47.1: Introduction to Angular Momentum

The rotational counterpart of momentum is called angular momentum. Just as linear momentum is defined as the product of mass and velocity $(p=m v)$, angular momentum $L$ is defined as the product of moment of inertia and angular velocity:

$$L=I \omega .$$

More generally, angular momentum, like linear momentum is a vector quantity:

$$\mathbf{L}=I \omega .$$

SI units for angular momentum are $\mathrm{kg} \mathrm{m}^{2} \mathrm{~s}^{-1}$, or $\mathrm{N} \mathrm{m} \mathrm{s}$.

Angular momentum $\mathbf{L}$ is related to linear momentum $\mathbf{p}$ according to

$$\mathbf{L}=\mathbf{r} \times \mathbf{p}$$

If you recall, Newton's second law of motion states that $F=d p / d t$, where $F$ is force and $p$ is momentum; in the special case where mass is constant, this reduces to $F=m a$, where $a$ is the acceleration. There are analogous formulæ in rotational motion, which can be derived by taking the time derivative of Eq. $\PageIndex{3}$:

$$\frac{d \mathbf{L}}{d t}=\mathbf{r} \times \frac{d \mathbf{p}}{d t}$$

The right-hand side is the torque; the result is the rotational form of Newton's second law:

$$\boldsymbol{\tau}=\frac{d \mathbf{L}}{d t}$$

where $\boldsymbol{\tau}$ is torque and $\mathbf{L}$ is angular momentum. In the case where the moment of inertia is constant, this reduces to $\boldsymbol{\tau}=I \boldsymbol{\alpha}$, where $\boldsymbol{\alpha}$ is the angular acceleration.