Concept #1: Finding Moment Of Inertia By Integrating

Example #1: Moment of Inertia of A Non-Uniform Disk

A non-uniform rod rotates about an axis through one end of the rod, perpendicular to its length. If the mass distribution of the rod is λm = βx3, what is the moment of inertia of the rod about the axis? What would the moment of inertia of the rod be about an aixs at the center of the rod, perpendicular to its length?

A nonuniform rod lies along the x-axis with its left end at x = y = 0 . It has a total mass M , length L and a mass distribution of dM / dx = B (L2 − x2), where B is a constant to be found in your analysis. What is the rotational inertia of the rod about the y-axis, pivoting about a vertical axis through its left end?
1. 1/2 ML2
2. ML2
3. 1/4 ML2
4. 2/3 ML2
5. 3/5 ML2
6. 1/7 ML2
7. 1/3 ML2
8. 1/6 ML2
9. 1/5 ML2
10. 3/4 ML2

A non-uniform disk with a radius R rotates about an axis through its center, perpendicular to the plane of the disk. If the disk has a radius R, a thickness h, and a mass distribution of ρm = a*r 2, where a is a constant, what is the moment of inertia of the disk about the axis?