Simple Harmonic Motion with Springs that have Mass

[simpleton]

Hi all,

I recently learned 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.

 

[simpleton]

anyone?

 

[Curl]

I think it is too difficult to model the forces on each spring element and write out a solvable differential equation.

What really happens in a spring is there is strain in the metal (twisting in the cross section) as well as moments in the cross section, which have linearly increasing forces as long as you stay in the elastic region of the material.

You can try to figure something out with a rubber band, since it is easier to think about. The rubber stretches but the mass/length changes. The tension is not constant throughout the band since it is accelerating.

I could probably come up with some models and equations but its late now and I'm a bit tired. I hope this helps a little bit.