Angular Velocity
A flywheel spinning at 3000 rpm and a car wheel turning at a few revolutions per second are both examples of the same underlying quantity: how quickly something rotates. Angular velocity turns that everyday idea of "how fast it's spinning" into a precise, signed rate that plugs directly into the rest of rotational mechanics.
Defining angular velocity
Just as linear velocity is the rate of change of displacement, angular velocity $\omega$ (omega) is the rate of change of angular displacement:
Angular velocity is measured in radians per second (rad s$^{-1}$). Because $\theta$ is signed (positive for one sense of rotation, negative for the other, following the convention set up in the previous lesson), $\omega$ is signed too — a body rotating anticlockwise has positive angular velocity, and one rotating clockwise has negative angular velocity, under the usual convention.
It's worth being precise about the distinction between angular speed and angular velocity, which parallels the distinction between linear speed and linear velocity exactly. Angular speed is the magnitude of the rate of rotation only — always positive, with no information about direction of turning. Angular velocity includes the sign (or, more generally, a direction associated with the rotation axis), so it fully specifies both how fast and which way something is rotating. In most calculations only the magnitude matters and the two terms get used loosely, but the distinction becomes important whenever a sign convention has been set and the direction of rotation is part of the question.
Connecting angular velocity to tangential speed
A rotating rigid body doesn't have a single "speed" in the linear sense — every point on it is moving at a different linear speed depending on how far that point sits from the rotation axis, even though every point shares the same angular velocity $\omega$. This linear speed of a point moving on a circular path is called its tangential speed (or linear speed), because at any instant the point's velocity is directed tangent to its circular path.
This relation is central to engineering applications: it's what lets you convert between "how fast is the shaft spinning" (an angular quantity, the same everywhere on the shaft) and "how fast is this particular point on the rim moving" (a linear quantity, different at different radii). A point on the axis itself ($r=0$) has zero tangential speed even while the body rotates at full angular velocity around it; a point on the outer rim of a large flywheel moves far faster, in metres per second, than a point close to the hub, even though both share the same $\omega$.
Rotational speeds are very often quoted in rev/min (rpm) in real engineering contexts — engine speeds, motor ratings, turbine specifications — but every rotational formula in this topic needs $\omega$ in rad s$^{-1}$. The conversion is $\omega \text{ (rad s}^{-1}\text{)} = \text{rpm} \times \dfrac{2\pi}{60}$. Forgetting this step, or mixing up rev/min with rev/s, is one of the most common sources of numerical error in this whole topic — always convert to rad s$^{-1}$ before substituting into a formula.
Key takeaways
- Angular velocity $\omega = \Delta\theta/\Delta t$ is the rate of change of angular displacement, measured in rad s$^{-1}$.
- Angular speed is the magnitude of $\omega$ only; angular velocity is signed and carries direction information, exactly paralleling linear speed versus linear velocity.
- Differentiating $s = r\theta$ with respect to time gives $v = \omega r$, connecting angular velocity to the tangential (linear) speed of a point at radius $r$.
- Every point on a rotating rigid body shares the same $\omega$, but tangential speed $v$ increases with distance from the rotation axis.
- Rotational speeds are often quoted in rpm; convert to rad s$^{-1}$ using $\omega = \text{rpm} \times 2\pi/60$ before using any rotational formula.
- This lesson builds directly on angular displacement and sets up angular acceleration, covered next.