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I recently learnt about simple harmonic motion. In all the questions I have done, the springs are massless. I would like to know what happens when the spring has mass. I think that if the spring has mass, then the force in the spring will also have to counteract the weight of the spring itself. And I think the extension of the spring is also not uniform over the length of the spring.

My idea is something like this. Let's say the length of the spring is AB, where A is the topmost point and B is the bottommost point. Let X is a point between A and B. At this point, the segment AX will extend by some amount e that support the weight of XB and the hanging mass by Hooke's Law. Then we can use this relationship to write out some differential equation.

However, I think I have made fundamental mistake in my reasoning above. It seems that as X moves towards B, the effective hanging mass (the hanging mass + the segment XB) becomes lighter, and yet the extension of the segment AX will increase.

Can someone help me out? I am getting more and more confused.

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# Simple Harmonic Motion with Springs that have Mass

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