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# Moment of Inertia

## How difficult it is to rotate an object.

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Moment of Inertia

Moment of inertia tells you how difficult it is to rotate an object. It is equivalent to the mass in linear problems. Moment of inertia is proportional to an objects mass and to it's distance from the rotational axis squared. The distance from the rotational axis dominates over the objects mass due to the square power. Thus the more mass an object has at it's 'edges' the more moment of inertia it has. Thus if you grab a big long pole and hold it at the center, it is fairly easy to rotate. However, if you hold the same pole at the end and try to rotate it -not so easy! Finally, moment of inertias can be added as long as the rotating parts in question are rotating around the same axis. For example, the moment of inertia of a pole with a mass at its end is the sum of the individual moment of inertias of each object involved. See example 2 for another problem where the moment of inertias are added together.

Key Equations

I=mir2i\begin{align*} I = \sum m_{\text{i}} r^2_{\text{i}} \end{align*}

#### Example

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?

For both parts of this problem we will be summing 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.

III=ΣIi=13mL2+ML2=13.5kg(3m)2+2kg(3m)2=19.5kg*m2\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.

III=112m(L2)2+2M(L2)2=112.5kg(3m2)2+22kg(3m2)2=9.1kg*m2\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*}

### Review

1. 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?
2. The wood plug, shown below, has a lower moment of inertia than the steel plug because it has a lower mass.
1. 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.

1. Which of the plugs would be harder to slow down? Explain.
3. Use the moment of inertia table in the Key Equations section in order to answer the following questions:
1. Calculate the moment of inertia of the Earth about its spin axis.
2. Calculate the moment of inertia of the Earth as it revolves around the Sun.
3. 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*}.
4. 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.
5. Repeat d., but calculate the moment of inertia about the center of the rod.
4. 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.)
5. 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?)

### Review (Answers)

1. Coin with hole in the middle because has more mass farther from rotational axis and thus a larger moment of inertia
2. a. wood b. plug on the right
3. 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*}
4. Lower
5. Iron ball, discuss why in class

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