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bgq

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Just for curiosity...

Is Hooke's law valid for a vibrating massive spring ?

I have done some calculations using both Newton's 2nd Law and the conservation of energy to a horizontal swinging spring connected to a small block in the absence of any friction. I have found that the tension of the spring depends on both the elongation and the

*acceleration.*However, the acceleration is multiplied by the mass of the spring, so if the spring is massless, the tension is reduced to T = kx.

Here is the outline of my work:

>> I have written the expression of the mechanical energy of the system (block-spring), and then set the derivative to zero. I concluded at the end that I can neglect the mass of the spring if I assume the mass of the block is M + m/3 where m is the mass of the spring and M is the mass of the block. I checked this out on the internet, and I found this conclusion true.

>> Next I have applied Newton's 2nd Law:

For a massless spring: T = Ma

For a massive spring, the tension is T':

T' = (M + m/3)a (Neglect the mass of the spring and add its third to the block)

T' = Ma + ma/3

T' = T + ma/3

T' = -kx + ma/3 (Note that the tension depends on the acceleration).

Is my work correct?

Thank you.