Considering conservation of linear momentum \(\left(\vec{p}\right)\) allows us to understand many phenomena that we observe, but it is insufficient to describe situations that involve rotation.

## 8.1 Angular momentum

The magnitude of linear momentum is given by

\[

p = mv

\]

or

\[

\textrm{momentum} = \textrm{mass}\times\textrm{velocity}

\]

Let’s break this down a little bit: velocity is the rate of change of an object’s position. When working with rotational motion, we consider angular velocity \(\omega\) (see chapter 5 for review). The mass of an object measures how much it resists changes in motion—that is, mass measures the object’s inertia. When working with rotational motion, we need to consider how much an object resists changes to its rotational motion, or it’s moment of inertia \(I\) (see section 6.3 for review). So, we could write this as

\[

\textrm{momentum} = \textrm{inertia}\times\textrm{rate of change of position}

\]

There is a quantity called *angular momentum*, denoted by the letter \(L\), whose magnitude is given by rotational inertia times rate of change of angular position or, in symbols:

\[ L = I\omega \tag{8.1} \]

The direction of angular momentum is the same as the direction of the angular velocity, and the unit of angular momentum is kg · m^{2}/s. Since we give the direction of rotation as simply counterclockwise or clockwise, we just use a plus or minus sign to note direction, as with angular velocity.

Refer to section 6.3.2 for a table of moments of inertia.