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I would like to calculate the moment of inertia of a helical spring.

When torque acts on the lower part of the spring hanging on the ceiling,

Every part of the spring will have different angular acceleration, so I don't think it is $mR^2$.

I have thought about it to integrate, but it failed.

How can I calculate?

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  • $\begingroup$ About which axis do you want to calculate the moment of inertia? If it is about the axis through the center of the helix along its long axis, no integral is necessary. $\endgroup$
    – Noah
    Commented Jan 26, 2021 at 6:09
  • $\begingroup$ @Noah I would like to calculate about the axis through the center. Isn't integral necessary? If not then there is no difference whether if we care about the mass of the spring or not. $\endgroup$
    – Sanghwa Lee
    Commented Jan 26, 2021 at 6:13
  • $\begingroup$ If the axis is the screw-axis of the spring, all the mass of of the spring is a distance R away and I = MR^2, just as it would be for a ring or pipe around this axis. $\endgroup$
    – Noah
    Commented Jan 26, 2021 at 6:17
  • $\begingroup$ I ment when the spring is held up on the ceiling so that the whole energy is given to the spring itself. $\endgroup$
    – Sanghwa Lee
    Commented Jan 26, 2021 at 6:18
  • $\begingroup$ What have you tried? You should show us any and all work you've attempted at a solution. $\endgroup$
    – Triatticus
    Commented Jan 26, 2021 at 8:07

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Not an answer, just a remark too big for comments

Actually, the moment of inertia about the central axis of the spring would be $mR^2$ just because of the reason @Noah stated. The problem you are dealing with is you are applying a torque that is producing a torsional strain rather than rotating it completely. The deformations are not uniform and so, even the correct expression of moment of inertia sounds this much misleading. Your body i.e., the spring behaves no more like a rotatable body, rather more like a pipe which is being twisted about one of its end. So moment of inertia would do no good to your further computations in the following case. In simple words, concept of moment of inertia of a rigid body is strictly incapable of explaining such cases where there is no rotation anymore.

Edit

Also, check this

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  • $\begingroup$ Hello Mr/Ms downvoter. Could you please tell my where I went wrong? $\endgroup$
    – SteelCubes
    Commented Jan 26, 2021 at 8:39

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