Moment of Inertia

The moment of inertia $I$ plays the role of mass in angular momentum. Thus, while $mv$ is the linear momentum associated with mass $m$ and velocity $v$, the angular momentum associated with rotational speed $\omega$ is $I\omega$.

The moment of inertia is given by summing all mass points times the square of their distance from the center of rotation. Thus, for a point mass $m$ orbiting along a circle of radius $r$, the moment of inertia is $I=mr^2$. For a set of point masses $m_i$ orbiting along circles of radii $r_i$, $i=1,\ldots,N$, the moment of inertia for the ensemble of masses is

$$I = m_1 r_1^2 + m_2 r_2^2 + \cdots + m_N r_N^2$$

For a continuous mass distribution, the moment of inertia is given by integrating the contribution of each differential mass element:

$$I = \int_M r^2 dm$$

where $r$ is the distance from the point of rotation to the mass element $dm$.

For example, the moment of inertia for a circular disk of total mass $M$ and radius $R$ rotating about its center is given by

$$I = \frac{M}{\pi R^2}\int_{-\pi}^\pi \int_0^R r^2 r\,dr\,d\theta ... ...c{2M}{R^2}\int_0^R r^3 dr = \frac{2M}{R^2}\frac{1}{4} R^4 = \frac{1}{2} M R^2.$$