document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197654>Question 1197654</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find a model for a simple harmonic motion that has initial displacement (t=0) of 0 inches, an amplitude of 5 inches, and a frequency  4/3 cycles per second. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #831013 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=831013\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A trig template I'll be using is<BR>\n" );
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document.write( "where,<BR>\n" );
document.write( "|A| = amplitude<BR>\n" );
document.write( "B = used to determine the period T<BR>\n" );
document.write( "C = phase shift<BR>\n" );
document.write( "D = vertical shift and midline\r<BR>\n" );
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document.write( "The initial displacement of 0 inches tells us we don't have any vertical shifting going on. We go with D = 0.\r<BR>\n" );
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document.write( "The amplitude of 5 means A = 5.\r<BR>\n" );
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document.write( "We won't have any phase shifts here, so use C = 0<BR>\n" );
document.write( "A useful property of the sine function is that its initial value, aka y intercept, is 0.\r<BR>\n" );
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document.write( "Now to find the value of B.\r<BR>\n" );
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document.write( "Frequency and the period are connected topics.<BR>\n" );
document.write( "It turns out that<BR>\n" );
document.write( "frequency = 1/period<BR>\n" );
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document.write( "period = 1/frequency\r<BR>\n" );
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document.write( "This means that we can apply the reciprocal to the frequency of \"4/3 cycles per second\" to get a period of \"3/4 seconds per cycle\".<BR>\n" );
document.write( "Take note that the units swap positions.<BR>\n" );
document.write( "The length of each cycle is 3/4 = 0.75 seconds.\r<BR>\n" );
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document.write( "Use the period T = 0.75 to find:<BR>\n" );
document.write( "B = 2pi/T<BR>\n" );
document.write( "B = 2pi/0.75<BR>\n" );
document.write( "B = 8pi/3\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "A = 5<BR>\n" );
document.write( "B = 8pi/3<BR>\n" );
document.write( "C = 0<BR>\n" );
document.write( "D = 0\r<BR>\n" );
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document.write( "So we go from<BR>\n" );
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document.write( "to this<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+5%2Asin%28expr%28%288pi%29%2F3%29%28x-0%29%29%2B0 ALIGN=MIDDLE  ALT=\"y+=+5%2Asin%28expr%28%288pi%29%2F3%29%28x-0%29%29%2B0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
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document.write( "which is a final answer. I say \"a\" instead of \"the\" because we could use cosine and have a nonzero phase shift involved, but it's easier to go with sine in my opinion. <BR>\n" );
document.write( "There are infinitely many possible phase shifts with either sine or cosine, but this one involves a phase shift of 0 which is easiest.\r<BR>\n" );
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document.write( "I recommend to use graphing software such as Desmos or GeoGebra or whichever tool you like to confirm the answer.\r<BR>\n" );
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document.write( "One way to measure the period is to look at the horizontal distance from peak to neighboring peak.\r<BR>\n" );
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document.write( "Side note: a related concept to period is wave length. <BR>\n" );
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