{"text_format":"3","workbook":"xlsx","n_sheets":3,"sheets":["No Damping","Damping","Sheet2"],"not_rendered":{"charts":5,"images":8,"defined_names":31,"comments":2},"read":"whole","date_system":"1900"}

{"sheet":"No Damping","index":0,"n_rows":37,"n_cols":11,"rows_kept":37,"merged":[[30,2,30,3]]}
["     The Spring: Simple Harmonic Motion"]
[]
[null,"This DIGMath module lets you investigate "]
[null,null,"simple harmonic motion:"]
[null,"        the behavior of a simple spring"]
[null,"with a bob of a given mass hanging from it "]
[null,"         with an initial displacement y0 "]
[]
["What is the initial displacement y0 (0 to 200 cm)?"]
[null,null,{"$f":"10*D10","v":50},5]
[]
[]
["What is the mass m (25-500 gms)?"]
[null,null,{"$f":"D14*25","v":300},12]
[]
[]
["What is the spring constant k (1 - 10 Newton/cm)?"]
[null,null,{"$f":"D18","v":5},5]
[{"$nulls":8},"Now trace a point along the path of the weight on the spring."]
[null,"The period of the oscillation = ",null,null,{"$f":"2*PI()*SQRT(L/32)","v":7.853981633974483},"seconds",{"$nulls":4},206]
[{"$nulls":8},"t =",{"$f":"K20*dt","v":41.2},"seconds"]
["Click each item below for suggestions and investigations",{"$nulls":7},"y =",{"$f":"LOOKUP(J$21+0.001,Sheet2!C3:C603,Sheet2!D3:D603)","v":{"v":28.500094209448605,"d":"28.50"}},"cm above/below the equilibrium point"]
[{"$nulls":3},"   Item 1",{"$nulls":4},"v =",{"$f":"LOOKUP(J$21,Sheet2!C3:C602,Sheet2!E3:E602)","v":{"v":5.350593483470583,"d":"5.35"}},"cm/sec"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["         Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Damping","index":1,"n_rows":43,"n_cols":10,"rows_kept":43,"merged":[[36,2,36,3]]}
["        Motion of a Spring with Damping"]
[]
[null,"This DIGMath module lets you investigate "]
[null,null,"the motion of a spring "]
[null,"with a bob of a given mass hanging from it,"]
[null,"         with an initial displacement y0 "]
[null," and with damping that retards the motion"]
[]
["What is the initial displacement y0 (0 to 200 cm)?"]
[null,null,{"$f":"10*D10","v":110},11]
[]
[]
["What is the mass m (25-500 gms)?"]
[null,null,{"$f":"D14*25","v":175},7,7]
[]
[]
["What is the spring constant k (1 - 10 Newton/cm)?"]
[null,null,{"$f":"D18","v":5},5]
[{"$nulls":7},"Now trace a point along the path of the weight on the spring."]
[{"$nulls":9},120]
["What is the viscous damping coefficient c (0 - 500 gms/sec)?",{"$nulls":6},"t =",{"$f":"J20*dtt","v":{"v":12,"d":"12.0"}},"seconds"]
[null,null,{"$f":"D22*25","v":175},7,{"$nulls":3},"y =",{"$f":"LOOKUP(I$21+0.001,Sheet2!I3:I603,Sheet2!M3:M603)","v":{"v":0.32703375264228085,"d":"0.33"}},"cm above/below the equilibrium point"]
[{"$nulls":7},"v =",{"$f":"LOOKUP(I$21,Sheet2!I3:I602,Sheet2!N3:N602)","v":{"v":-0.15477902664241705,"d":"-0.15"}},"cm/sec"]
["This combination produces",null,null,{"$f":"IF(case=1,\"critical damping\",IF(case=2,\"over-damping\",\"under-damping\"))","v":"over-damping"}]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[{"$nulls":3},"   Item 6"]
[]
[]
["To access/download all the DIGMath Apps for Calculus"]
["        Click this link:",null,"DIGMath for Calculus"]
[]
["Created by:  Sheldon P. Gordon   "]
["                      Farmingdale State College    "]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sheet2","index":2,"n_rows":603,"n_cols":15,"rows_kept":603}
[null,null,"t","y","v",{"$nulls":3},"t","y1","y2","y3","y","v"]
[]
[null,null,0,{"$f":"y0*COS(SQRT(k/m)*t)","v":50},{"$f":"(D4-D3)/dq","v":-0.0833287038065933},{"$nulls":3},0,{"$f":"(H$27+H$28*tt)*EXP(-omega0*tt)","v":110},{"$f":"H$31*EXP(gamma2*tt)+H$32*EXP(gamma2*tt)","v":110},{"$f":"EXP(-cc/2/mm*tt)*(H$36*COS(SQRT(-Disc)/2/mm*tt)+H$37*SIN(SQRT(-Disc)/2/mm*tt))","v":{"$error":"#NUM!"}},{"$f":"IF(case=1,J3,IF(case=2,K3,L3))","v":110},{"$f":"(M4-M3)/dqq","v":-52.06096555204553},0]
