×

## Moment of Inertia

A sewer pipe rolls more slowly down an incline than a bowling ball with the same mass. Understand this and more by learning about Moment of Inertia, a measure of how compact objects are.

# Inertia - Mass Distributions

A disk of mass $$M = 9 \text{ kg}$$ and radius $$R = 8 \text{ m}$$ rotates about the $$y$$-axis, as shown in the figure above. Find its moment of inertia.

Consider a solid cylinder of mass $$M = 9 \text{ kg}$$ with homogeneous density that has a circular base of radius $$R = 5 \text{ m},$$ and a height of $$H = 5 \text{ m}.$$ If the cylinder rotates about the diameter of the circular base, what is its moment of inertia?

A ring of mass $$M = 4 \text{ kg}$$ and radius $$R = 6 \text{ m}$$ rotates about the $$y$$-axis, as shown in the figure above. Find its moment of inertia.

Consider a rectangular sheet of metal with width $$W=2~\mbox{m}$$ and length $$L=5~\mbox{m}$$. The sheet is in the x-y plane, with the origin right in the geometric middle of the sheet. The x-axis is parallel to the short edge, while the y-axis is parallel to the long edge. The moment of inertia about the z-axis is $$I_z=10~\mbox{kg}\cdot\mbox{m}^2$$ and the moment of inertia about an axis that passes diagonally through the sheet (i.e. corner to corner) in the xy plane is $$I=5~\mbox{kg}\cdot\mbox{m}^2$$. What is the moment of inertia about the y-axis in $$\mbox{kg}\cdot\mbox{m}^2$$?

Estimate the moment of inertia of a die along an axis that passes through one of the die's edges in $$g~cm^2$$. The mass of the die is $$m=30~\mbox{g}$$ and the length of each edge is $$a=1~\mbox{cm}$$.

Details and assumptions

Assume that the die is a perfect cube and its mass is evenly distributed.

×