Problem :
What is the moment of inertia of a hoop of mass M and radius R rotated about a cylinder axis, as shown below?
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?
To solve this problem we split the cylinder into small hoops of mass dm , and width dr :
We also know that the total volume of the entire cylinder is given by: 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:
Substituting this into our equation for dm ,
Now that we have 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} | ||
= |