:D!>z\"F*F]is7$$\"3!******\\^->%=F*F]is F_gsFggs-%)POLYGONSG6%7&7$$\"+:D!>%=!\"*$\"\"$!\"#7$F`js$\"#)*Fejs7$$ \"+:D!>#>FbjsFgjs7$FjjsFcjs-F`gs6&Fbgs$\"\")FfgsF_[tF_[t-%&STYLEG6#%&P ATCHG-F$6$7$7$$\"3'******fY\\QI\"F*F_hs7$Fi[t$!3/+++++++5F--%'COLOURG6 &FbgsFcgsFcgsFcgs-F$6$7$7$$\"3%******\\^->)=F*F_hs7$Fe\\tF\\\\tF^\\t-F \\js6&7&7$Fjjs$\"++++]\\!#57$Fjjs$\"++++]^F^]t7$$\"+:D]@?FbjsF`]t7$Fc] tF\\]t7%7$Fc]t$\"+++++cF^]t7$$\"+:D!>/#Fbjs$\"$0&!\"$7$Fc]t$\"+++++XF^ ]t-Fb[t6#%,PATCHNOGRIDGF^\\t-F$6#7*F[]tF_]tFb]tFg]tFj]tF`^tFe]tF[]t-%% TEXTG6$7$$\"+m%\\QI\"Fbjs$FejsFfgsQ\"06\"-Fj^t6$7$$\"+:D!>)=Fbjs$F-Fej sQ\"xFa_t-Fj^t6$7$$\"+vD^4&*F^]t$\"#7FfgsQ0stretched~stateFa_t-Fj^t6$7 $$\"+:D!>%>Fbjs$\"$D\"FejsQ+F(x)~=~k~xFa_t-%+AXESLABELSG6%Q!Fa_tF\\at- %%FONTG6#%(DEFAULTG-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$F`atF`at" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" }}{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 102 "The work done in stretching the string from a sit uation where the extension from the natural state is " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 39 " to a situation where the extensio n is " }{XPPEDIT 18 0 "x=b" "6#/%\"xG%\"bG" }{TEXT -1 5 " is: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` W` = Int(F(x),x = a .. b);" "6#/%#~WG-%$IntG6$-%\"FG6#%\"xG/F+;%\"aG%\"bG" }{XPPEDIT 18 0 "``=Int(k*x,x=a..b)" "6#/%!G-%$IntG6$*&%\"kG\"\"\"%\"xGF*/F+;%\"aG% \"bG" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=k*x^2/2" "6#/%!G*(%\"kG\"\"\"*$%\"xG\"\"#F'F*!\"\"" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([b, ``],[``, ``],[a, ``]);" "6#-%*PIECE WISEG6%7$%\"bG%!G7$F(F(7$%\"aGF(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=k/2" "6#/%!G*&%\"kG\"\"\"\"\"#!\"\" " }{XPPEDIT 18 0 "``(b^2-a^2)" "6#-%!G6#,&*$%\"bG\"\"#\"\"\"*$%\"aGF)! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 293 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 124 "A force of 35 N is required to hold a spring t hat has been stretched to a length of 25 cm from its natural length of 20 cm. " }}{PARA 0 "" 0 "" {TEXT -1 68 "How much work is done in stre tching the spring from 25 cm to 28 cm? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 75 "Holding the spring stretched by 5 cm = 0.05 m r equires a force of 35 N, so " }{XPPEDIT 18 0 "F(x)=k*x" "6#/-%\"FG6#% \"xG*&%\"kG\"\"\"F'F*" }{TEXT -1 12 ", satisfies " }{XPPEDIT 18 0 "F(` 0.05`) = 35;" "6#/-%\"FG6#%%0.05G\"#N" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }{TEXT 295 1 "k" }{TEXT -1 1 " " }{TEXT 296 1 "." }{TEXT -1 4 " 0.0" }{XPPEDIT 18 0 "5=35" "6#/\"\"&\"#N" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "k = 35/`0.05`;" "6#/%\"kG*&\"#N\"\" \"%%0.05G!\"\"" }{XPPEDIT 18 0 "``=700" "6#/%!G\"$+(" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 63 "The work done in stretching the sprin g from 25 cm to 28 cm is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "` W` = Int(F(x),x = `0.05` .. `0.08`);" "6#/%#~WG-%$Int G6$-%\"FG6#%\"xG/F+;%%0.05G%%0.08G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = \+ Int(700*x,x = `0.05` .. `0.08`);" "6#/%!G-%$IntG6$*&\"$+(\"\"\"%\"xGF* /F+;%%0.05G%%0.08G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 350*x^2;" "6#/%! G*&\"$]$\"\"\"*$%\"xG\"\"#F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS E([`0.08`, ``],[`0.05`, ``]);" "6#-%*PIECEWISEG6$7$%%0.08G%!G7$%%0.05G F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``=350*(`0.08`^2-`0.05`^2)" "6#/%! G*&\"$]$\"\"\",&*$%%0.08G\"\"#F'*$%%0.05GF+!\"\"F'" }{XPPEDIT 18 0 "`` = 350;" "6#/%!G\"$]$" }{TEXT -1 7 " ( 0.00" }{XPPEDIT 18 0 "64*` - `* 0" "6#*(\"#k\"\"\"%$~-~GF%\"\"!F%" }{TEXT -1 8 ".0025 ) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 350*`.`*0.;" "6#/%!G*(\"$] $\"\"\"%\".GF'-%&FloatG6$\"\"!F,F'" }{TEXT -1 4 "0039" }{XPPEDIT 18 0 "``=1.365" "6#/%!G-%&FloatG6$\"%l8!