.MCAD 306000000 Z  docDocument<mcObjectI d2_graph_format graphData axisFormatLLtrace2D      dim_formatCmasslengthtimecharge temperature luminosity substanceNumericalFormat@dii shpRectEmcDocumentObjectStateJ mcPageModel:????mcHeaderFooter99 ComputeEngine=BuiltInsB7 SerialAnyvalH@8@H @9@H:@HMbP?;@H units_classA TextState0 TextStyle/@ Arial0,0,128 Normalfont_style_list> font_style?  VariablesTimes New Roman?  ConstantsTimes New Roman? TextArial? Greek VariablesSymbol? User^1Arial? User^2 Courier New? User^3System? User^4Script? User^5Roman? User^6Modern? User^7Times New Roman? SymbolsSymbol? Current Selection FontArial? Undefined Font? HeaderArial? FooterArial? Rotated Math FontTimes New Roman! TextRegion docRegion7shpBoxDp1pC<*<* CharacterMapRangeMap,1UNIVERSITY PHYSICS I, PHY241/243 Computer Lab #3 ChrPropMap(/,1 RangeElem-,  ChrPropData) RangeData.lArial0,0,128 - )| - )|   ParPropMap*1-1 ParPropData+EmbedMap"-LinkMap 1-1LinkData!@NormalArial @DT,`($J$JA.In this lab we simulate the sliding of a small mass m on a spherical surface of radius R. We assume no friction. Hence the mechanical energy E = K + U is constant. The kinetic energy is: K = 0.5mv2 where v = Rw. The potential energy is U = mgRcosq. So the total energy is: E = 0.5mR2w2 + mgRcosq.(-.-)-)_- )-)Symbol-%)-)Symbol -&!)"-#)_ $-%)Symbol"&-')_$(-))_&*-+)(,--)Symbol*.-/),.*.0-.1+"2- .3-.4!@NormalArial 5MetafileRegionU PictureRegion6@D``k(6 mc_metafileS mcad_metafileT*)I $ (&WordMicrosoft Word    @Times New Roman- - "-- !hkm-- "-9 j@Times New Roman- .2  mguK. 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(79@V-@W)@U@X-@Y)@USymbol@V@Z-@[)@U_@X@\-@])@U@Z@^-@_)@USymbol@\@`-@a)@U_@^@b-@c)@U@`@b@X*9@d-9@e+"@f- 9@g-9@h!@NormalArial @i3@DFaX@j1 p@k1 @j@l1d@kE@m1@k@n1@@m@o1@@n@p1@@o@q1t@p0.5@r1@pm@s1@o@t1d@sR@u1@s2@v1@n@w1d@v\w.0@x1@v2@y1@m@z1@@y@{1@@z@|1d@{m@}1@{g@~1@zR@1@y@1d@cos@1p@@1@\Q.0@@DL[XLP+This is the energy: kinetic plus potential.(+*+@-+@+"@- +@-+@!@NormalArial @3@Dm@1 p@1@@1d@E@1@@1+@@Serial_DisplayNodeF@1@ _n_u_l_l_@3@D @1 p@1 @@1d@\Q@1@@1 @@@1d@\Q.0@1@@1@@@1t@0.01@1@rad@1@\Q.0@1@@1@@@1d@\p@1@2@1@rad@3@D4@1 p@1 @@1@@@1d@\w@1p@@1@\Q@1{@@1@@1t@2@1@@1@@@1d@E@1@@1@@@1@@@1d@m@1@g@1@R@1@@1d@cos@1p@@1@\Q@1@@1d@m@1@@1d@R@1@2@@D)P))@bThis equation is obtained by solving for the angular velocity in the energy conservation equation.(b*b@-b@+"@- b@-b@!@NormalArial @@Dx'2x0 Copyright @ Miron Kaufman, 1998(@-@)@nArial0,0,128*@-@+"@- @-@!