["sqrt(g/L) =",{"$f":"SQRT(32/L)","v":0.8},{"$f":"C3+dt","v":0.2},49.98333425923868,-0.2499305620366954,null,"sqrt(k/m) =",{"$f":"SQRT(kk/mm)","v":0.1690308509457033},{"$f":"I3+dtt","v":0.1},113.56410109614188,104.79390344479545,{"$error":"#NUM!"},104.79390344479545,-49.59701633912616,0]
["period =",{"$f":"2*PI()*omega","v":5.026548245743669},0.4,49.93334814683134,-0.4163658091486866,{"$nulls":3},0.2,116.97782365328557,99.83420181088283,{"$error":"#NUM!"},99.83420181088283,-47.24968128538492,0]
[null,null,0.6000000000000001,49.850074985001605,-0.5825234944751045,{"$nulls":3},0.30000000000000004,120.24520739924152,95.10923368234434,{"$error":"#NUM!"},95.10923368234434,-45.01344125834507,0]
[null,null,0.8,49.733570286106584,-0.7482928523796772,null,"gamma =",{"$f":"cc/2/SQRT(mm*kk)","v":2.958039891549808},0.4,123.37019888798658,90.60788955650983,{"$error":"#NUM!"},90.60788955650983,-42.88303833586369,0]
[null,null,1,49.58391171563065,-0.91356337609664,null,"disc=",{"$f":"cc^2-4*mm*kk","v":27125},0.5,126.3566534881533,86.31958572292346,{"$error":"#NUM!"},86.31958572292346,-40.85346344352686,0]
[null,null,1.2,49.40119904041132,-1.078224891397781,{"$nulls":3},0.6,129.20833733103692,82.23423937857078,{"$error":"#NUM!"},82.23423937857078,-38.91994457714944,0]
[null,null,1.4,49.185554062131764,-1.2421676300380469,null,"v0 =",0,0.7,131.92892921891757,78.34224492085583,{"$error":"#NUM!"},78.34224492085583,-37.07793558267795,0]
[null,null,1.5999999999999999,48.937120536124155,-1.4052823029300754,{"$nulls":3},0.7999999999999999,134.52202249448172,74.63445136258804,{"$error":"#NUM!"},74.63445136258804,-35.32310546712296,0]
["delta t = ",0.2,1.7999999999999998,48.65606407553814,-1.5674601729998727,null,"delta t = ",0.1,0.8999999999999999,136.9911268721098,71.10214081587574,{"$error":"#NUM!"},71.10214081587574,-33.65132821538168,0]
[null,null,1.9999999999999998,48.342572040938165,-1.7285931276737898,{"$nulls":3},0.9999999999999999,139.33967023178394,67.73700799433757,{"$error":"#NUM!"},67.73700799433757,-32.058673089016736,0]
[null,null,2.1999999999999997,47.99685341540341,-1.8885737509497957,null,"gamma1 =",{"$f":"(-kk+SQRT(cc^2-4*mm*kk))/2/mm","v":0.4562762597714458},1.0999999999999999,141.57100037635408,64.5311406854359,{"$error":"#NUM!"},64.5311406854359,-30.541395384169903,0]
[null,null,2.4,47.61913866521345,-2.0472953950040917,null,"gamma2 =",{"$f":"(-kk-SQRT(cc^2-4*mm*kk))/2/mm","v":-0.4848476883428744},1.2,143.68838675288671,61.47700114701891,{"$error":"#NUM!"},61.47700114701891,-29.09592762689101,0]
[null,null,2.6,47.20967958621263,-2.20465225128585,{"$nulls":3},1.3,145.69502213880676,58.56740838432981,{"$error":"#NUM!"},58.56740838432981,-27.718871185174407,0]
[null,null,2.8000000000000003,46.76874913595546,-2.360539421052046,{"$nulls":3},1.4000000000000001,147.59402429352926,55.79552126581237,{"$error":"#NUM!"},55.79552126581237,-26.406988277980687,0]
["Left extreme",null,3.0000000000000004,46.29664125174505,-2.514852985296301,null,"Left extreme",null,1.5000000000000002,149.38843757626364,53.1548224380143,{"$error":"#NUM!"},53.1548224380143,-25.157194362459663,0]
[{"$f":"0","v":0},{"$f":"L","v":50},3.2000000000000006,45.79367065468579,-2.667490074024279,null,{"$f":"0","v":0},{"$f":"LL","v":110},1.6000000000000003,151.08123453066108,50.63910300176833,{"$error":"#NUM!"},50.63910300176833,-23.966550881468578,0]
[null,null,3.400000000000001,45.260172639880935,-2.8183489348300483,null,{"$f":"L3","v":{"$error":"#NUM!"}},{"$f":"M3","v":110},1.7000000000000004,152.67531743696176,48.242447913621476,{"$error":"#NUM!"},48.242447913621476,-22.832258354340098,0]