\"$" }{TEXT -1 2 " J" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(700 *x,x=0.05..0.08);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Int G6$,$*&\"$+(\"\"\"%\"xGF)F)/F*;$\"\"&!\"#$\"\")F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++++l8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 287 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 33 "When a particle is at a distance " }{TEXT 289 1 "x" }{TEXT -1 39 " metres from a fixed point \+ a force of " }{XPPEDIT 18 0 "F(x) = 10*sin(Pi*x/3);" "6#/-%\"FG6#%\"x G*&\"#5\"\"\"-%$sinG6#*(%#PiGF*F'F*\"\"$!\"\"F*" }{TEXT -1 21 " newton s acts on it. " }}{PARA 0 "" 0 "" {TEXT -1 132 "Find the work done in \+ moving the particle from a distance of 1 metre from the fixed point to a distance of 2 metres from the point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 18 "The work done is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` W` = Int(F(x),x = 1 .. 2);" "6#/%#~WG -%$IntG6$-%\"FG6#%\"xG/F+;\"\"\"\"\"#" }{XPPEDIT 18 0 "`` = Int(10*sin (Pi*x/3),x = 1 .. 2);" "6#/%!G-%$IntG6$*&\"#5\"\"\"-%$sinG6#*(%#PiGF*% \"xGF*\"\"$!\"\"F*/F0;F*\"\"#" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=3/Pi" "6#/%!G*&\"\"$\"\"\"%#PiG!\"\" " }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(-10*cos(Pi/3*x))" "6#-%!G6#,$*&\" #5\"\"\"-%$cosG6#*(%#PiGF)\"\"$!\"\"%\"xGF)F)F0" }{TEXT -1 1 " " } {XPPEDIT 18 0 "PIECEWISE([2, ``],[``, ``],[1, ``]);" "6#-%*PIECEWISEG6 %7$\"\"#%!G7$F(F(7$\"\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 3/Pi " "6#/%!G*&\"\"$\"\"\"%#PiG!\"\"" }{XPPEDIT 18 0 "``(-10*cos(2*Pi/3)-(-1 0*cos(Pi/3)))" "6#-%!G6#,&*&\"#5\"\"\"-%$cosG6#*(\"\"#F)%#PiGF)\"\"$! \"\"F)F1,$*&F(F)-F+6#*&F/F)F0F1F)F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3/Pi" "6#/%!G*&\"\"$\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(5+5)" "6#-%!G6#,&\"\"&\"\"\"F'F(" }{XPPEDIT 18 0 " ``= 30/Pi" "6 #/%!G*&\"#I\"\"\"%#PiG!\"\"" }{TEXT -1 1 " " }{TEXT 290 1 "~" }{TEXT -1 14 " 9.55 joules. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Int(10*sin(Pi*x/3),x=1..2);\nvalue(%);\ne valf(evalf(%,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,$*&\"# 5\"\"\"-%$sinG6#,$*(\"\"$!\"\"%#PiGF)%\"xGF)F)F)F)/F2;F)\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#I\"\"\"%#PiG!\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+'e'H\\&*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT 305 8 " Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 165 "A spherical tank of radius 4 metres is filled with water to a height of 6 metres. Find the work required to pump the water out through a hole in the to p of the tank." }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 45 ": T ake the density of water to be 1000 kg / m" }{XPPEDIT 18 0 "``^3" "6#* $%!G\"\"$" }{TEXT -1 52 ", and the gravitational acceleration to be 9. 81 m / " }{XPPEDIT 18 0 "s^2;" "6#*$%\"sG\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution " }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 506 436 436 {PLOTDATA 2 "6=-%'CURVESG6$7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPi j!**!#=$\"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5 ')yke[#*F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G ._U!zF/$\"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>j ubk6'F/$\"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3 nW'R,#QF/$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB ^hAo8X8F/$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!3 3)\\T#fB[i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3% *yV1J*3Jx$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[i gNw%*[RgF/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/ U!zF/$\"3u3HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2** F/$\"31V?