Math 2280 - Explanation for Step 6Dylan ZwickSpring 2009Class, there has been some confusion about how we handle step 6 ofthe project, as it requires some chain rule jiu jitsu. So, here’s an explanationof how it works.If we define r = 1/z then the chain rule tells us that:drdt=drdzdzdt= −1z2dzdt.If we then use the relation from the textbook:r2dθdt= hwe get:−1z2dzdt= −r2dzdt= −hdtdθdzdt= −hdzdθ.And so we derive the relation:drdt= −hdzdθ.If we differentiate again with respect to t and again use the chain rulewe get:1d2rdt2= −hd2zdθ2dθdt.Now, if we agin use our relation:r2dθdt= hthen we get:d2rdt2= −hd2zdθ2hr2= −h2r2d2zdθ2.If we then equate this with our relation from the textbook:d2rdt2−h2r3= −kr2we get:−h2r2d2zdθ2−h2r3= −kr2which simplifies tod2zdθ2+1r=kh2.If we then use our de fining relation for z, namely z = 1/r, then we getthe relation:d2zdθ2+ z =kh2.Which is what we wa nt to
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