<HTML> <HEAD> <META http-equiv=Content-Type content='text/html; charset=ISO-8859-1'> <TITLE>Simple harmonic movement</TITLE> <LINK REL="stylesheet" TYPE="text/css" HREF="../css/newton.css"> <script type="text/javascript" language="JavaScript" src="../js/objNavegador.js"></script> <script type="text/javascript" language="JavaScript" src="../js/objLayer.js"></script> <script type="text/javascript" language="JavaScript" src="../js/newton.js"></script> <style type="text/css"> <!-- .Estilo1 {font-size: 14px} .Estilo2 { color: #0033CC; font-weight: bold; } --> </style> </HEAD> <body onload="posicionaMenu(5)" bgcolor="#ffffff"> <div class="clsCurso" id="CapaCurso" contenteditable="false" gblcoloron="FFACC2" gblcolor="E48CA2">2<sup>nd</sup> year of post-compulsory secondary education</div> <div id="CapaFondo"><img src="images/fondo.jpg" height="429" width="770"></div> <div id="CapaImg"><img id="imgUnidad" src="images/mas.gif" name="imgUnidad"></div> <div id="eltoMenuHead" 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width="50"><a class="clsMenuH" id="inicioUrl" onmouseover="rollOver('imgInicio', 'CapaMenuH', 'images/menuH/inicio_on.jpg')" onmouseout="rollOver('imgInicio', 'CapaMenuH', 'images/menuH/inicio.jpg')" href="../../index.html">Home</a></td> <td width="2"><img src="images/menuH/separaFondo.jpg" height="21" width="2"></td> </tr> </tbody> </table> </div> <div class="" id="CapaCont" contenteditable="false"> <TABLE id=oTblCont0 borderColor=#e48ca2 cellSpacing=0 cellPadding=2 width=606 bgColor=#ffffff border=0> <TR height=26> <TD class=clsCabecera bgColor=#ffacc2><span class="Estilo2">2.4 The relation between the S.H.M. and a Uniform Circular Movement</span></TD> </TR> <TR> <TD class=clsTexto vAlign=top> <P align="center"><br> &nbsp; <applet name="Relaci&oacute;n" code="Descartes.class" codebase="../.." archive="Descartes3.jar" MAYSCRIPT width=400 height=340> <param name="name" value="Relaci&oacute;n"> <param name="Version" value="3.506"> <param name="Language" value="english"> <param name="Buttons" value="about=no config=no init=yes clear=yes"> <param name="E_00" value="fixed='yes' bg_display='topleft' background='d7d7d7' net='no' net10='no' axes='a5a5a5' text='white' y-axis='Y' numbers='yes'"> <param name="C_00" value="type='num&eacute;rico' id='Ox' region='external' value='-50' decimals='0' fixed='yes' name='O.x' incr='32' visible='no'"> <param name="C_01" value="type='num&eacute;rico' id='Oy' gui='spinner' region='external' fixed='yes' incr='32'"> <param name="C_02" value="type='num&eacute;rico' id='A' value='6' decimals='0' fixed='yes' name='A' incr='1' inf='0' sup='15'"> <param name="C_03" value="type='num&eacute;rico' id='T' value='5.0' decimals='1' fixed='yes' name='T' incr='0.2' inf='1' sup='7'"> <param name="C_04" value="type='num&eacute;rico' id='Escena' gui='button' region='north' fixed='yes' name='Visual' tooltip='Description (left-click for more information)' Explanation='Description (left-click for more information)' action='message' parameter='Click on play to start the movement with the default amplitude and period. Observe the fact that the position of the real body coincides with the shadow of the auxiliary body which would be caused by spotlights on either side of the visual on the vertical diameter of the circumference.'"> <param name="C_05" value="type='num&eacute;rico' id='A1' gui='button' region='north' fixed='yes' name='A1' tooltip='Activity 1 (left-click for more information)' Explanation='Activity 1 (left-click for more information)' action='message' parameter='Change the values of the amplitude and the period and observe the changes in the movement. Use the zoom controls if necessary. Observe the fact that the product É" t coincides with the angle covered by the auxiliary body at every instant.'"> <param name="C_06" value="type='num&eacute;rico' id='escala' region='north' value='17' decimals='0' fixed='yes' name='zoom' incr='1' inf='1' sup='1000000' visible='no'"> <param name="A_00" value="id='pi' expresion='3.14159' constant='yes'"> <param name="A_01" value="id='w' expresion='(2*pi)/T'"> <param name="A_02" value="id='f' expresion='1/T'"> <param name="A_03" value="id='a(x)' expresion='((x*17)-100-Ox)/escala'"> <param name="A_04" value="id='b(x)' expresion='((x*17)+Oy)/escala'"> <param name="A_05" value="id='grad(x)' expresion='(x*180)/pi'"> <param name="A_06" value="id='nv' expresion='ent((grad(w*t))/360)'"> <param name="A_07" value="id='ya' expresion='A*(sen(w*t))'"> <param name="A_08" value="id='xa' expresion='A*(cos(w*t))'"> <param