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3** F/$!3q#4*>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4 &=j`Bsw$F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H! pUCrgF/7$$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF /7$$!3g@UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E* F/7$$!30yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/ 7$$!3EzyOHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$ $\"3EKRqX8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$ \"3O7w4w0I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\" 33/wY'Q+$*4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mG Ap))oa'*F/$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36Y KhSr8/#)!#F-%'COLOURG6&%$RGBGF*F*F*-F$6$7S7$$\"3a+++SSDg')F/$\"3++++++ ++]F/7$$\"3%R0!3x9;z&)F/$\"35mdD=y_O^F/7$$\"3GN$\\3PexP)F/$\"3g#\\ff*) GLD&F/7$$\"3=mXyT3^4!)F/$\"3!Hj&[K5J!Q&F/7$$\"3M-8\"z&F/7$$\"3l\\GShaXRVF/$\"3'*>p (Qh-a'eF/7$$\"3InC6zzL3LF/$\"3[.g5^l:CfF/7$$\"3W^27q'[q=#F/$\"3-Q\"[M \"oenfF/7$$\"3oQ0CgE#\\;\"F/$\"3CgVPO<\"4*fF/7$$!39KsP8\")4r^Fao$\"318 ;t@)*****fF/7$$!3c\"*\\gNV%*z6F/$\"3iY>wGZn!*fF/7$$!3%\\k?e'4m\"H#F/$ \"37#>=\\D`V'fF/7$$!3MP5O'=3wE$F/$\"3gxAe`m3EfF/7$$!3C%HM>7tnO%F/$\"3s kyI2&oN'eF/7$$!3<3]r88NI_F/$\"3G;)RQ+Bqz&F/7$$!3g4.H`!G&RhF/$\"3Dcn.1s F0dF/7$$!33u@TS;CXoF/$\"3aq6VQZc7cF/7$$!3W@Tq<'z\"*\\(F/$\"34H$\\4/k,] &F/7$$!3A^7C:`A%*zF/$\"3Qm[*>HvXQ&F/7$$!3#oTNQXV*o$)F/$\"3[p9m#Q%=d_F/ 7$$!3K:d+v'R(z&)F/$\"3h/_]+D/O^F/7$$!3k1w@#R7-m)F/$\"3y\"\\O%485.]F/7$ $!3=\"ft\"Q!f4e)F/$\"3i!4!Q9o)\\'[F/7$$!3c12gak)zP)F/$\"3iYC<(prnu%F/7 $$!3vp**R+/AA!)F/$\"34\\\"ojkxKi%F/7$$!3)fg@%\\b17vF/$\"3hZYN64U-XF/7$ $!3QvS$=b'\\\")oF/$\"3ODq4tb(GR%F/7$$!3Ia\"Q$Q2$z:'F/$\"3sxJz=_'oH%F/7 $$!3GB79nO?%F/7$$!3VLnl`H1AVF/$\"3ba'Q)ewVLTF/7$$!3' >!3\"pgCtE$F/$\"3Spk)f++R2%F/7$$!3G8(RRmq\"fAF/$\"3z0GSI^iMSF/7$$!3eZ7 KQ'H&=6F/$\"3!3;$p+eP3SF/7$$!3QB7g>LSF/7$$\"3'[J:l%) )*=K$F/$\"3eHHy!G#\\wSF/7$$\"3un$[FS&)HL%F/$\"3D(RW*Q_;MTF/7$$\"3#HP%* yk_3H&F/$\"3g/^\"3F>$3UF/7$$\"3SjzI&[u\"[hF/$\"37V6%RzFdH%F/7$$\"3KX]4 &=b<%oF/$\"3Qpq#[9;pQ%F/7$$\"3$RC3>l'>7vF/$\"35=?u-tW-XF/7$$\"3gF\"4u< &3*)zF/$\"3W!GHB9.Sh%F/7$$\"3SOIYGe?h$)F/$\"3u$\\Kgh\"[RZF/7$$\"3MGYWq ^nz&)F/$\"3Q-`QpK!R'[F/7$Fa[l$\"3u8/#3++++&F/-Fjz6&F\\[l$\")!\\DP\"!\" )Fejl$\")viobFgjl-F$6$7S7$$\"3m*****fOeCo*F/$!3++++++++DF/7$$\"3Y>(QW7 %z\"f*F/$!3iLUu\"=sMO#F/7$$\"3;@yF1'=mO*F/$!3S20//6nYAF/7$$\"3i]Af_`! \\&*)F/$!3`mV^n*)o>@F/7$$\"3#efH4*)*Hy$)F/$!3U[Y01zv)*>F/7$$\"3!)e$oA7 fKl(F/$!3i/x*HVTu)=F/7$$\"3qbd>^HndoF/$!3sj2\"*Hd/%z\"F/7$$\"3'e$=76GB AfF/$!3^yy/&4o)3XCF/$!3Vh=b'=8C`\"F/7$$\"3+YW:GJU-8F /$!3IScij#)34:F/7$$!3iFYw[MY\"y&Fao$!3%pQo#y,++:F/7$$!3+Elxry@>8F/$!3# G0Q7FD$4:F/7$$!3[^&o*\\[:iDF/$!3(y!=3XnkN:F/7$$!3;h8a8qH`OF/$!3'=sF/7$$!3H\"H%Q9c\"y$*)F/$!3KL^+3ZU:@F/7$ $!3W8-XHKwc$*F/$!3_I&Qth:GC#F/7$$!3=0c=8]j#F/7$$!3%\\ 3k;htoO*F/$!3Q`v#GIGKv#F/7$$!3u^M5r]6p*)F/$!3#4&=j`BswGF/7$$!3[7wH3Yu) R)F/$!3Q_`k)3zv*HF/7$$!39EK)o.ZPp(F/$!3juH!pUCr5$F/7$$!3M`%eW'ex%)oF/$ !3FAo?\"yMJ?$F/7$$!3))eoFG'=l&eF/$!3Kj2yeGL'H$F/7$$!32*[UFG8A$[F/$!3YX 8;TBcmLF/7$$!3]]v2i*zHl$F/$!3gIN,%***4EMF/7$$!3$3;o\")eHe_#F/$!3?%>(fp [PlMF/7$$!3+\">=G:a0D\"F/$!3?RoI*>C;\\$F/7$$!3Gz_Pw()HgHFas$!3_gB>E`** *\\$F/7$$\"3o7C-zE$oC\"F/$!3wu%3X=u;\\$F/7$$\"3o$p=`93RZ#F/$!3Y[o%H'z! oY$F/7$$\"3iAM&o\"e*Rr$F/$!3Uqq@>x]BMF/7$$\"3m%*[1_\\UW[F/$!3w-c0hZ$eO $F/7$$\"3hV]j)3``\"fF/$!3S&*[=H2o\"H$F/7$$\"3?G+(>/oQ(oF/$!3)o&)eg?sU? 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This layer can be regarded as being a disc of r adius " }{TEXT 308 1 "r" }{TEXT -1 13 " metres where" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r^2+(4-x)^2=4^2" "6#/,&*$%\"rG\" \"#\"\"\"*$,&\"\"%F(%\"xG!\"\"F'F(*$F+F'" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "r^2=16-(4-x)^2" "6#/*$%\"rG\"\"#,&\"#;\"\"\"*$,&\"\"%F) %\"xG!\"\"F&F." }{XPPEDIT 18 0 "``=16-(16-8*x+x^2)" "6#/%!G,&\"#;\"\" \",(F&F'*&\"\")F'%\"xGF'!