What the heck is a   moment,   anyway?

---

Our goal in this section is to learn about a specific aspect of rotational motion, the moment of inertia. But before we do that, it might help to learn a bit about the concept of a "moment" in mathematics. In the physics sense, it doesn't mean "a bit of time."

In mathematics, the word moment is a measure – or one of a set of measures or properties – that describe the shape of some distribution. A distribution can be a probability distribution or (often) a mass distribution. The moment of inertia describes how mass is distributed in a rotating object.

Moments in probability

---

Here is the familiar Gaussian (bell-shaped curve) distribution. It could represent any group of experimental measurements of some value, such as a length, a mass or a temperature.

The first moment of the distribution is the sum of all measurements divided by the number of measurements, or the mean (or average).

$$\bar{x} = \frac{1}{n} \sum_{i = 1}^n x_i$$

The second moment is called the variance (sigma², σ²),

$$\sigma^2 = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^2$$

In the figure, the square root of the variance, called the standard deviation (σ), is shown. It is a measure of the width of the probability distribution. A small standard deviation means that most of the data are clustered closely around the mean, or that the data is relatively precise. For a Gaussian distribution like the one shown, 68% of the measurements will lie between ±σ of the mean. We sometimes worry about what's between ±2σ and ±3σ as well.

The third moment of a probability distribution is called skewness. It is rarely used in common statistical analysis, but it is also a measure of the shape of a probability distribution, particularly when it's not as symmetric as the one shown above.

Center of mass, centroid

---

The center of mass of an object is the balance point, a single point from which, if suspended, the object would balance. The center of mass is the first moment of the mass distribution of a stationary object.

For point masses on a line, the center of mass is the sum of the linear position of each mass multiplied by its mass, and then that sum is divided by the total mass. Here's the formula for n masses.

$$c = \frac{1}{M} \sum_{i = 1}^n m_i r_i$$

Here's an example, three masses in a line, and for simplicity we'll assume that they're point masses (all of the mass of each concentrated at the center of the sphere):

The center of mass formula gives us:

$$ \require{cancel} c = \frac{1}{13 \, \cancel{Kg}} [4(0) + 1(5) + 8(22)] \, \cancel{Kg}\cdot m$$

Reduced mass

---

The moment of inertia of two point masses turning about their common center of mass (see above) is

$$I = m_1 r_1^2 + m_2r_2^2$$

This can be reduced to a simpler form, that of a single mass rotating by using the reduced mass, μ:

$$I = \mu R^2$$

The reduced mass of two point masses is defined as:

$$\mu = \left( \frac{1}{m_1} + \frac{1}{m_2} \right)^{-1} = \frac{m_1 m_2}{m_1 + m_2}$$

The second equality is obtained by finding a common denominator for the two fractions inside the parenthesis, then taking the reciprocal.

Often, no rationale is given for how the reduced mass is obtained, so here is some. You can skip this, but it can also be enlightening. Your choice:

First let's define our system with these coordinates and masses. The point c is the center of mass of the system.

We first want to locate that center of mass. That is defined below, and we use coordinates 0 and R for the locations of our masses; those are the r_i in the sum:

$$ \begin{align} c &= \frac{1}{m_1 + m_2} \sum m_i r_i \\[5pt] &= \frac{1}{m_1 + m_2} (m_1 r_1 + m_2 r_2) \\[5pt] &= \frac{1}{m_1 + m_2} (m_1 (0) + m_2 (R)) \\[5pt] &= \frac{m_2R}{m_1 + m_2} \end{align}$$

Note that R is the distance between the masses. Now that we have the center of mass, we go back to the moment of inertia for two point masses:

$$I = \sum m_i r_i^2 = m_1 r_1^2 + m_2 r_2^2$$

Moments of inertia – more complicated objects

---

The moment of inertia of two masses connected by a mass-less rod is the sum of the masses multiplied by the square of the distance between each mass and the center of mass.

The moment of inertia of the two-mass system shown is

$$I = m_1 r_1^2 + m_2 r_2^2$$

and in general, the moment of inertia of n masses in a line, where the r_i are the distances from the center of mass, is the sum:

$$I = \sum_{i = 1}^n \, m_i r_i^2$$

The moments of inertia of more complicated objects are more difficult to compute. They rely on integral calculus, and I won't present their derivations here. I'll save that for another section. Here, however, are some of the results.

The moment of inertia of a cylindrical rod thin enough that its width is a small fraction of its length, and rotating about its center of mass, is

If we rotate that same rod about one of its ends, the moment of inertia is:

In these figures, we assume that the solids are of uniform density throughout. Changes in density would change the moment of inertia.