Angular Velocity, Angular Momentum

Angular speed

Rotational motion is a different type of motion than linear motion. Angular velocity defines how fast the angular orientation of an object changes with time (i.e., rotates). Its magnitude is called angular speed.

$$\omega=\lim_{\Delta t\to 0}\frac{\Delta\theta}{\Delta t}$$

where $\omega$ is the instantaneous angular speed, $\theta$ is the angle with respect to a certain point. The SI unit for $\omega$ is $rad/s$ (radians per second). According to the equation, instantaneous angular speed is the rate of change of the angle in a very short period of time.

Tangential velocity

Consider a spinning (bike or car) wheel. It is fair to say every point on the wheel has the same rate of change of angles, except for the center of the wheel. Indeed, if you set any start time and end time for any two points on the wheel, they turn the same amount of angles with respect to the center, in the same amount of time.

Now we can consider the velocity of points along the radial direction. As you can see, three arrows on the wheel represents tangential velocities at those points. If the point is more outward, it seems to move faster.

Think about a full circle of motion with constant angular speed and tangential speed. Let the overall travel time be $t$, the overall travel distance be $2\pi r$, the overall travel angle be $2\pi$ (in radians). Then

$$v=\frac{2\pi r}{t},\quad w=\frac{2\pi}{t}$$

Therefore,

$$v=\frac{2\pi}{t}r=wr$$

This means the tangential velocity is proportional to the distance between the point of interest and the center of rotation, when the angular velocity is fixed.

Angular momentum

We know linear momentum is defined as $\vec{p}=m\vec{v}$. Angular momentum is defined as the (cross) product of the linear momentum and the distance between the point of interest and the center of rotation. In the simplest case, where a single point of mass rotates around a center point in circle, we can write

$$L=pr=mvr=mwr^2$$

where $p$ is the linear momentum, $r$ is the distance between the point of interest and the center of rotation, $v$ is tangential speed, and $w$ is angular speed. The SI unit of angular momentum is $kg\cdot m^2/s$.