I was asked such a question:(adsbygoogle = window.adsbygoogle || []).push({});

It is known that the effective mass of the spring in this vertical spring-mass system(figure can be viewed by the following Wiki link) is 1/3 of the mass of the spring, for example, if the mass of spring is m, and the mass of the block is M, then the period of the SHO is [tex]2\pi\sqrt{\frac{M+m/3}{k}}[/tex] where [tex]k[/tex] is the spring constant.

And I found this question can be solved by the energy consideration, for example, in the Wiki demonstration:

http://en.wikipedia.org/wiki/Effective_mass_(spring-mass_system)#Vertical_spring-mass_system

Notice that there is a key step that the velocity of each position of the spring is directly proportional to its length.(which I don't know how to prove it, this is my first question.)

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However, my second question is, Wiki said that the spring of the horizontal spring-mass system has the effective mass 0!

But I cannot understand why the above argument for solving vertical spring-mass system is not applicable here?!

This confuses me a lot.

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Moreover, then how about the effective masses of the such following systems? (Please see the attached plots)

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# Effective masses of spring-mass systems

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