GuanyinBuddha
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Hello, I wish you happiness and wellbeing.

Thank you to all who helped with the last problem I had, but I'm facing an even more confusing problem. I've attached the question to this post, but I am completely lost as to what they mean by 'elements of length dl.' How would this help me find the kinetic energy of the spring and more basically what is the hint trying to say?
Thanks again

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Protoxylic
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dl I presume means delta l, so a small change in l. So the overall energy of the system is governed by the sum of potential and kinetic energies, the only potential energy that is considered is the elastic potential energy coming from the spring, so E=1/2 K(delta)l. The kinetic energy of the mass is just 1/2 Mv^2. And now the kinetic energy of the spring varies across the spring, that is, the kinetic energy of a section of the spring nearest the mass is at it's maximum, whereas the kinetic energy of the spring nearest the top of the spring is near zero as it doesn't move. So the kinetic energy varies across the length of the spring which is where you consider the kinetic energies of an infinite number of portions of the spring, dl, and sum them up over l.
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