name="G_00" value="type='arc' draw-if='nv=10*s' colour='006200' radius='(A/2)+(3*s)' end='grad(w*t)-(nv*360)' trace='3c4141' family='s' s.interval='[0,1]' s.steps='10' fixed='yes'"> <param name="G_01" value="type='equation' colour='gray' expresion='y^2=A^2-x^2' fill+='green' visible='no'"> <param name="G_02" value="type='polygon' background='yes' colour='a24040' expresion='(a(11),b(-6.3))(a(17),b(-6.3))(a(17),b(-8))(a(11),b(-8))(a(11),b(-6.3))' fill='white' width='4'"> <param name="G_03" value="type='segment' expresion='(0,0)(0,ya)' text=' y' fixed='yes' width='3'"> <param name="G_04" value="type='segment' colour='a24040' expresion='(-1-A,0)(-1-A,A)' text='A' fixed='yes'"> <param name="G_05" value="type='segment' colour='white' expresion='(0,ya)(xa,ya)' fixed='yes' width='3'"> <param name="G_06" value="type='segment' colour='006000' expresion='(0,0)(xa,ya)' fixed='yes'"> <param name="G_07" value="type='text' colour='a24040' expresion='[290,262]' text='t=[t]s' font='Courier,BOLD,14' fixed='yes'"> <param name="G_08" value="type='text' expresion='[70,260]' text='y=A&middot;É&middot;sin(É&middot;t)=[ya]m' font='SantSerif,BOLD,16' fixed='yes'"> <param name="G_09" value="type='text' colour='black' expresion='[270,10]' text='f= [f] Hz' font='Monospaced,PLAIN,14' fixed='yes'"> <param name="G_10" value="type='text' colour='black' expresion='[270,30]' text='É= [w] rad/s' font='Monospaced,PLAIN,14' fixed='yes'"> <param name="G_11" value="type='text' colour='006200' expresion='[270,50]' text='É&middot;t= [w*t] rad' font='Monospaced,BOLD,14' fixed='yes'"> <param name="G_12" value="type='text' colour='red' expresion='[270,200]' text='&middot; Real body' fixed='yes'"> <param name="G_13" value="type='text' colour='006000' expresion='[270,220]' text='&middot; Auxiliary body' fixed='yes'"> <param name="G_14" value="type='point' colour='006000' expresion='(xa,ya)' fixed='yes' size='5'"> <param name="G_15" value="type='point' colour='red' expresion='(0,ya)' fixed='yes' size='6'"> <param name="Animation" value="delay='100' controls='no' auto='no' init='n=0; t=0' do='n=n+1;t=n/10' while='n<=199'"> </applet> </P> <p></P></TD> </TR> </TABLE> <p>&nbsp;</p> <TABLE id=oTblSep0 height=1 cellSpacing=0 cellPadding=0 border=0> <TR height=1> <TD height=1></TD> </TR> </TABLE> </DIV> <div class="clsCapasMenu" id="eltoMenu0" nombre="eltoMenu0"> <table border="0" cellpadding="0" cellspacing="0" width="138"> <tbody> <tr> <td><img src="images/menu/menu_19.jpg" height="2" width="10"></td> <td><img src="images/menu/menu_20.jpg" height="2" width="112"></td> <td><img src="images/menu/menu_21.jpg" height="2" width="16"></td> </tr> <tr> <td background="images/menu/menu_22.jpg" height="25" width="10"><img src="images/spacer.gif" height="1" width="1"></td> <td class="clsMenu" id="eltoMenuHead0" align="left" bgcolor="#e48ca2" height="25" width="112"><a class="clsMenu" href="javascript:gestionMenu(0)">Introduction</a></td> <td background="images/menu/menu_24.jpg" height="25" width="16"><img src="images/spacer.gif" height="1" width="1"></td> </tr> <tr> <td><img src="images/menu/menu_25.jpg" height="4" width="10"></td> <td><img src="images/menu/menu_26.jpg" height="4" width="112"></td> <td><img src="images/menu/menu_27.jpg" height="4" width="16"></td> </tr> </tbody> </table> <table align="center" border="0" bordercolor="#ffffff" cellpadding="2" cellspacing="0" width="138"> <tbody> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead00" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('11_mas.html', 0, 0)">Definitions</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead01" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('12_mas.html', 0, 1)">Representing the S.H.M.</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> </tbody> </table> </div> <div class="clsCapasMenu" id="eltoMenu1" nombre="eltoMenu1"> <table border="0" cellpadding="0" cellspacing="0" width="138"> <tbody> <tr> <td><img src="images/menu/menu_19.jpg" height="2" width="10"></td> <td><img src="images/menu/menu_20.jpg" height="2" width="112"></td> <td><img src="images/menu/menu_21.jpg" height="2" width="16"></td> </tr> <tr> <td background="images/menu/menu_22.jpg" height="25" width="10"><img src="images/spacer.gif" height="1" width="1"></td> <td class="clsMenu" id="eltoMenuHead1" align="left" bgcolor="#e48ca2" height="25" width="112"><a class="clsMenu" href="javascript:gestionMenu(1)">The kinematics of an S.H.M.