\"\"*$F+\"\"#F'F," }{XPPEDIT 18 0 "``=8*x-x^2 " "6#/%!G,&*&\"\")\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 27 "The volume of this disc is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*r^2*Delta*x = Pi*(8*x-x^2)*D elta*x;" "6#/**%#PiG\"\"\"*$%\"rG\"\"#F&%&DeltaGF&%\"xGF&**F%F&,&*&\" \")F&F+F&F&*$F+F)!\"\"F&F*F&F+F&" }{TEXT -1 16 " cubic metres. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 150 "The mass of this disc is given by its volume times its density, and its weight is obtained by multiplying its mass by the gravitational acceleration . " }}{PARA 0 "" 0 "" {TEXT -1 41 "Hence the weight of the disc elemen t is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*x*(8-x)* Delta*x*9810;" "6#*.%#PiG\"\"\"%\"xGF%,&\"\")F%F&!\"\"F%%&DeltaGF%F&F% \"%5)*F%" }{TEXT -1 11 " newtons. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 71 "The distance of the disc element from th e top of the spherical tank is " }{XPPEDIT 18 0 "8 - x" "6#,&\"\")\"\" \"%\"xG!\"\"" }{TEXT -1 149 " metres. Hence the work required to raise it to the top of the spherical tank is obtained by multiplying its we ight by the distance moved to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Pi*x*(8-x)*Delta*x*9810*(8-x);" "6#*0%#PiG\"\"\" %\"xGF%,&\"\")F%F&!\"\"F%%&DeltaGF%F&F%\"%5)*F%,&F(F%F&F)F%" }{TEXT -1 10 " joules. " }}{PARA 0 "" 0 "" {TEXT -1 28 "This \"element of wo rk\" is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "9810*Pi *x*(8-x)^2*Delta*x = 9810*Pi*x*(64-16*x+x^2)*Delta*x;" "6#/*.\"%5)*\" \"\"%#PiGF&%\"xGF&,&\"\")F&F(!\"\"\"\"#%&DeltaGF&F(F&*.F%F&F'F&F(F&,( \"#kF&*&\"#;F&F(F&F+*$F(F,F&F&F-F&F(F&" }{XPPEDIT 18 0 "`` = 9810*Pi*x *(64-16*x+x^2)*Delta*x;" "6#/%!G*.\"%5)*\"\"\"%#PiGF'%\"xGF',(\"#kF'*& \"#;F'F)F'!\"\"*$F)\"\"#F'F'%&DeltaGF'F)F'" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 9810*Pi*(64*x-16*x^2+x ^3)*Delta*x;" "6#/%!G*,\"%5)*\"\"\"%#PiGF',(*&\"#kF'%\"xGF'F'*&\"#;F'* $F,\"\"#F'!\"\"*$F,\"\"$F'F'%&DeltaGF'F,F'" }{TEXT -1 10 " joules. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "The wo rked required to raise all the water in the tank to the level of the t op of the tank (in joules) is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Limit(``,Delta*x=0)" "6#-%&LimitG6$%!G/*&%&DeltaG\"\"\" %\"xGF*\"\"!" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(9810*Pi*(64*x-16*x^2 +x^3)*Delta*x,x = 0 .. 6);" "6#-%$SumG6$*,\"%5)*\"\"\"%#PiGF(,(*&\"#kF (%\"xGF(F(*&\"#;F(*$F-\"\"#F(!\"\"*$F-\"\"$F(F(%&DeltaGF(F-F(/F-;\"\"! \"\"'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=9810*Pi*Int(``(64*x-16*x^2+x^3),x = 0 .. 6)" "6#/%!G*(\"%5)* \"\"\"%#PiGF'-%$IntG6$-F$6#,(*&\"#kF'%\"xGF'F'*&\"#;F'*$F1\"\"#F'!\"\" *$F1\"\"$F'/F1;\"\"!\"\"'F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 9810*Pi*(32*x^2-16*x^3/3+x^4/4);" "6#/%!G*(\"%5)*\"\"\"%#PiGF',(*&\"#KF'*$%\"xG\"\"#F'F'*(\"#;F'*$F-\"\" $F'F2!\"\"F3*&F-\"\"%F5F3F'F'" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEW ISE([6, ``],[``, ``],[0, ``]);" "6#-%*PIECEWISEG6%7$\"\"'%!G7$F(F(7$\" \"!F(" }{XPPEDIT 18 0 "`` = 9810*Pi*x^2*(32-16*x/3+x^2/4);" "6#/%!G** \"%5)*\"\"\"%#PiGF'%\"xG\"\"#,(\"#KF'*(\"#;F'F)F'\"\"$!\"\"F0*&F)F*\" \"%F0F'F'" }{TEXT -1 2 " " }{XPPEDIT 18 0 "PIECEWISE([6,``],[``,``],[ 0,``])" "6#-%*PIECEWISEG6%7$\"\"'%!G7$F(F(7$\"\"!F(" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=9180*Pi*`.`*36*(32-32+9)" "6#/%!G*,\"%!=*\"\"\"%#PiG F'%\".GF'\"#OF',(\"#KF'F,!\"\"\"\"*F'F'" }{XPPEDIT 18 0 "``=3178440*Pi " "6#/%!G*&\"(S%yJ\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Hence the work required t o pump out all the water is approximately 9.985 " }{TEXT 309 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 9 " jou les. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "9810*Pi*Int(64*x-16*x^2+ x^3,x=0..6);\n``=value(%);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"%5)*\"\"\"%#PiGF&-%$IntG6$,(*&\"#kF&%\"xGF&F&*&\" #;F&)F.