["Right extreme",null,3.600000000000001,44.696502852914925,-2.9673290007268704,null,"Right extreme",null,1.8000000000000005,154.17351983228588,45.959222078187466,{"$error":"#NUM!"},45.959222078187466,-21.751649794648316,0]
[{"$f":"0","v":0},{"$f":"L","v":50},3.800000000000001,44.10303705276955,-3.114330957188507,null,{"$f":"0","v":0},{"$f":"LL","v":110},1.9000000000000006,155.57860799970112,43.784057098722634,{"$error":"#NUM!"},43.784057098722634,-20.72218443950362,0]
[null,null,4.000000000000001,43.48017086133185,-3.2592568083550333,null,{"$f":"-L3","v":{"$error":"#NUM!"}},{"$f":"M3","v":110},2.0000000000000004,156.8932824266852,41.71183865477227,{"$error":"#NUM!"},41.71183865477227,-19.741441775623514,0]
[null,null,4.200000000000001,42.82831949966084,-3.4020099423598893,{"$nulls":3},2.1000000000000005,158.12017923359187,39.73769447720992,{"$error":"#NUM!"},39.73769447720992,-18.807115848142928,0]
[null,null,4.400000000000001,42.147917511188865,-3.542495195734041,{"$nulls":3},2.2000000000000006,159.26187157271573,37.85698289239563,{"$error":"#NUM!"},37.85698289239563,-17.917009838776323,0]
[null,null,4.600000000000001,41.43941847204206,-3.680618916845013,null,"1. Critical damping",null,2.3000000000000007,160.32087099854056,36.065281908517996,{"$error":"#NUM!"},36.065281908517996,-17.069030900583257,0]
[null,null,4.800000000000002,40.703294688673054,-3.8162890283275885,null,"A =",{"$f":"yy0","v":110},2.400000000000001,161.29962880974335,34.35837881845967,{"$error":"#NUM!"},34.35837881845967,-16.26118523719974,0]
[null,null,5.000000000000002,39.94003688300754,-3.9494150884655,null,"B =",{"$f":"v0+cc/2/mm*yy0","v":55},2.500000000000001,162.200537363517,32.732260294739696,{"$error":"#NUM!"},32.732260294739696,-15.4915734149548,0]
[null,null,5.200000000000002,39.15015386531444,-4.0799083514825085,{"$nulls":3},2.600000000000001,163.02593136276155,31.183102953244216,{"$error":"#NUM!"},31.183102953244216,-14.7583858968612,0]
[null,null,5.400000000000002,38.334172195017935,-4.207681826702903,null,"2. Over-damped",null,2.700000000000001,163.77808911668492,29.707264363558096,{"$error":"#NUM!"},29.707264363558096,-14.05989878797012,0]
[null,null,5.600000000000002,37.492635829677354,-4.332650336542052,null,"A =",{"$f":"(v0-gamma2*yy0)/(gamma1-gamma2)","v":56.66973603697702},2.800000000000001,164.4592337753425,28.301274484761084,{"$error":"#NUM!"},28.301274484761084,-13.394469782092138,0]
[null,null,5.8000000000000025,36.626105762368944,-4.454730573288472,null,"B =",{"$f":"yy0-H31","v":53.33026396302298},2.9000000000000012,165.07153453863535,26.96182750655187,{"$error":"#NUM!"},26.96182750655187,-12.760534300352617,0]
[null,null,6.000000000000003,35.73515964771125,-4.573841154639098,{"$nulls":3},3.0000000000000013,165.61710784027622,25.68577407651661,{"$error":"#NUM!"},25.68577407651661,-12.156601812501435,0]
[null,null,6.200000000000003,34.82039141678343,-4.689902677951032,{"$nulls":3},3.1000000000000014,166.09801850722283,24.470113895266465,{"$error":"#NUM!"},24.470113895266465,-11.581252332328162,0]
[null,null,6.400000000000003,33.88241088119322,-4.802837773173891,null,"3. Under-damped",null,3.2000000000000015,166.5162808950685,23.31198866203365,{"$error":"#NUM!"},23.31198866203365,-11.033133078943749,0]
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[null,null,6.800000000000003,31.93932909567306,-5.019029670186725,null,"B =",{"$f":"1/H39*(gamma*omega0*yy0+v0)","v":{"$error":"#NUM!"}},3.4000000000000017,167.17267254688358,21.157579824531638,{"$error":"#NUM!"},21.157579824531638,-10.013491221905468,0]
[null,null,7.0000000000000036,30.935523161635714,-5.122142352052226,{"$nulls":3},3.5000000000000018,167.41458805667116,20.15623070234109,{"$error":"#NUM!"},20.15623070234109,-9.53957120230605,0]
[null,null,7.200000000000004,29.91109469122527,-5.221840462054423,null,"omegaD =",{"$f":"omega0*SQRT(1-gamma^2)","v":{"$error":"#NUM!"}},3.600000000000002,167.6014298890292,19.202273582110486,{"$error":"#NUM!"},19.202273582110486,-9.088080940720076,0]
[null,null,7.400000000000004,28.866726598814385,-5.318057538479195,null,"gamma =",null,3.700000000000002,167.7349762651843,18.29346548803848,{"$error":"#NUM!"},18.29346548803848,-8.657958878184502,0]
[null,null,7.600000000000004,27.803115091118546,-5.4107294401722505,{"$nulls":3},3.800000000000002,167.81696126869053,17.427669600220028,{"$error":"#NUM!"},17.427669600220028,-8.248193697358985,0]
[null,null,7.800000000000004,26.720969203084096,-5.499794389298245,{"$nulls":3},3.900000000000002,167.84907582545586,16.60285023048413,{"$error":"#NUM!"},16.60285023048413,-7.8578219446818665,0]
[null,null,8.000000000000004,25.621010325224447,-5.585193012522858,null,"case =",{"$f":"IF(Disc<0,3,IF(Disc>0,2,1))","v":2},4.000000000000002,167.83296866331548,15.817068036015943,{"$error":"#NUM!"},15.817068036015943,-7.48592576506713,0]
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[null,null,8.400000000000002,23.37059804660117,-5.744766046289875,{"$nulls":3},4.200000000000001,167.6624787208392,14.355312385128002,{"$error":"#NUM!"},14.355312385128002,-6.794103850645428,0]
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[null,null,8.8,21.057878021199247,-5.8890231079366195,{"$nulls":3},4.4,167.31787251680362,13.028646851955777,{"$error":"#NUM!"},13.028646851955777,-6.166217590449428,0]
[null,null,9,19.880073399611923,-5.9552863378554655,{"$nulls":3},4.5,167.0839754234075,12.412025092910834,{"$error":"#NUM!"},12.412025092910834,-5.874381916301363,0]
[null,null,9.2,18.68901613204083,-6.017579597443579,{"$nulls":3},4.6,166.81091407937703,11.824586901280698,{"$error":"#NUM!"},11.824586901280698,-5.596358284861314,0]
[null,null,9.399999999999999,17.485500212552115,-6.0758613601684,{"$nulls":3},4.699999999999999,166.50006706023402,11.264951072794567,{"$error":"#NUM!"},11.264951072794567,-5.331492997693008,0]
[null,null,9.599999999999998,16.270327940518435,-6.1300927736801025,{"$nulls":3},4.799999999999999,166.15277773130606,10.731801773025266,{"$error":"#NUM!"},10.731801773025266,-5.079163294698592,0]
[null,"t3",9.799999999999997,15.044309385782414,-6.180237685711356,null,null,"t3",4.899999999999999,165.77035504075826,10.223885443555407,{"$error":"#NUM!"},10.223885443555407,-4.8387758898636335,0]
[null,{"$f":"L3","v":{"$error":"#NUM!"}},9.999999999999996,13.808261848640143,-6.226262668178162,null,null,{"$f":"S3","v":0},4.999999999999998,165.3540742959333,9.740007854569043,{"$error":"#NUM!"},9.740007854569043,-4.609765576303353,0]
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[null,null,11.199999999999992,6.227909658272333,-6.414370546005408,{"$nulls":3},5.599999999999996,162.2127373439023,7.2814739769253745,{"$error":"#NUM!"},7.2814739769253745,-3.446186962552975,0]
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[null,null,11.59999999999999,3.6588649326495166,-6.443048620998264,{"$nulls":3},5.799999999999995,160.94746114613693,6.608546749936906,{"$error":"#NUM!"},6.608546749936906,-3.12770295179587,0]
[null,null,11.79999999999999,2.370255208449864,-6.4509490327667525,{"$nulls":3},5.899999999999995,160.2786598015999,6.295776454757319,{"$error":"#NUM!"},6.295776454757319,-2.979674555768179,0]
[null,null,11.99999999999999,1.0800654018965132,-6.454549050765352,{"$nulls":3},5.999999999999995,159.58706274547015,5.997808999180501,{"$error":"#NUM!"},5.997808999180501,-2.8386520699462103,0]
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