**Problem 1:**

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 2:**

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

*Πr*)(

*L*)(

*dr*) , where

*dr*is the width of the hoop. Thus the mass of this element can be expressed in terms of volume and density:

*dm*=

*ρV*=

*ρ*(2

*ΠrLdr*)

*V*=

*AL*=

*ΠR*

^{2}

*L*. In addition, our density is given by the total mass of the cylinder divided by the total volume of the cylinder. Thus:

*ρ*= =

*dm*,

*dm*= = 2

*rdr*

*dm*in terms of

*r*, we simply have to integrate over all possible values of

*r*to get our rotational inertia:

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

*kMR*

^{2}, where

*k*is some constant less than one.