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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.

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