\"\"#F&!\"\"*$)F.\"\"$F&F&/F.;\"\"!\"\"'F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G,$*&\"(S%yJ\"\"\"%#PiGF(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+bPO&)**!\"$" }}}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT 297 8 "Question " }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 82 "A uniform cable of l ength 100 metres weighs 400 newtons and is hanging vertically." }} {PARA 0 "" 0 "" {TEXT -1 54 "(a) Find the work done in winding up the \+ whole cable. " }}{PARA 0 "" 0 "" {TEXT -1 62 "(b) Find the work done i n winding up 60 metres of this cable. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 298 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 54 "First note that the cable weighs 4 newtons per me tre. " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 362 371 371 {PLOTDATA 2 "65-%'CURVESG6$7in7$$\"\" \"\"\"!$F*F*7$$\"3A+++a*=_%**!#=$\"3/+++KYGX5F/7$$\"3a*****p+w9y*F/$\" 3#******z!p6z?F/7$$\"3))*****H;l0^*F/$\"39+++V*p,4$F/7$$\"3_*****pdaa8 *F/$\"35+++IkOnSF/7$$\"3Q+++QSDg')F/$\"3#*******)*******\\F/7$$\"3m*** **R%*p,4)F/$\"3'******>__y(eF/7$$\"3[*****fD[9V(F/$\"3')*****H118p'F/7 $$\"3-+++lgI\"p'F/$\"3U+++a#[9V(F/7$$\"39+++CD&y(eF/$\"3e*****H%*p,4)F /7$$\"3;+++-+++]F/$\"3I+++PSDg')F/7$$\"3')*****RVmt1%F/$\"3Z+++vXXN\"* F/7$$\"3++++[*p,4$F/$\"3z*****>;l0^*F/7$$\"3')*****R\"p6z?F/$\"3c+++1g Z\"y*F/7$$\"31+++RYGX5F/$\"39+++`*=_%**F/7$$\"3y*****>>m*[z!#FF(7$$!3% ******Rj%GX5F/F-7$$!3#******z!p6z?F/F37$$!31+++U*p,4$F/F87$$!3#******z Umt1%F/$\"3g*****zdaa8*F/7$$!3u*****p*******\\F/$\"3a+++SSDg')F/7$$!3s ******=D&y(eF/$\"3#******p%*p,4)F/7$$!3_******egI\"p'F/$\"3t******e#[9 V(F/7$$!34+++]#[9V(F/$\"3O+++pgI\"p'F/7$$!3O+++R*p,4)F/$\"3_******HD&y (eF/7$$!3Q+++QSDg')F/$\"3++++++++]F/7$$!3b+++wXXN\"*F/$\"3E+++KkOnSF/7 $$!3z*****>;l0^*F/$\"3%)*****f%*p,4$F/7$$!3a*****p+w9y*F/$\"3)******> \"p6z?F/7$$!39+++`*=_%**F/$\"3/+++PYGX5F/7$$!\"\"F*$\"3I+++&QKz*eF`p7$ $!3A+++a*=_%**F/$!3'******fi%GX5F/7$$!3q******3gZ\"y*F/$!35++++p6z?F/7 $$!38+++m^c5&*F/$!3/+++N*p,4$F/7$$!3&)*****4eaa8*F/$!3!******4Umt1%F/7 $$!3w*****R/a-m)F/$!3G+++!*******\\F/7$$!3%)*****f%*p,4)F/$!3!)******> D&y(eF/7$$!3k*****zD[9V(F/$!3g******fgI\"p'F/7$$!3E+++ogI\"p'F/$!3<+++ ^#[9V(F/7$$!3Y+++GD&y(eF/$!3W+++S*p,4)F/7$$!3Z*****p++++&F/$!3/+++MSDg ')F/7$$!3s******QkOnSF/$!3I+++tXXN\"*F/7$$!3%******R&*p,4$F/$!3j****** f^c5&*F/7$$!3++++>p6z?F/$!3[+++0gZ\"y*F/7$$!3-+++XYGX5F/$!31+++_*=_%** F/7$$!3#)*****zd)*o%QF`pFbt7$$\"3++++GYGX5F/Fgt7$$\"3)******>!p6z?F/F \\u7$$\"3?+++P*p,4$F/$!3/+++l^c5&*F/7$$\"31+++BkOnSF/$!3x******zXXN\"* F/7$$\"3!)*****4*******\\F/$!3o*****H/a-m)F/7$$\"3U+++9D&y(eF/$!3;+++] *p,4)F/7$$\"3I+++bgI\"p'F/$!31+++j#[9V(F/7$$\"3w*****fC[9V(F/$!3m***** R218p'F/7$$\"3/+++N*p,4)F/$!3%******\\`_y(eF/7$$\"3s******HSDg')F/$!30 +++9+++]F/7$$\"3Q+++uXXN\"*F/$!37+++PkOnSF/7$$\"3j******f^c5&*F/$!3y** ***>&*p,4$F/7$$\"3[+++0gZ\"y*F/$!37+++F+7$ $\"3m*****R+Q2*[F]^l$\"32+++a%e&R5F]^l7$$\"39+++)GFxc%F]^l$\"37+++:KoL ?F]^l7$$\"37+++s\\3XSF]^l$\"38+++hi#*QHF]^l7$$\"3q*****>.`cM$F]^l$\"3@ +++FTs:PF]^l7$$\"3))*****4+++]#F]^l$\"3p*****z,F,L%F]^l7$$\"3)******R( \\3X:F]^l$\"3w*****4e#GbZF]^l7$$\"3n*****\\>BkA&!#?$\"3!******fZ4E(\\F ]^l7$$!3?+++qJUE_F_`l$\"3)******pZ4E(\\F]^l7$$!33+++r\\3X:F]^l$\"3%)** ***>e#GbZF]^l7$$!3'******z******\\#F]^l$\"3')******>q7IVF]^l7$$!3A+++I IlXLF]^l$\"3w******HTs:PF]^l7$$!3'*******p\\3XSF]^l$\"36+++li#*QHF]^l7 $$!3)******fGFxc%F]^l$\"3%)******>KoL?F]^l7$$!3?+++-!Q2*[F]^l$\"30+++g %e&R5F]^l7$$!3G+++++++]F]^l$\"3O+++!>m*[z!#G7$$!3m*****R+Q2*[F]^l$!3)* *****\\Xe&R5F]^l7$$!39+++)GFxc%F]^l$!37+++:KoL?F]^l7$$!3?+++t\\3XSF]^l $!3/+++gi#*QHF]^l7$$!3')*****R.`cM$F]^l$!38+++ETs:PF]^l7$$!37+++/+++DF ]^l$!3J+++ \+ " 0 "" {MPLTEXT 1 0 101 "\{a=6.37*10^6,b=7.37*10^6,m[1]=5.98*10^24,m[2 ]=10^3,G=6.67*10^(-11)\};\nsubs(%,G*m[1]*m[2]*(b-a)/(a*b));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<'/%\"GG$\"++++qm!