# Moment of Inertia of A Cylinder with axis through the perpendicular bisector of the length

Shouldn't the moment of inertia of a cylinder, wit the axis passing through the perpendicular bisector of the length, be equal to the moment of inertia of a disk with the axis passing through the diameter> as a cylinder is nothing but a disk, extended on both sides?

I thought this would be true as, for a cylinder's MOI, for the axis passing through centre of the circular area is equal to the MOI of a regular disk, with axis passing through the centre.

• And what is the moment of inertia of a disk separated by a distance $x$ from the rotation axis? A cylinder also includes those disks, in addition to the one rotating about one centered on the rotation axis. You have to integrate the disks over the entire length of the cylinder. Jul 10, 2021 at 7:47

## 4 Answers

The moment of inertia depends on the distance of the mass elements from the axis, so a long cylinder has a higher moment of inertia than a disk.

There is a table of moments of inertia here

https://en.wikipedia.org/wiki/List_of_moments_of_inertia

and you could also look at the 'parallel axis theorem'

https://en.wikipedia.org/wiki/Parallel_axis_theorem

Considering the cylinder to be solid, (made by several discs joined end to end) its moment of inertia with respect to an axis which is the perpendicular bisector of its length cannot be equal to that of disc with respect to a diametrical axis. This is because the same mass in a cylinder is spread out on both sides of the central disc, increasing the net moment of inertia. By the way the increase is $${+Mh^2\over 12}$$, where h is the length or height of the cylinder.

The derivation of its MOI uses Calculus. It is not possible here to directly use perpendicular axis theorem as it is applicable for laminar(2D) bodies only. As the axis passes through its COM, even parallel axis theorem is not directly applicable.

The MOI of the cylinder is $${MR^2\over 4}$$ + $${Mh^2\over 12}$$.

Obviously, cylinder is nothing but a disk, extended on both sides. For simplicity, let's consider a cylinder consists of three disks as the given diagram below.

If you rotate this system about the axis which goes through the diameter of the middle disk, you can see that the middle disk performs a rotational motion while the two other disks perform a circular motion about the axis. Thus the moment of inertia of disks at both ends are equal but it is not the MOI of the middle disk. So it is clear that cylinder's MOI and disk's MOI are not equal when cylinder and disk have same diameter.

I disagree with the other answers which claim we have to model a cylinder as a stack of discs, with one purely rotating and others in circular translation plus rotation. The exercise is interesting, and they are not wrong in their calculations, but the calculations are not necessary because..

Yes, a disc is a cylinder. The only difference is the general notion that discs have a length that is low compared to diameter. But the formulas for disc MOI and for cylinder MOI are identical if we replace “thickness, t” with “length, l”.

To explain in simple, qualitative terms why the formulas ever differ:

If you find formulas for the two MOI’s that differ, it is because the disc formula assumes zero thickness for the disc and models it as a shaded circle with mass at points in two dimensions distributed by the area.