**Problem : **

What is the moment of inertia of a hoop of mass *M* and radius *R* rotated about a
cylinder
axis, as shown below?

A hoop of radius *R*

Fortunately, we do not need to use calculus to solve this problem. Notice that
all the mass is the same distance *R* from the axis of rotation. Thus we do not
need to integrate over a range, but can calculate the total moment of inertia.
Each small element *dm* has a rotational inertia of *R*^{2}*dm*, where *r* is constant.
Summing over all elements, we see that *I* = *R*^{2}*dm* = *R*^{2}*M*. The sum of all the
small elements of mass is simply the total mass. This value for *I* of *MR*^{2}
agrees with experiment, and is the accepted value for a hoop.

**Problem : **

What is the rotational inertia of a solid cylinder with length *L* and radius *R*,
rotated about its central axis, as shown below?

A cylinder being rotated about its axis

To solve this problem we split the cylinder into small hoops of mass *dm*, and
width *dr*:

A cylinder being rotated about its axis, shown with a small element of mass
from the cylinder

I | = | r^{2}dm | |

= | 2r^{3}dr | ||

= | [r^{4}/2]_{0}^{R} | ||

= |

Thus the rotational inertia of a cylinder is simply . Once again, it has the form of

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