6.3 Kinetic energy
The energy associated with motion is called kinetic energy. Since there are two kinds of motion (translational and rotational), there are two kinds of kinetic energy.
6.3.1 Translational kinetic energy
For an object with a mass \(m\), traveling in a straight line with a speed \(v\), the translational kinetic energy is given by the expression
\[
K_t = \frac{1}{2}mv^2 \tag{6.3}
\]
From this expression, we see how to express the joule in base SI units: 1 J = 1 kg · m2/s2
Remember, speed is the magnitude of velocity. This means energy is a scalar quantity. Kinetic energy is not associated with any particular direction of motion,* only how fast the object is going.
Practice 6.1
6.3.2 Rotational kinetic energy
Now consider an object with mass \(m\) being swung on a string, in a circular path. The object is moving in a circle, so it has some tangential speed \(v_t\). And, since it has mass and is moving, it has kinetic energy. Let’s see what happens when we start from the expression for kinetic energy (given above) and make some substitutions:
\[
\begin{align*}
K &= \frac{1}{2}mv_t^2 \\
&= \frac{1}{2}m\left(r\omega\right)^2 \\
&= \frac{1}{2}mr^2\omega^2
\end{align*}
\]
For convenience, we group the mass and radius factors together:
\[
K = \frac{1}{2}\left(mr^2\right)\omega^2
\]
This is similar to the original expression of kinetic energy, which gave an object’s kinetic energy as: the fraction one-half, times the object’s inertia (mass, \(m\)), times the object’s speed squared. There are two differences between what the expression for translational kinetic energy says and what we have derived for rotation: we’re now dealing with rotation, so we use angular speed \(\omega\) instead of the linear speed \(v\); and we have an additional factor \(r\) that corresponds to the distance from the mass to the axis of rotation.
The distance squared, multiplied by the mass, measures the object’s rotational inertia, or resistance to changes in rotational motion. This is called the moment of inertia; we use the letter \(I\) to represent moment of inertia.**
Rotational kinetic energy is given by the expression
\[
K_r = \frac{1}{2}I\omega^2 \tag{6.4}
\]
While the expression we initially derived \(\left(K = \frac{1}{2}mr^2\omega^2\right)\) is only valid for a point mass traveling around a circular path, the expression here (in terms of the moment of inertia \(I\)) is valid in general.
Moment of inertia
Notice that the moment of inertia depends on the distance of the mass from the axis of rotation. For a point mass \(m\) on a taut string, a distance \(r\) from the axis of rotation, we have \(I = mr^2\).
Now, imagine taking ten small blocks of wood with the same mass, and glue them together in a straight line. If you hold onto one end and let the rod swing like a pendulum, the blocks at the end far from your hand will have a larger moment of inertia than the blocks closer to your hand, but all of the blocks will affect the motion of the object. On the other hand, if you hold the rod in the middle and let it rotate end-over-end, each block as a different moment of inertia than when the rod was swinging like a pendulum—each block affects the object’s motion in a different way than before.
In general, an object’s moment of inertia depends on both the shape of the object, and the axis of rotation (how the object is rotating). The following table lists the moment of inertia of several common objects. The column labeled “type of motion” is a sort of generalization, to describe how the object is rotating. For example, if the type of motion is “rolling,” visualize the object spinning around a central axis as if it were wheel. An object rotating like this may be spinning in place, or rolling forward, but the moment of inertia would be the same—the moment of inertia depends on the shape of the object and its axis of rotation.
For a specific example, a disk (which is simply a very short cylinder) that is rolling forward would have the same moment of inertia as a disk that is acting as a pulley and rotating in place. However, if the disk is flipping end-over-end, like you would flip a coin, the moment of inertia would be different.
| Shape | Type of motion | Moment of inertia |
|---|---|---|
| Hoop or hollow cylinder | Rolling | \(I = MR^2 \) |
| Disk or solid cylinder | Rolling | \(I = \frac{1}{2}MR^2 \) |
| Solid ball | Rolling | \(I = \frac{2}{5}MR^2 \) |
| Hollow ball | Rolling | \(I = \frac{2}{3}MR^2\) |
| Rod | Flipping end-over-end | \(I = \frac{1}{12}ML^2\) |
| Rod | Swinging like a pendulum | \(I = \frac{1}{3}ML^2\) |
Note that the moment of inertia for each object is some constant times mass times length squared (\(MR^2\) or \(ML^2\)); it is only the constant that changes from one object to another
Practice 6.2
Practice 6.3
In this problem you’ll need to consider how an object’s mass distribution affects its moment of inertia.
*This is one of the limitations of energy analysis. Just like any tool, it is not necessarily the best for every situation.
**The term “moment” has a mathematical meaning that is entirely unrelated to time; it refers to a physical quantity times distance. Since the physical quantity here is inertia, mass times distance is the moment of inertia.