Moment of Inertia: Integration, Axis Shifts, and Rotating Systems

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Point-mass definition of moment of inertia

For point masses, $I=\sum_i m_i r_i^2$, where $r_i$ is each mass's perpendicular distance from the chosen axis.

Moment of inertia for continuous mass

For a continuous object, $I=\int r^2\,dm$, integrating over its infinitesimal mass elements.

Why moment of inertia depends on the axis

Changing the axis changes the mass elements' perpendicular distances, which are squared in the calculation.

Parallel-axis theorem

For parallel axes, one through the center of mass, $I=I_{cm}+Md^2$, where $d$ is their separation.

Uniform thin disk about its central perpendicular axis

A uniform thin disk of mass $M$ and radius $R$ has $I=\frac{1}{2}MR^2$.

Quiz questions

1. What is the moment of inertia of a uniform thin rod of mass $M$ and length $L$ about a perpendicular axis through its midpoint?Answer: $\frac{1}{12}ML^2$
2. A uniform thin rod has moment of inertia $I_{cm}$ about a perpendicular axis through its center of mass. What is its moment of inertia about a parallel axis a distance $d$ away?Answer: $I_{cm}+Md^2$
3. A uniform thin disk of mass $M$ and radius $R$ rotates about an axis through its center and perpendicular to its face. What is its moment of inertia?Answer: $\frac{1}{2}MR^2$
4. When calculating the total moment of inertia of a compound object, how must the component moments be expressed before they are added?Answer: About the same physical rotation axis
5. A uniform thin rod of length $L$ is released from rest at angle $\theta$ from the downward vertical and swings about one end. What is its angular speed at the bottom, assuming no energy losses?Answer: $\sqrt{\frac{3g(1-\cos\theta)}{L}}$