Classical Mechanics

Moment of Inertia

Moment of Inertia of Mass Distributions

         

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

Consider a solid cylinder of mass M=9 kg M = 9 \text{ kg} with homogeneous density that has a circular base of radius R=5 m, R = 5 \text{ m}, and a height of H=5 m. 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 kg M = 4 \text{ kg} and radius R=6 m R = 6 \text{ m} rotates about the yy-axis, as shown in the figure above. Find its moment of inertia.

Consider a rectangular sheet of metal with width W=2 mW=2~\mbox{m} and length L=5 mL=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 Iz=10 kgm2I_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 kgm2I=5~\mbox{kg}\cdot\mbox{m}^2. What is the moment of inertia about the y-axis in kgm2\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 cm2g~cm^2. The mass of the die is m=30 gm=30~\mbox{g} and the length of each edge is a=1 cma=1~\mbox{cm}.

Details and assumptions

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

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