#?/&%\"mG6#\"\"#\"%+5/&F+6 #\"\"\"$\"++++!)f\"#:/%\"bG$\"*+++P(!\"#/%\"aG$\"*+++P'F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+M%4h\\)\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }} {PARA 0 "" 0 "" {TEXT -1 33 "When a particle is at a distance " } {TEXT 291 1 "x" }{TEXT -1 39 " metres from a fixed point a force of \+ " }{XPPEDIT 18 0 "F(x) = 5*sqrt(x);" "6#/-%\"FG6#%\"xG*&\"\"&\"\"\"-%% sqrtG6#F'F*" }{TEXT -1 21 " newtons acts on it. " }}{PARA 0 "" 0 "" {TEXT -1 133 "Find the work done in moving the particle from a distanc e of 2 metres from the fixed point to a distance of 4 metres from the \+ point. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int (5*sqrt(x),x=2..4)=20/3" "6#/-%$IntG6$*&\"\"&\"\"\"-%%sqrtG6#%\"xGF)/F -;\"\"#\"\"%*&\"#?F)\"\"$!\"\"" }{XPPEDIT 18 0 "``(4-sqrt(2))" "6#-%!G 6#,&\"\"%\"\"\"-%%sqrtG6#\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 292 1 "~" }{TEXT -1 15 " 17.24 joules. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 " }}{PARA 0 "" 0 "" {TEXT -1 135 "A force of 120 N is sufficient to keep a spring with a natural length of 18 cm extende d by 2 cm so that it has a total length of 20 cm." }}{PARA 0 "" 0 "" {TEXT -1 65 "Find the work required to extend the spring from 20 cm to 22 cm. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "In t(6000*x,x = `0.02` .. `0.04`) = 3.6;" "6#/-%$IntG6$*&\"%+g\"\"\"%\"xG F)/F*;%%0.02G%%0.04G-%&FloatG6$\"#O!\"\"" }{TEXT -1 4 " J. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________ _________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "2 joules of work is need ed to stretch a spring from its natural length of 30 cm to a length of 42 cm. " }}{PARA 0 "" 0 "" {TEXT -1 61 "How much work is required to stretch it from 35 cm to 40 cm? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(2500/9*x,x=1/20..1/10)=25/24" "6#/-%$IntG6$ *(\"%+D\"\"\"\"\"*!\"\"%\"xGF)/F,;*&F)F)\"#?F+*&F)F)\"#5F+*&\"#DF)\"#C F+" }{TEXT -1 1 " " }{TEXT 282 1 "~" }{TEXT -1 10 " 1.042 J. " }}} {PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q4 " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The force acting on a body at a distance " }{TEXT 316 1 "s" }{TEXT -1 23 " from a fixed point is " }{XPPEDIT 18 0 "F=2*s+1/s^2" "6#/%\"FG,&*&\"\"#\" \"\"%\"sGF(F(*&F(F(*$F)F'!\"\"F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "(a) Find the work done when the body moves from the posit ion where " }{XPPEDIT 18 0 "s=1/2" "6#/%\"sG*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 6 " to " }{XPPEDIT 18 0 "s = 2;" "6#/%\"sG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 65 "(b) What is the average value of the for ce during this movement? " }}{PARA 0 "" 0 "" {TEXT -1 6 " " } {TEXT 259 4 "Note" }{TEXT -1 46 ": The average value of a continuous f unction " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 17 " on a n interval " }{XPPEDIT 18 0 "[a,b]" "6#7$%\"aG%\"bG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "1/(b-a)" "6#*&\"\"\"F$,&%\"bGF$%\"aG!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(f(x),x=a..b)" "6#-%$IntG6$-%\"fG6#% \"xG/F);%\"aG%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "Int(``(2*s+1 /(s^2)),s = 1/2 .. 5/2) = 38/5;" "6#/-%$IntG6$-%!G6#,&*&\"\"#\"\"\"%\" sGF-F-*&F-F-*$F.F,!\"\"F-/F.;*&F-F-F,F1*&\"\"&F-F,F1*&\"#QF-F6F1" } {XPPEDIT 18 0 "``=7" "6#/%!G\"\"(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "3/5 " "6#*&\"\"$\"\"\"\"\"&!\"\"" }{TEXT -1 9 ". (b) " }{XPPEDIT 18 0 " 19/5=3" "6#/*&\"#>\"\"\"\"\"&!