10.4 Moment of inertia and rotational kinetic energy  (Page 2/7)

and since ${\omega }_j=\omega$ for all masses,

The units of [link] are joules (J). The equation in this form is complete, but awkward; we need to find a way to generalize it.

Moment of inertia

If we compare [link] to the way we wrote kinetic energy in Work and Kinetic Energy , $\left(\frac{1}{2}m{v}^2\right)$ , this suggests we have a new rotational variable to add to our list of our relations between rotational and translational variables. The quantity ${\displaystyle \sum _j{m}_j{r}_j^2}$ is the counterpart for mass in the equation for rotational kinetic energy. This is an important new term for rotational motion. This quantity is called the moment of inertia     I , with units of $\text{kg}·{\text{m}}^2$ :

For now, we leave the expression in summation form, representing the moment of inertia of a system of point particles rotating about a fixed axis. We note that the moment of inertia of a single point particle about a fixed axis is simply $m{r}^2$ , with r being the distance from the point particle to the axis of rotation. In the next section, we explore the integral form of this equation, which can be used to calculate the moment of inertia of some regular-shaped rigid bodies.

The moment of inertia is the quantitative measure of rotational inertia, just as in translational motion, and mass is the quantitative measure of linear inertia—that is, the more massive an object is, the more inertia it has, and the greater is its resistance to change in linear velocity. Similarly, the greater the moment of inertia of a rigid body or system of particles, the greater is its resistance to change in angular velocity about a fixed axis of rotation. It is interesting to see how the moment of inertia varies with r, the distance to the axis of rotation of the mass particles in [link] . Rigid bodies and systems of particles with more mass concentrated at a greater distance from the axis of rotation have greater moments of inertia than bodies and systems of the same mass, but concentrated near the axis of rotation. In this way, we can see that a hollow cylinder has more rotational inertia than a solid cylinder of the same mass when rotating about an axis through the center. Substituting [link] into [link] , the expression for the kinetic energy of a rotating rigid body becomes

We see from this equation that the kinetic energy of a rotating rigid body is directly proportional to the moment of inertia and the square of the angular velocity. This is exploited in flywheel energy-storage devices, which are designed to store large amounts of rotational kinetic energy. Many carmakers are now testing flywheel energy storage devices in their automobiles, such as the flywheel, or kinetic energy recovery system, shown in [link] .

The rotational and translational quantities for kinetic energy and inertia are summarized in [link] . The relationship column is not included because a constant doesn’t exist by which we could multiply the rotational quantity to get the translational quantity, as can be done for the variables in [link] .