Students will learn what is momentum of inertia, what determines a large or small moment of inertia for an object and how to calculate moment of inertia.

### Key Equations

\begin{align*} I = \sum m_{\text{i}} r^2_{\text{i}} \end{align*}

#### Example 1

#### Example 2

Two 2 kg masses is placed at either end of a rod that has a mass of .5 kg and a length of 3 m. What is the moment of inertia if the system it is rotated about (a) one end of the rod and (b) the center of the rod?

##### Solution

For both parts of this problem we will be summing the the moment of inertia's for each component of the system based on the table above. We will treat the masses as satellites when calculating their moment of inertia's.

(a): For this part, we'll only add the value for the rod being rotated about one end and one of the masses because one of the masses is at the axis of rotation.

\begin{align*} I&=\Sigma I_i = \frac{1}{3}mL^2 + ML^2\\ I&=\frac{1}{3}.5\text{kg} * (3\;\text{m})^2 + 2\;\text{kg} * (3\;\text{m})^2\\ I&=19.5\;\text{kg*m}^2\\ \end{align*}

(b):For the second part, we'll sum the moment of inertia's of both masses and the rod being rotated about it's center.

\begin{align*} I&=\frac{1}{12}m(\frac{L}{2})^2 + 2M(\frac{L}{2})^2\\ I&=\frac{1}{12}.5\;\text{kg}*(\frac{3\text{m}}{2})^2 + 2 * 2\;\text{kg} * (\frac{3\text{m}}{2})^2\\ I&=9.1\;\text{kg*m}^2\\ \end{align*}

### Watch this Explanation

### Time for Practice

- You have two coins; one is a standard U.S. quarter, and the other is a coin of equal mass and size, but with a hole cut out of the center. Which coin has a higher moment of inertia?
- The wood plug, shown below, has a lower moment of inertia than the steel plug because it has a lower mass.
- Which of these plugs would be easier to spin on its axis? Explain.

Even though they have the same mass, the plug on the right has a higher moment of inertia (I), than the plug on the left, since the mass is distributed at greater radius.

- Which of the plugs would be harder to slow down? Explain.

- Use the moment of inertia table in the
*Key Equations*section in order to answer the following questions:- Calculate the moment of inertia of the Earth about its spin axis.
- Calculate the moment of inertia of the Earth as it revolves around the Sun.
- Calculate the moment of inertia of a hula hoop with mass \begin{align*}2 \;\mathrm{kg}\end{align*} and radius \begin{align*}0.5 \;\mathrm{m}\end{align*}.
- Calculate the moment of inertia of a rod \begin{align*}0.75 \;\mathrm{m}\end{align*} in length and mass \begin{align*}1.5 \;\mathrm{kg}\end{align*} rotating about one end.
- Repeat d., but calculate the moment of inertia about the center of the rod.

- If most of the mass of the Earth were concentrated at the core (say, in a ball of dense iron), would the moment of inertia of the Earth be higher or lower than it is now? (Assume the total mass stays the same.)
- Two spheres of the same mass are spinning in your garage. The first is \begin{align*}12 \;\mathrm{cm}\end{align*} in diameter and made of solid plastic. The second is \begin{align*}10 \;\mathrm{cm}\end{align*} in diameter but is a thin sphere shell of iron filled with air. Which is harder to slow down? Why? (And why are two spheres spinning in your garage?)

**Answers**

- coin with hole in the middle because has more mass farther from rotational axis and thus a larger moment of inertia
- a. wood b. plug on the right
- a. \begin{align*}9.74 \times 10^{37} \;\mathrm{kg \ m}^2\end{align*} b. \begin{align*}1.33 \times 10^{47} \;\mathrm{kg \ m}^2\end{align*} c. \begin{align*} 0.5 \;\mathrm{kg \ m}^2\end{align*} d. \begin{align*}0.28 \;\mathrm{kg \ m}^2\end{align*} e. \begin{align*}0.07 \;\mathrm{kg \ m}^2\end{align*}
- Lower
- Iron ball, discuss why in class