</a></td> <td background="images/menu/menu_24.jpg" height="25" width="16"><img src="images/spacer.gif" height="1" width="1"></td> </tr> <tr> <td><img src="images/menu/menu_25.jpg" height="4" width="10"></td> <td><img src="images/menu/menu_26.jpg" height="4" width="112"></td> <td><img src="images/menu/menu_27.jpg" height="4" width="16"></td> </tr> </tbody> </table> <table align="center" border="0" bordercolor="#ffffff" cellpadding="2" cellspacing="0" width="138"> <tbody> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead10" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('21_mas.html', 1, 0)">Position</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead11" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('22_mas.html', 1, 1)">Velocity</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead12" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('23_mas.html', 1, 2)">Acceleration</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead13" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('24_mas.html', 1, 3)">S.H.M. and Uniform Circular Movement</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead14" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('25_mas.html', 1, 4)">Phase</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead15" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('26_mas.html', 1, 5)">Conclusions</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> </tbody> </table> </div> <div class="clsCapasMenu" id="eltoMenu2" nombre="eltoMenu2"> <table border="0" cellpadding="0" cellspacing="0" width="138"> <tbody> <tr> <td><img src="images/menu/menu_19.jpg" height="2" width="10"></td> <td><img src="images/menu/menu_20.jpg" height="2" width="112"></td> <td><img src="images/menu/menu_21.jpg" height="2" width="16"></td> </tr> <tr> <td background="images/menu/menu_22.jpg" height="25" width="10"><img src="images/spacer.gif" height="1" width="1"></td> <td class="clsMenu" id="eltoMenuHead2" align="left" bgcolor="#e48ca2" height="25" width="112"><a class="clsMenu" href="javascript:gestionMenu(2)">The dynamics of an S.H.M.</a></td> <td background="images/menu/menu_24.jpg" height="25" width="16"><img src="images/spacer.gif" height="1" width="1"></td> </tr> <tr> <td><img src="images/menu/menu_25.jpg" height="4" width="10"></td> <td><img src="images/menu/menu_26.jpg" height="4" width="112"></td> <td><img src="images/menu/menu_27.jpg" height="4" width="16"></td> </tr> </tbody> </table> <table align="center" border="0" bordercolor="#ffffff" cellpadding="2" cellspacing="0" width="138"> <tbody> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead20" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('31_mas.html', 2, 0)">Elastic force</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead21" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('32_mas.html', 2, 1)">Frequency</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead22" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('33_mas.html', 2, 2)">Conclusions</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> </tbody> </table> </div> <div class="clsCapasMenu" id="eltoMenu3" nombre="eltoMenu3"> <table border="0" cellpadding="0" cellspacing="0" width="138"> <tbody> <tr> <td><img src="images/menu/menu_19.jpg" height="2" width="10"></td> <td><img src="images/menu/menu_20.jpg" height="2" width="112"></td> <td><img src="images/menu/menu_21.jpg" height="2" width="16"></td> </tr> <tr> <td background="images/menu/menu_22.jpg" height="25" width="10"><img src="images/spacer.gif" height="1" width="1"></td> <td class="clsMenu" id="eltoMenuHead3" align="left" bgcolor="#e48ca2" height="25" width="112"><a class="clsMenu" href="javascript:gestionMenu(3)">The energy of an S.H.M.</a></td> <td background="images/menu/menu_24.jpg" height="25" width="16"><img src="images/spacer.gif" height="1" width="1"></td> </tr> <tr> <td><img src="images/menu/menu_25.jpg" height="4" width="10"></td> <td><img src="images/menu/menu_26.jpg" height="4" width="112"></td> <td><img src="images/menu/menu_27.jpg" height="4" width="16"></td> </tr> </tbody> </table> <table align="center" border="0" bordercolor="#ffffff" cellpadding="2" cellspacing="0" width="138"> <tbody> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead30" align="left" bgcolor="#ffffff" width="123"><a class="clsMenu" href="javascript:enlace('41_mas.html', 3, 0)">Conservation of energy</a></td> <td width="5"><img src="images/spacer.gif" height="1" width="10"></td> </tr> <tr> <td width="5"><img src="images/spacer.gif" height="1" width="5"></td> <td class="clsSubMenu" id="eltoSubMenuHead31" align="left" bgcolor="#f0f0f0" width="123"><a class="clsMenu" href="javascript:enlace('42_mas.html', 3, 1)">Graphic representation</a></td> <td width="5"><img src="images/spacer.gif" height="1" 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