\"\"\"\"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/5" "6#*&\"\"%\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}}{PARA 0 "" 0 " " {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "____________________ _________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 247 "A tank has the shape of an inverted circular cone with a height of 10 metres and a base radius of 4 metre s. It is filled with water to a height of 8 metres. Find the work requ ired to empty the tank by pumping all the water to the top of the tank . " }}{PARA 0 "" 0 "" {TEXT -1 43 "Take the density of water to be 100 0 kg / m" }{XPPEDIT 18 0 "``^3" "6#*$%!G\"\"$" }{TEXT -1 52 ", and the gravitational acceleration to be 9.81 m / " }{XPPEDIT 18 0 "s^2;" "6# *$%\"sG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "9810*Pi*Int(4/25*x^2*(10-x),x = 0 .. 8)=9810*P i*``(8192/75)" "6#/*(\"%5)*\"\"\"%#PiGF&-%$IntG6$**\"\"%F&\"#D!\"\"%\" xG\"\"#,&\"#5F&F/F.F&/F/;\"\"!\"\")F&*(F%F&F'F&-%!G6#*&\"%#>)F&\"#vF.F &" }{TEXT -1 1 " " }{TEXT 314 1 "~" }{TEXT -1 6 " 3.37 " }{TEXT 315 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^6" "6#*$\"#5\"\"'" }{TEXT -1 9 " joules. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q6 " }}{PARA 0 "" 0 "" {TEXT -1 112 "Find \+ the work done in winding up 20 metres of a uniform rope which is 25 me tres long and weighs of 150 newtons. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(6*(25-x) ,x = 0 .. 20) = 1800;" "6#/-%$IntG6$*&\"\"'\"\"\",&\"#DF)%\"xG!\"\"F)/ F,;\"\"!\"#?\"%+=" }{TEXT -1 9 " joules. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q7 " }}{PARA 0 "" 0 "" {TEXT -1 75 "Some s prings behave in a different way than that suggested by Hooke's law. \+ " }}{PARA 0 "" 0 "" {TEXT -1 94 "The force required to extend the spri ng may increase more rapidly as the extension increases. " }}{PARA 0 " " 0 "" {TEXT -1 63 "Suppose that the force needed to extend a particul ar spring by " }{TEXT 286 1 "x" }{TEXT -1 35 " metres from its natural length is " }{XPPEDIT 18 0 "F(x) = 800*x+400*x^3;" "6#/-%\"FG6#%\"xG, &*&\"$+)\"\"\"F'F+F+*&\"$+%F+*$F'\"\"$F+F+" }{TEXT -1 10 " newtons. " }}{PARA 0 "" 0 "" {TEXT -1 78 "Find the work done in extending the spr ing by 20 cm, from its natural length. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(800*x+400*x^3,x = 0 .. 1/5)=404/25 " "6#/-%$IntG6$,&*&\"$+)\"\"\"%\"xGF*F**&\"$+%F**$F+\"\"$F*F*/F+;\"\"! *&F*F*\"\"&!\"\"*&\"$/%F*\"#DF5" }{XPPEDIT 18 0 "`` = 16.16" "6#/%!G-% &FloatG6$\"%;;!\"#" }{TEXT -1 9 " joules. " }}}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 37 "_______________________________ ______" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for pictures" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 30 "work done b y a variable force " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 585 "f := x -> 2+sin(x)/5-cos(1.3*x+1)/7+arctan (x):\na := 0: b := 5.5:\np1 := plot(f(x),x=-1..7,color=red,thickness=2 ):\np2 := plot(f(x),x=a..b,filled=true,color=COLOR(RGB,.9,.9,.95)):\np 3 := plot([[[a,0],[a,f(a)]],[[b,0],[b,f(b)]]],color=black):\np4 := plo ttools[arrow]([-1.2,0],[7,0],0,.1,.02,arrow,color=black):\np5 := plott ools[arrow]([-1,-.2],[-1,4.3],0,.15,.03,arrow,color=black):\nt1 := plo ts[textplot]([[7,-.15,`x`],[-1.5,4.1,`F(x)`],\n [a,-.2,`x = a`],[ b,-.2,`x = b`],[2.75,1.75,`area = work done`]],color=black):\nplots[di splay]([p1,p2,p3,p4,p5,t1],axes=none,view=[-1.5..7,-.4..4.3]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 16 "spring pictures " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "natural s tate " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 664 "a := evalf(-Pi/2): b := evalf(22.5*Pi):\nr1 := 0.015:\nf := t -> -0.04*sin(t)-r1*t:\ne : = .05-f(b): \nc := f(a)-f(b)+.1:\nzz := c+.04:\np1 := plot([e+f(t),.5+ .3*cos(t),t=a..b],\n color=COLOR(RGB,0,.6,0),thickness=2): \np2 := plot([[0,1],[0,0],[2.1,0]],thickness=2,color=COLOR(RGB,.2,.2,. 