Tutorial on Energy

<BODY LANG="EN" > <HR><center> <table border=1> <td><A HREF="node13.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="diffeq.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node15.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node14.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> <H1>Tutorial on Energy</H1> <P> <OL> <LI>Study your solutions to the oscillator problem for <EM>energy flow</EM>, and in so doing use the energy conservation principle to test your solutions for long-term <EM>stability</EM>. <OL type="a"> <LI> Plot the <EM>kinetic energy</EM> and <EM>potential energy</EM> <P> <IMG WIDTH=500 HEIGHT=34 ALIGN=BOTTOM ALT="[PE, KE]" SRC="img123.gif"> <P> as functions of time and notice the correlation between the two. <P> <LI> If a particle is essentially ``free'' most of the time when bound by a highly anharmonic oscillator, its average <I>KE</I> should exceed its average <I>PE</I>. Check the veracity of this statement against your solution. <P> <LI>Energy conservation tells us that the sum <P> <IMG WIDTH=500 HEIGHT=15 ALIGN=BOTTOM ALT="E = KE(t) + PE(t)" SRC="img124.gif"> <P> should be a constant for this oscillator (there is no external force adding energy, and no friction removing it). Check the <EM>stability</EM> of your solution by plotting the relative deviation <NOBR>(<I>E</I>(<I>t</I>) - <I>E</I>(<I>t</I>=0))/<I>E</I>(<I>t</I>=0)</NOBR> for a large number of cycles. <P> <LI>Check how different choices for the time step <IMG WIDTH=47 HEIGHT=12 ALIGN=BOTTOM ALT="delta t = h" SRC="img125.gif"> affects the stability of your answer. </OL> <P> <LI><B>Stability:</B> Include <EM>friction</EM> into your oscillator and see how this affects the time dependence of the total energy. Since the energy lost to friction is <IMG WIDTH=82 HEIGHT=23 ALIGN=MIDDLE ALT="dW = F_f dx" SRC="img126.gif">, the energy lost per unit time is <P> <DD> <IMG WIDTH=500 HEIGHT=39 ALIGN=BOTTOM ALT="[dW/dt]" SRC="img127.gif"> <P> Thus we expect the energy loss is not a monotonic function, but instead is greatest when the velocity is greatest. <P> <OL type="a"> <LI>Use your differential equation program to compute and plot the energy versus time for a harmonic oscillator with <EM>sliding</EM> friction. (Make sure to go out real far in time, say to where the solution has died to <IMG WIDTH=30 HEIGHT=13 ALIGN=BOTTOM ALT="10^(-8)" SRC="img128.gif"> times its initial value.) <P> <LI>Compute and plot the energy versus time for a harmonic oscillator with <EM>viscous</EM> friction. </OL> <P> <LI>Now for the even more industrious, consider the harmonic oscillator driven by an external sinusoidal force. For simplicity you may leave out friction. Plot the total energy of the system versus time for an external driving force below, above, and ``at'' resonance, that is, for <IMG WIDTH=156 HEIGHT=15 ALIGN=TOP ALT="w=(0.75,1.0,1.25)w0" SRC="img129.gif">, where <IMG WIDTH=16 HEIGHT=14 ALIGN=MIDDLE ALT="w0" SRC="img32.gif"> is the natural frequency of the system. <P> <OL type="a"> <LI>Make the driving force relatively <EM>weak</EM> so that the system still has a tendency to oscillate at its own frequency; if the driving force is very <EM>strong</EM> it will force the system to oscillate at the driving frequency. (<EM>Hint</EM>: The scale for forces is set by the spring's maximum force <IMG WIDTH=70 HEIGHT=24 ALIGN=MIDDLE ALT="|F|=k A^p" SRC="img130.gif">.) <P> <LI>You should find that unless the relative phase of the driving force and the oscillator is ``just right'', the force will not ``pump'' energy into the system -- a fact familiar to those who have played with yo-yo's. </OL> </OL> <P> <HR><center> <table border=1> <td><A HREF="node13.html"><img height=36 width=36 alt="PREVIOUS" src="/UCESIcons/tools/previous.gif"></A></td> <td><A HREF="diffeq.html"><img height=36 width=36 alt="UP" src="/UCESIcons/tools/up.gif"></A></td> <td><A HREF="node15.html"><img height=36 width=36 alt="NEXT" src="/UCESIcons/tools/next.gif"></A></td> <td><a href="contents-node14.html"><img height=36 width=36 alt="CONTENTS" src="/UCESIcons/tools/toc.gif"></a></td> </table> </center> <HR> </BODY>