@NormalArial @@DS>s`0P> > @OSince the angular velocity w = dq/dt then by integrating 1/w one gets the time.(<O@-@)@@-@)@Symbol@@-@)@@@-@)@Symbol@@-@)@@@-@)@Symbol@@-@)@@@@*O@-O@+"@- O@-O@!@NormalArial @3@DG0h@1 p@1 @@1@@@1d@t@1p@@1@\Q@1%@@1@@@1d@\Q.0@1@\Q@1@@1d@\q@1@@1t@1@1@@1d@\w@1p@@1@\q@@D>UH>$>$@\The angular acceleration is obtained from at = Ra, and from Newton's 2'nd law: mgsinq = mat.([+\@-+@)@@-@)@_@@-@)@@@-@)@Symbol@@-#@)@@@-@)@Symbol@@-A)@@A-A)@_@A-A)@AA@*\A-\A+"A- \A-\A !@NormalArial A 3@Dr4 A 1 pA 1 A A 1@A A1dA \aA1pA A1A\QA1A A1@AA1dAgA1ARA1AA1dAsinA1pAA1A\QA@D=3==0=0@The normal force N is obtained from:centripetal force = mgcosq - N, and centripetal force = mRw2 . When N = 0 the object loses contact with the surface.(`$A-$A)AA-A)A}AA-A)AAA -A!)ASymbolAA"- A#)AA A$-A%)A}A"A&-A')AA$A(-A))ASymbolA&A*-A+)A_A(A,-=A-)AA*A,A*A.-A/+"A0- A1-A2!@NormalArial A33@D7!A41 pA51 A4A61@A5A71dA6NA81pA6A91A8\QA:1A5A;1@A:A<1@A;A=1dA<mA>1A<gA?1A;A@1dA?cosAA1pA?AB1AA\QAC1A:AD1@ACAE1dADmAF1ADRAG1ACAH1@AGAI1dAH\wAJ1pAHAK1AJ\QAL1AG2AM3@D;4pAN1 pAO1ANAP1@AOAQ1@APAR1@AQAS1fAR 9.806647AT1AR0AU1AQAV1dAU _n_u_l_l_AW1AU _n_u_l_l_AX1 APAY1 @AXAZ1 @AYA[1@AZA\1dA[\QA]1A[radA^1AZA_1@A^A`1dA_\wAa1pA_Ab1Aa\QAc1A^Ad1dAcradAe1AcAf1dAesecAg1KAeAh1Ag1Ai1AYAj1@AiAk1dAj\aAl1pAjAm1Al\QAn1AiAo1dAnradAp1AnAq1dApsecAr1KApAs1Ar2At1AXAu1@AtAv1dAuNAw1pAuAx1Aw\QAy1AtnewtonAz1AOA{1@AzA|1@A{A}1fA| 1.482162A~1A|0A1A{A1dA _n_u_l_l_A1A _n_u_l_l_A1AzA1@AA1dAtA1pAA1A\QA1AsecA< NN     A@DKjX  @The critical angle Qc is where N = 0 so the object loses contact with the surface. We solve numericaly the equation N = 0 by using the root function.(A-A)AA-A)ASymbolAA-A)A_AA-A)AAAA*A-A+"A- A-A!@NormalArial A3@D8A1 pA1 AA1dA\qA1A0.5A3@DpA1 pA1 AA1dA\Q.cA1AA1dArootA1pAA1 AA1@AA1dANA1pAA1A\qA1A\qA3@DKA1 pA1AA1dA\Q.cA1AA1+@A@FA1A _n_u_l_l_A@D`5`@CThis is in radians. To get the answer in degrees multiply by 180/p.(BACA-AA)AA-A)A@ Symbol0,0,128AA-A)A@ Arial0,0,128AAA*CA-CA+"A- CA-CA!@NormalArial A3@D`A1 pA1AA1@AA1tA0.841A1AA1tA180A1A\pA1AA1+@A@FA1AA@D;833@ t is the time interval when the object is on the surface. We also compute the angular velocity and acceleration when the object loses contact with the surface.(*A-A+"A- A-A!@NormalArial A3@D"|980A1 pA1AA1@AA1dAtA1pAA1A\Q.cA1AA1+@A@FA1A _n_u_l_l_A3@D 90A1 pA1AA1@AA1dA\wA1pAA1A\Q.cA1AA1+@A@FA1A _n_u_l_l_A3@D89d0A1 pA1AA1@AA1dA\aA1pAA1A\Q.cA1AA1+@A@FA1A _n_u_l_l_A@Dx_jxh@ Copyright @Miron Kaufman, 1998(A-A)AnArial0,0,128*A-A+"A- A-A!@NormalArial