2)):\np3 := plot([[[0,.5],[.053,.5]],[[c-.05,.5],[c,.5]]],color=COLOR( RGB,0,.6,0),thickness=2):\np4 := plots[polygonplot]([[c,.03],[c,.98],[ c+.08,.98],[c+.08,.03]],\n color=COLOR(RGB,.8,.8,.8),style=patch): \np5 := plot([[c+.04,0],[c+.04,-.1]],color=black):\nt1 := plots[textpl ot]([[c+.04,-.2,`0`],[c/2+.03,1.2,`natural state`]]):\nplots[display]( [p1,p2,p3,p4,p5,t1],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 16 "stretched state " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 798 "a := evalf(-Pi/2): b := evalf(22.5*Pi):\nr1 := \+ 0.023:\ng := t -> -0.04*sin(t)-r1*t:\ne := .05-f(b): \nc := f(a)-f(b)+ .1:\np1 := plot([e+f(t),.5+.3*cos(t),t=a..b],\n color=COLO R(RGB,0,.6,0),thickness=2):\np2 := plot([[0,1],[0,0],[2.1,0]],thicknes s=2,color=COLOR(RGB,.2,.2,.2)):\np3 := plot([[[0,.5],[.053,.5]],[[c-.0 5,.5],[c,.5]]],color=COLOR(RGB,0,.6,0),thickness=2):\np4 := plots[poly gonplot]([[c,.03],[c,.98],[c+.08,.98],[c+.08,.03]],\n color=COLOR(R GB,.8,.8,.8),style=patch):\np5 := plot([[[zz,0],[zz,-.1]],[[c+.04,0],[ c+.04,-.1]]],color=black):\np6 := plottools[arrow]([c+.08,.505],[c+.2, .505],.02,.11,.17,color=black):\nt1 := plots[textplot]([[zz,-.2,`0`],[ c+.04,-.18,`x`],[c/2+.03,1.2,`stretched state`],\n [c+.1,1.25,` F(x) = k x`]]):\nplots[display]([p1,p2,p3,p4,p5,p6,t1],axes=none);" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 15 "spherical tank " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1460 "r1 := evalf(sqrt(3)/2): \nx2 := .25: r2 := evalf(sqrt(1-x2^2)) :\nx3 := .2: r3 := evalf(sqrt(1-x3^2)):\nx4 := .225: r4 := evalf(sqrt( 1-x4^2)):\np1 := plot([cos(t),sin(t),t=0..2*Pi],color=black):\np2 := p lot([r1*cos(t),.5+.1*sin(t),t=0..2*Pi],color=navy):\np3 := plot([r2*co s(t),-.25+.1*sin(t),t=0..2*Pi],color=black):\np4 := plot([r3*cos(t),-x 3+.1*sin(t),t=0..2*Pi],color=black):\np5 := plot([[[r4,-x4],[r4+.25,-x 4]],[[0,-1],[1.5,-1]],\n [[-r3,-x3],[-r3-.2,-x3]],[[-r2,-x2],[-r2-.2 1,-x2]],\n [[r1,.5],[r1+.6,.5]]],color=black,linestyle=2):\np6 := \+ plottools[arrow]([1.1,-.69],[1.1,-1],0,.07,.15,arrow,color=black):\np7 := plottools[arrow]([1.1,-.54],[1.1,-x4],0,.07,.15,arrow,color=black) :\np8 := plottools[arrow]([-1.1,0],[-1.1,-x3],0,.07,.2,arrow,color=bla ck):\np9 := plottools[arrow]([-1.1,-.42],[-1.1,-x2],0,.07,.2,arrow,col or=black):\np10 := plottools[arrow]([1.4,-.15],[1.4,.5],0,.07,.1,arrow ,color=black):\np11 := plottools[arrow]([1.4,-.35],[1.4,-1],0,.07,.1,a rrow,color=black):\np12 := plot([[0,0],[r2,-x2],[0,-x2],[0,0]],color=r ed,linestyle=1):\nt1 := plots[textplot]([[1.1,-.6,`x`],[-1.25,-x4,`x`] ],color=black,font=[HELVETICA,10]):\nt2 := plots[textplot]([[-1.32,-x4 +.005,`D`]],color=black,font=[SYMBOL,11]):\nt3 := plots[textplot]([1.4 ,-.24,`6`],color=black,font=[HELVETICA,9]):\nt4 := plots[textplot]([[. 5,-.05,`4`],[-.1,-.1,`4-x`],[.35,-.18,`r`]],\n color=red,f ont=[HELVETICA,10]):\nplots[display]([p||(1..12),t||(1..4)],axes=none, scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 19 "winding up a cable " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 852 "d := evalf(Pi/30): r1 := .0 5:\np1 := plot([seq([cos(i*d),sin(i*d)],i=0..60)],color=black):\np2 := plot([seq([r1*cos(2*i*d),r1*sin(2*i*d)],i=0..30)],color=black):\np3 : = plot([[1,0],[1,-7]],color=black,thickness=2):\np4 := plot([[[1,0],[1 .7,0]],[[1,-4],[1.9,-4]],[[1,-4.2],[1.9,-4.2]],\n [[1,-7],[1.7,-7]] ],color=black,linestyle=2):\np5 := plottools[arrow]([1.5,-1.7],[1.5,0] ,0,.15,.1,arrow,color=black):\np6 := plottools[arrow]([1.5,-2.3],[1.5, -4],0,.15,.1,arrow,color=black):\np7 := plottools[arrow]([1.5,-5.35],[ 1.5,-4.2],0,.15,.12,arrow,color=black):\np8 := plottools[arrow]([1.5,- 5.75],[1.5,-7],0,.15,.12,arrow,color=black):\nt1 := plots[textplot]([[ 1.6,-2,`100 - x`],[1.5,-5.51,`x`],\n [2.3,-4.08,`x`]],color=black): \nt2 := plots[textplot]([[2.1,-4.07,`D`]],color=black,font=[SYMBOL,11] ):\nplots[display]([p||(1..8),t1,t2],axes=none,scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0 " 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }